Motion Mountain Physics Textbook Volume 1 - Fall, Flow and Heat

Authors Christoph Schiller

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Christoph Schiller

MOTION MOUNTAIN
the adventure of physics – vol.i
fall, flow and heat

www.motionmountain.net
Christoph Schiller

Motion Mountain

The Adventure of Physics
Volume I

Fall, Flow and Heat

Edition 31, available as free pdf
with films at www.motionmountain.net
Editio trigesima prima.

Proprietas scriptoris © Chrestophori Schiller
primo anno Olympiadis trigesimae secundae.

Omnia proprietatis iura reservantur et vindicantur.
Imitatio prohibita sine auctoris permissione.
Non licet pecuniam expetere pro aliqua, quae
partem horum verborum continet; liber
pro omnibus semper gratuitus erat et manet.

Thirty-first edition.

Copyright © 1990–2023 by Christoph Schiller,
from the third year of the 24th Olympiad
to the first year of the 32nd Olympiad.

This pdf file is licensed under the Creative Commons
Attribution-Noncommercial-No Derivative Works 3.0 Germany
Licence, whose full text can be found on the website
creativecommons.org/licenses/by-nc-nd/3.0/de,
with the additional restriction that reproduction, distribution and use,
in whole or in part, in any product or service, be it
commercial or not, is not allowed without the written consent of
the copyright owner. The pdf file was and remains free for everybody
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electronically, but only in unmodified form and only at no charge.
To Britta, Esther and Justus Aaron

τῷ ἐμοὶ δαὶμονι
Die Menschen stärken, die Sachen klären.
PR E FAC E

“ ”
Primum movere, deinde docere.*
Antiquity

T
his book series is for anybody who is curious about motion in nature. How do
hings, people, animals, images and empty space move? The answer leads
o many adventures; this volume presents the best ones about everyday mo-

Motion Mountain – The Adventure of Physics
tion. Carefully observing everyday motion allows us to deduce six essential statements:
everyday motion is continuous, conserved, relative, reversible, mirror-invariant – and
lazy. Yes, nature is indeed lazy: in every motion, it minimizes change. This text explores
how these six results are deduced and how they fit with all those observations that seem
to contradict them.
In the structure of modern physics, shown in Figure 1, the results on everyday motion
form the major part of the starting point at the bottom. The present volume is the first of
a six-volume overview of physics. It resulted from a threefold aim I have pursued since
1990: to present motion in a way that is simple, up to date and captivating.
In order to be simple, the text focuses on concepts, while keeping mathematics to the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
necessary minimum. Understanding the concepts of physics is given precedence over
using formulae in calculations. The whole text is within the reach of an undergraduate.
In order to be up to date, the text is enriched by the many gems – both theoretical and
empirical – that are scattered throughout the scientific literature.
In order to be captivating, the text tries to startle the reader as much as possible. Read-
ing a book on general physics should be like going to a magic show. We watch, we are
astonished, we do not believe our eyes, we think, and finally we understand the trick.
When we look at nature, we often have the same experience. Indeed, every page presents
at least one surprise or provocation for the reader to think about. Numerous interesting
challenges are proposed.
The motto of the text, die Menschen stärken, die Sachen klären, a famous statement
on pedagogy, translates as: ‘To fortify people, to clarify things.’ Clarifying things – and
adhering only to the truth – requires courage, as changing the habits of thought produces
fear, often hidden by anger. But by overcoming our fears we grow in strength. And we
experience intense and beautiful emotions. All great adventures in life allow this, and
exploring motion is one of them. Enjoy it.

Christoph Schiller
* ‘First move, then teach.’ In modern languages, the mentioned type of moving (the heart) is called motiv-
ating; both terms go back to the same Latin root.
8 preface

Final, unified description of motion: upper limit c4/4Ghbar
Adventures: describing precisely all motion, understanding
the origin of colours, space -time and particles, enjoying
extreme thinking, calculating masses and couplings,
catching a further, tiny glimpse of bliss (vol. VI).

PHYSICS: An arrow indicates an
Describing motion with precision, increase in precision by
i.e., using the least action principle. adding a motion limit.

upper limit: Quantum theory
General relativity: 1/4G hbar with classical gravity Quantum field theory
upper limit c4/4G Adventures: bouncing (the ‘standard model’):
Adventures: the neutrons, under- upper limit c/hbar
night sky, measu- standing tree Adventures: building

Motion Mountain – The Adventure of Physics
ring curved and growth (vol. V). accelerators, under-
wobbling space, standing quarks, stars,
exploring black bombs and the basis of
holes and the life, matter & radiation
universe, space (vol. V).
and time (vol. II).

Classical gravity: upper limit: c Special relativity Quantum theory:
upper limit 1/4G Adventures: light, upper limit 1/hbar
Adventures: magnetism, length Adventures: biology,
climbing, skiing, c contraction, time birth, love, death,
space travel, limits dilation and chemistry, evolution,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the wonders of G fast E0 = mc2 h, e, k enjoying colours, art,
astronomy and limits motion (vol. II). limit paradoxes, medicine
geology (vol. I). uniform tiny and high-tech business
motion motion (vol. IV and vol. V).

Galilean physics, heat and electricity: no limits
The world of everyday motion: human scale, slow and weak.
Adventures: sport, music, sailing, cooking, describing
beauty and understanding its origin (vol. I);
using electricity, light and computers,
understanding the brain and people (vol. III).
F I G U R E 1 A complete map of physics, the science of motion, as ﬁrst proposed by Matvei Bronshtein
(b. 1907 Vinnytsia, d. 1938 Leningrad). The Bronshtein cube starts at the bottom with everyday motion,
and shows the connections to the ﬁelds of modern physics. Each connection increases the precision of
the description and is due to a limit to motion that is taken into account. The limits are given for
uniform motion by the gravitational constant G, for fast motion by the speed of light c, and for tiny
motion by the Planck constant h, the elementary charge e and the Boltzmann constant k.
preface 9

Using this b o ok
Marginal notes refer to bibliographic references, to other pages or to challenge solutions.
In the colour edition, marginal notes, pointers to footnotes and links to websites are
typeset in green. Over time, links on the internet tend to disappear. Most links can be
recovered via www.archive.org, which keeps a copy of old internet pages. In the free
pdf edition of this book, available at www.motionmountain.net, all green pointers and
links are clickable. The pdf edition also contains all films; they can be watched directly
in Adobe Reader.
Solutions and hints for challenges are given in the appendix. Challenges are classified
as easy (e), standard student level (s), difficult (d) and research level (r). Challenges for
which no solution has yet been included in the book are marked (ny).

Advice for learners
Learning allows us to discover what kind of person we can be. Learning widens know-
ledge, improves intelligence and provides a sense of achievement. Therefore, learning

Motion Mountain – The Adventure of Physics
from a book, especially one about nature, should be efficient and enjoyable. Avoid bad
learning methods like the plague! Do not use a marker, a pen or a pencil to highlight
or underline text on paper. It is a waste of time, provides false comfort and makes the
text unreadable. Add notes and comments instead! And do not learn from a screen. In
particular, do not learn from videos, from games or from a smartphone. All games and
almost all videos are drugs for the brain. Smartphones are drug dispensers that make
people addicted and prevent learning. Learn from paper – at your speed, and allow your
mind to wander! Nobody marking paper or looking at a screen is learning efficiently.
In my experience as a pupil and teacher, one learning method never failed to trans-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
form unsuccessful pupils into successful ones: if you read a text for study, summarize
every section you read, in your own words and images, aloud. If you are unable to do
so, read the section again. Repeat this until you can clearly summarize what you read in
your own words and images, aloud. And enjoy the telling aloud! You can do this alone
or with friends, in a room or while walking. If you do this with everything you read, you
will reduce your learning and reading time significantly; you will enjoy learning from
good texts much more and hate bad texts much less. Masters of the method can use it
even while listening to a lecture, in a low voice, thus avoiding to ever take notes.

Advice for teachers
A teacher likes pupils and likes to lead them into exploring the field he or she chose. His
or her enthusiasm is the key to job satisfaction. If you are a teacher, before the start of a
lesson, picture, feel and tell yourself how you enjoy the topic of the lesson; then picture,
feel and tell yourself how you will lead each of your pupils into enjoying that topic as
much as you do. Do this exercise consciously, every day. You will minimize trouble in
your class and maximize your teaching success.
This book is not written with exams in mind; it is written to make teachers and stu-
dents understand and enjoy physics, the science of motion.
10 preface

Feedback
The latest pdf edition of this text is and will remain free to download from the internet.
I would be delighted to receive an email from you at fb@motionmountain.net, especially
on the following issues:
Challenge 1 s — What was unclear and should be improved?
— What story, topic, riddle, picture or film did you miss?
Also help on the specific points listed on the www.motionmountain.net/help.html web
page is welcome. All feedback will be used to improve the next edition. You are welcome
to send feedback by mail or by sending in a pdf with added yellow notes, to provide
illustrations or photographs, or to contribute to the errata wiki on the website. If you
would like to translate a chapter of the book in your language, please let me know.
On behalf of all readers, thank you in advance for your input. For a particularly useful
contribution you will be mentioned – if you want – in the acknowledgements, receive a
reward, or both.

Motion Mountain – The Adventure of Physics
Support
Your donation to the charitable, tax-exempt non-profit organisation that produces, trans-
lates and publishes this book series is welcome. For details, see the web page www.
motionmountain.net/donation.html. The German tax office checks the proper use of
your donation. If you want, your name will be included in the sponsor list. Thank you in
advance for your help, on behalf of all readers across the world.
The paper edition of this book is available, either in colour or in black and white,
from www.amazon.com, in English and in certain other languages. And now, enjoy the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
reading.
C ON T E N T S

7 Preface
Using this book 9 • Advice for learners 9 • Advice for teachers 9 • Feedback 10 •
Support 10
11 Contents
15 1 Why should we care about motion?
Does motion exist? 16 • How should we talk about motion? 18 • What are

Motion Mountain – The Adventure of Physics
the types of motion? 20 • Perception, permanence and change 25 • Does
the world need states? 27 • Galilean physics in six interesting statements 29 •
Curiosities and fun challenges about motion 30 • First summary on motion 33
34 2 From motion measurement to continuity
What is velocity? 35 • What is time? 40 • Clocks 44 • Why do clocks go
clockwise? 48 • Does time flow? 48 • What is space? 49 • Are space
and time absolute or relative? 52 • Size – why length and area exist, but volume
does not 52 • What is straight? 59 • A hollow Earth? 60 • Curiosities and fun
challenges about everyday space and time 61 • Summary about everyday space

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
and time 74
75 3 How to describe motion – kinematics
Throwing, jumping and shooting 78 • Enjoying vectors 80 • What is rest? What
is velocity? 82 • Acceleration 85 • From objects to point particles 85 • Legs
and wheels 89 • Curiosities and fun challenges about kinematics 92 • Summary
of kinematics 96
98 4 From objects and images to conservation
Motion and contact 99 • What is mass? 100 • Momentum and mass 102 • Is
motion eternal? – Conservation of momentum 108 • More conservation – en-
ergy 110 • The cross product, or vector product 115 • Rotation and angular mo-
mentum 116 • Rolling wheels 121 • How do we walk and run? 122 • Curiosities
and fun challenges about mass, conservation and rotation 123 • Summary on
conservation in motion 134
135 5 From the rotation of the earth to the relativity of motion
How does the Earth rotate? 145 • Does the Earth move? 150 • Is velocity absolute?
– The theory of everyday relativity 156 • Is rotation relative? 158 • Curiosities and
fun challenges about rotation and relativity 158 • Legs or wheels? – Again 168 •
Summary on Galilean relativity 172
173 6 Motion due to gravitation
Gravitation as a limit to uniform motion 173 • Gravitation in the sky 174 • Grav-
itation on Earth 178 • Properties of gravitation: 𝐺 and 𝑔 182 • The gravitational
12 contents

potential 186 • The shape of the Earth 188 • Dynamics – how do things move in
various dimensions? 189 • The Moon 190 • Orbits – conic sections and more 192 •
Tides 197 • Can light fall? 201 • Mass: inertial and gravitational 202 • Curios-
ities and fun challenges about gravitation 204 • Summary on gravitation 225
226 7 Classical mechanics, force and the predictability of motion
Should one use force? Power? 227 • Forces, surfaces and conservation 231 •
Friction and motion 232 • Friction, sport, machines and predictability 234 •
Complete states – initial conditions 237 • Do surprises exist? Is the future de-
termined? 238 • Free will 241 • Summary on predictability 242 • From
predictability to global descriptions of motion 242
248 8 Measuring change with action
The principle of least action 253 • Lagrangians and motion 256 • Why is motion
so often bounded? 257 • Curiosities and fun challenges about Lagrangians 261 •
Summary on action 264
266 9 Motion and symmetry
Why can we think and talk about the world? 270 • Viewpoints 271 • Sym-

Motion Mountain – The Adventure of Physics
metries and groups 272 • Multiplets 273 • Representations 275 • The symmet-
ries and vocabulary of motion 276 • Reproducibility, conservation and Noether’s
theorem 280 • Parity inversion and motion reversal 284 • Interaction symmet-
ries 285 • Curiosities and fun challenges about symmetry 285 • Summary on sym-
metry 286
288 10 Simple motions of extended bodies – oscillations and waves
Oscillations 288 • Damping 289 • Resonance 291 • Waves: general and har-
monic 293 • Water waves 295 • Waves and their motion 300 • Why can we talk to
each other? – Huygens’ principle 304 • Wave equations 305 • Why are music and

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
singing voices so beautiful? 307 • Measuring sound 310 • Is ultrasound imaging
safe for babies? 313 • Signals 314 • Solitary waves and solitons 316 • Curiosities
and fun challenges about waves and oscillation 318 • Summary on waves and
oscillations 331
333 11 Do extended bodies exist? – Limits of continuity
Mountains and fractals 333 • Can a chocolate bar last forever? 333 • The case
of Galileo Galilei 335 • How high can animals jump? 337 • Felling trees 338 •
Little hard balls 339 • The sound of silence 340 • How to count what cannot
be seen 340 • Experiencing atoms 342 • Seeing atoms 344 • Curiosities and fun
challenges about solids and atoms 345 • Summary on atoms 351
354 12 Fluids and their motion
What can move in nature? – Flows of all kinds 354 • The state of a fluid 357 •
Laminar and turbulent flow 361 • The atmosphere 364 • The physics of blood and
breath 367 • Curiosities and fun challenges about fluids 370 • Summary on flu-
ids 382
383 13 On heat and motion reversal invariance
Temperature 383 • Thermal energy 387 • Why do balloons take up space? – The
end of continuity 389 • Brownian motion 391 • Why stones can be neither smooth
nor fractal, nor made of little hard balls 394 • Entropy 395 • Entropy from
particles 398 • The characteristic entropy of nature – the quantum of informa-
tion 399 • Is everything made of particles? 400 • The second principle of
contents 13

thermodynamics 402 • Why can’t we remember the future? 404 • Flow of en-
tropy 404 • Do isolated systems exist? 405 • Curiosities and fun challenges about
reversibility and heat 405 • Summary on heat and time-invariance 413
415 14 Self-organization and chaos – the simplicity of complexity
Appearance of order 418 • Self-organization in sand 420 • Self-organization of
spheres 422 • Conditions for the appearance of order 422 • The mathematics of
order appearance 423 • Chaos 424 • Emergence 426 • Curiosities and
fun challenges about self-organization 427 • Summary on self-organization and
chaos 434
435 15 From the limitations of physics to the limits of motion
Research topics in classical dynamics 435 • What is contact? 436 • What determ-
ines precision and accuracy? 437 • Can all of nature be described in a book? 437
• Something is wrong about our description of motion 438 • Why is measure-
ment possible? 439 • Is motion unlimited? 439
441 a Notation and conventions
The Latin alphabet 441 • The Greek alphabet 443 • The Hebrew alphabet and

Motion Mountain – The Adventure of Physics
other scripts 445 • Numbers and the Indian digits 446 • The symbols used in the
text 447 • Calendars 449 • People Names 451 • Abbreviations and eponyms or
concepts? 451
453 b Units, measurements and constants
SI units 453 • The meaning of measurement 456 • Curiosities and fun challenges
about units 456 • Precision and accuracy of measurements 459 • Limits to preci-
sion 460 • Physical constants 460 • Useful numbers 468
469 c Sources of information on motion
475 Challenge hints and solutions

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
523 Bibliography
556 Credits
Acknowledgements 556 • Film credits 557 • Image credits 557
564 Name index
577 Subject index
Fall, Flow and Heat

In our quest to learn how things move,
the experience of hiking and other motion
leads us to introduce the concepts of
velocity, time, length, mass and temperature.
We learn to use them to measure change
and find that nature minimizes it.
We discover how to float in free space,
why we have legs instead of wheels,
why disorder can never be eliminated,
and why one of the most difficult open issues
in science is the flow of water through a tube.
Chapter 1

W H Y SHOU L D W E C A R E A B OU T
MOT ION ?

“ ”
All motion is an illusion.
Zeno of Elea**

W
ham! The lightning striking the tree nearby violently disrupts our quiet forest
alk and causes our hearts to suddenly beat faster. In the top of the tree

Motion Mountain – The Adventure of Physics
e see the fire start and fade again. The gentle wind moving the leaves around
us helps to restore the calmness of the place. Nearby, the water in a small river follows
its complicated way down the valley, reflecting on its surface the ever-changing shapes
of the clouds.
Motion is everywhere: friendly and threatening, terrible and beautiful. It is funda-
mental to our human existence. We need motion for growing, for learning, for thinking,
for remaining healthy and for enjoying life. We use motion for walking through a forest,
for listening to its noises and for talking about all this. Like all animals, we rely on motion
to get food and to survive dangers. Like all living beings, we need motion to reproduce,
to breathe and to digest. Like all objects, motion keeps us warm.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Motion is the most fundamental observation about nature at large. It turns out that
everything that happens in the world is some type of motion. There are no exceptions.
Motion is such a basic part of our observations that even the origin of the word is lost in
the darkness of Indo-European linguistic history. The fascination of motion has always
made it a favourite object of curiosity. By the fifth century b ce in ancient Greece, its
Ref. 1 study had been given a name: physics.
Motion is also important to the human condition. What can we know? Where does
the world come from? Who are we? Where do we come from? What will we do? What
should we do? What will the future bring? What is death? Where does life lead? All these
questions are about motion. And the study of motion provides answers that are both
deep and surprising.
Ref. 2 Motion is mysterious. Though found everywhere – in the stars, in the tides, in our
eyelids – neither the ancient thinkers nor myriads of others in the 25 centuries since then
have been able to shed light on the central mystery: what is motion? We shall discover
that the standard reply, ‘motion is the change of place in time’, is correct, but inadequate.
Just recently a full answer has finally been found. This is the story of the way to find it.
Motion is a part of human experience. If we imagine human experience as an island,
then destiny, symbolized by the waves of the sea, carried us to its shore. Near the centre of

** Zeno of Elea (c. 450 bce), one of the main exponents of the Eleatic school of philosophy.
16 1 why should we care about motion?

Astronomy

Theory
of motion
Materials science
Quantum
Chemistry field theory
Geosciences
Medicine Quantum
Biology theory
Motion
Electromagnetism Mountain
Relativity
Thermodynamics
Engineering
Physics
Mechanics

Emotion Bay
Mathematics

Motion Mountain – The Adventure of Physics
The humanities

Social Sea

F I G U R E 2 Experience Island, with Motion Mountain and the trail to be followed.

the island an especially high mountain stands out. From its top we can see over the whole
landscape and get an impression of the relationships between all human experiences, and

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in particular between the various examples of motion. This is a guide to the top of what
I have called Motion Mountain (see Figure 2; a less symbolic and more exact version is
Page 8 given in Figure 1). The hike is one of the most beautiful adventures of the human mind.
The first question to ask is:

Does motion exist?

“
Das Rätsel gibt es nicht. Wenn sich eine Frage
überhaupt stellen läßt, so kann sie beantwortet

”
werden.*
Ludwig Wittgenstein, Tractatus, 6.5

To sharpen the mind for the issue of motion’s existence, have a look at Figure 3 or Fig-
Ref. 3 ure 4 and follow the instructions. In all cases the figures seem to rotate. You can exper-
ience similar effects if you walk over cobblestone pavement that is arranged in arched
patterns or if you look at the numerous motion illusions collected by Kitaoka Akiyoshi
Ref. 4 at www.ritsumei.ac.jp/~akitaoka. How can we make sure that real motion is different
Challenge 2 s from these or other similar illusions?
Many scholars simply argued that motion does not exist at all. Their arguments deeply
Ref. 5 influenced the investigation of motion over many centuries. For example, the Greek

* ‘The riddle does not exist. If a question can be put at all, it can also be answered.’
1 why should we care about motion? 17

F I G U R E 3 Illusions of motion: look at the ﬁgure on the left and slightly move the page, or look at the
white dot at the centre of the ﬁgure on the right and move your head back and forward.

philosopher Parmenides (born c. 515 b ce in Elea, a small town near Naples) argued that
since nothing comes from nothing, change cannot exist. He underscored the perman-
ence of nature and thus consistently maintained that all change and thus all motion is an
illusion.

Motion Mountain – The Adventure of Physics
Ref. 6
Heraclitus (c. 540 to c. 480 b ce) held the opposite view. Plato describes Heraclitus
as making the famous statement πάντα ῥεῖ ‘panta rhei’ or ‘everything flows’.* He saw
change as the essence of nature, in contrast to Parmenides. These two equally famous
opinions induced many scholars to investigate in more detail whether in nature there
are conserved quantities or whether creation is possible. We will uncover the answer later
Challenge 3 s on; until then, you might ponder which option you prefer.
Parmenides’ collaborator Zeno of Elea (born c. 500 b ce) argued so intensely against
motion that some people still worry about it today. In one of his arguments he claims –
in simple language – that it is impossible to slap somebody, since the hand first has to

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
travel halfway to the face, then travel through half the distance that remains, then again
so, and so on; the hand therefore should never reach the face. Zeno’s argument focuses
on the relation between infinity and its opposite, finitude, in the description of motion.
Ref. 7 In modern quantum theory, a related issue is a subject of research up to this day.
Zeno also stated that by looking at a moving object at a single instant of time, one
cannot maintain that it moves. He argued that at a single instant of time, there is no
difference between a moving and a resting body. He then deduced that if there is no
difference at a single time, there cannot be a difference for longer times. Zeno therefore
questioned whether motion can clearly be distinguished from its opposite, rest. Indeed,
in the history of physics, thinkers switched back and forward between a positive and a
negative answer. It was this very question that led Albert Einstein to the development of
general relativity, one of the high points of our journey. In our adventure, we will explore
all known differences between motion and rest. Eventually, we will dare to ask whether
single instants of time do exist at all. Answering this question is essential for reaching
the top of Motion Mountain.
When we explore quantum theory, we will discover that motion is indeed – to a cer-
tain extent – an illusion, as Parmenides claimed. More precisely, we will show that mo-
tion is observed only due to the limitations of the human condition. We will find that we
experience motion only because
* Appendix A explains how to read Greek text.
18 1 why should we care about motion?

F I G U R E 4 Zoom this image to large
size or approach it closely in order

Motion Mountain – The Adventure of Physics
to enjoy its apparent motion
(© Michael Bach after the discovery
of Kitaoka Akiyoshi).

— we have a finite size,
— we are made of a large but finite number of atoms,
— we have a finite but moderate temperature,
— we move much more slowly than the speed of light,
— we live in three dimensions,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
— we are large compared with a black hole of our own mass,
— we are large compared with our quantum mechanical wavelength,
— we are small compared with the universe,
— we have a working but limited memory,
— we are forced by our brain to approximate space and time as continuous entities, and
— we are forced by our brain to approximate nature as made of different parts.
If any one of these conditions were not fulfilled, we would not observe motion; motion,
then, would not exist! If that were not enough, note that none of the conditions requires
human beings; they are equally valid for many animals and machines. Each of these con-
ditions can be uncovered most efficiently if we start with the following question:

How should we talk ab ou t motion?

“
Je hais le mouvement, qui déplace les lignes,

”
Et jamais je ne pleure et jamais je ne ris.
Charles Baudelaire, La Beauté.*

Like any science, the approach of physics is twofold: we advance with precision and with
curiosity. Precision is the extent to which our description matches observations. Curios-
* Charles Baudelaire (b. 1821 Paris, d. 1867 Paris) Beauty: ‘I hate movement, which changes shapes, and
Ref. 8 never do I weep and never do I laugh.’ Beauty.
1 why should we care about motion? 19

Anaximander Empedocles Eudoxus Ctesibius Strabo Frontinus Cleomedes
Anaximenes Aristotle Archimedes Varro Maria Artemidor
the Jewess
Pythagoras Heraclides Konon Athenaius Josephus Sextus Empiricus
Almaeon Philolaus Theophrastus Chrysippos Eudoxus Pomponius Dionysius Athenaios Diogenes
of Kyz. Mela Periegetes of Nauc. Laertius
Heraclitus Zeno Autolycus Eratosthenes Sosigenes Marinus
Xenophanes Anthistenes Euclid Dositheus Virgilius Menelaos Philostratus
Thales Parmenides Archytas Epicure Biton Polybios Horace Nicomachos Apuleius

Alexander Ptolemy II Ptolemy VIII Caesar Nero Trajan

600 BCE 500 400 300 200 100 1 100 200

Socrates Plato Ptolemy I Cicero Seneca

Anaxagoras Hicetas Aristarchus Asclepiades Livius Dioscorides Ptolemy
Leucippus Pytheas Archimedes Seleukos Vitruvius Geminos Epictetus
Protagoras Erasistratus Diocles Manilius Demonax Diophantus
Oenopides Aristoxenus Aratos Philo Dionysius Diodorus Valerius Theon Alexander
of Byz. Thrax Siculus Maximus of Smyrna of Aphr.
Hippocrates Berossos

Motion Mountain – The Adventure of Physics
Herodotus Herophilus Apollonius Theodosius Plinius Rufus Galen
Senior
Democritus Straton Hipparchus Lucretius Aetius Arrian
Hippasos Speusippos Dikaiarchus Poseidonius Heron Plutarch Lucian
Naburimannu Kidinnu

F I G U R E 5 A timeline of scientiﬁc and political personalities in antiquity. The last letter of the name is
aligned with the year of death. For example, Maria the Jewess is the inventor of the bain-marie process
and Thales is the ﬁrst mathematician and scientist known by name.

ity is the passion that drives all scientists. Precision makes meaningful communication

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
possible, and curiosity makes it worthwhile. Take an eclipse, a beautiful piece of music
Ref. 9 or a feat at the Olympic games: the world is full of fascinating examples of motion.
If you ever find yourself talking about motion, whether to understand it more pre-
cisely or more deeply, you are taking steps up Motion Mountain. The examples of Fig-
ure 6 make the point. An empty bucket hangs vertically. When you fill the bucket with
a certain amount of water, it does not hang vertically any more. (Why?) If you continue
adding water, it starts to hang vertically again. How much water is necessary for this last
Challenge 4 s transition? The second illustration in Figure 6 is for the following puzzle. When you pull
a thread from a reel in the way shown, the reel will move either forwards or backwards,
depending on the angle at which you pull. What is the limiting angle between the two
possibilities?
High precision means going into fine details. Being attuned to details actually in-
creases the pleasure of the adventure.* Figure 7 shows an example. The higher we get
on Motion Mountain, the further we can see and the more our curiosity is rewarded.
The views offered are breathtaking, especially from the very top. The path we will follow
– one of the many possible routes – starts from the side of biology and directly enters the
Ref. 10 forest that lies at the foot of the mountain.

Challenge 6 s * Distrust anybody who wants to talk you out of investigating details. He is trying to deceive you. Details
are important. Be vigilant also during this journey.
20 1 why should we care about motion?

F I G U R E 6 How much water is required to make a bucket hang vertically? At what angle does the reel
Challenge 5 s (drawn incorrectly, with too small rims) change direction of motion when pulled along with the thread?
(© Luca Gastaldi).

Motion Mountain – The Adventure of Physics
F I G U R E 7 An example of how precision of observation can lead to the discovery of new effects: the
deformation of a tennis ball during the c. 6 ms of a fast bounce (© International Tennis Federation).

Intense curiosity drives us to go straight to the limits: understanding motion re-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
quires exploration of the largest distances, the highest velocities, the smallest particles,
the strongest forces, the highest precision and the strangest concepts. Let us begin.

What are the t ypes of motion?

“ ”
Every movement is born of a desire for change.
Antiquity

A good place to obtain a general overview on the types of motion is a large library. Table 1
shows the results. The domains in which motion, movements and moves play a role are
indeed varied. Already the earliest researchers in ancient Greece – listed in Figure 5 –
had the suspicion that all types of motion, as well as many other types of change, are
related. Three categories of change are commonly recognized:
1. Transport. The only type of change we call motion in everyday life is material trans-
port, such as a person walking, a leaf falling from a tree, or a musical instrument
playing. Transport is the change of position or orientation of objects, fluids included.
To a large extent, the behaviour of people also falls into this category.
2. Transformation. Another category of change groups observations such as the dis-
solution of salt in water, the formation of ice by freezing, the rotting of wood, the
cooking of food, the coagulation of blood, and the melting and alloying of metals.
1 why should we care about motion? 21

TA B L E 1 Content of books about motion found in a public library.

Motion topics Motion topics

motion pictures and digital effects motion as therapy for cancer, diabetes, acne and
depression
motion perception Ref. 11 motion sickness
motion for fitness and wellness motion for meditation
motion control and training in sports and motion ability as health check
singing
perpetual motion motion in dance, music and other performing
arts
motion as proof of various gods Ref. 12 motion of planets, stars and angels Ref. 13
economic efficiency of motion the connection between motional and emotional
habits
motion as help to overcome trauma motion in psychotherapy Ref. 14
locomotion of insects, horses, animals and motion of cells and plants

Motion Mountain – The Adventure of Physics
robots
collisions of atoms, cars, stars and galaxies
growth of multicellular beings, mountains,
sunspots and galaxies
motion of springs, joints, mechanisms, motion of continents, bird flocks, shadows and
liquids and gases empty space
commotion and violence motion in martial arts
motions in parliament movements in art, sciences and politics
movements in watches movements in the stock market
movement teaching and learning movement development in children Ref. 15

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
musical movements troop movements Ref. 16
religious movements bowel movements
moves in chess cheating moves in casinos Ref. 17
connection between gross national product and citizen mobility

These changes of colour, brightness, hardness, temperature and other material prop-
erties are all transformations. Transformations are changes not visibly connected with
transport. To this category, a few ancient thinkers added the emission and absorption
of light. In the twentieth century, these two effects were proven to be special cases of
transformations, as were the newly discovered appearance and disappearance of mat-
ter, as observed in the Sun and in radioactivity. Mind change, such as change of mood,
Ref. 18 of health, of education and of character, is also (mostly) a type of transformation.
Ref. 19 3. Growth. This last and especially important category of change, is observed for an-
imals, plants, bacteria, crystals, mountains, planets, stars and even galaxies. In the
nineteenth century, changes in the population of systems, biological evolution, and
in the twentieth century, changes in the size of the universe, cosmic evolution, were
added to this category. Traditionally, these phenomena were studied by separate sci-
ences. Independently they all arrived at the conclusion that growth is a combination
of transport and transformation. The difference is one of complexity and of time scale.
22 1 why should we care about motion?

F I G U R E 8 An
example of
transport, at
Mount Etna
(© Marco Fulle).

Motion Mountain – The Adventure of Physics
At the beginnings of modern science during the Renaissance, only the study of transport
was seen as the topic of physics. Motion was equated to transport. The other two domains
were neglected by physicists. Despite this restriction, the field of enquiry remains large,
Page 16 covering a large part of Experience Island.
Early scholars differentiated types of transport by their origin. Movements such as
those of the legs when walking were classified as volitional, because they are controlled
by one’s will, whereas movements of external objects, such as the fall of a snowflake,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
which cannot be influenced by will-power, were classified as passive. Young humans, es-
pecially young male humans, spend considerable time in learning elaborate volitional
movements. An example is shown in Figure 10.
The complete distinction between passive and volitional motion is made by children
by the age of six, and this marks a central step in the development of every human to-
wards a precise description of the environment.* From this distinction stems the histor-
ical but now outdated definition of physics as the science of motion of non-living things.
The advent of machines forced scholars to rethink the distinction between volitional
and passive motion. Like living beings, machines are self-moving and thus mimic voli-
tional motion. However, careful observation shows that every part in a machine is moved
by another, so their motion is in fact passive. Are living beings also machines? Are human
actions examples of passive motion as well? The accumulation of observations in the last
100 years made it clear that volitional movement** indeed has the same physical prop-

* Failure to pass this stage completely can result in a person having various strange beliefs, such as believing
in the ability to influence roulette balls, as found in compulsive players, or in the ability to move other bod-
ies by thought, as found in numerous otherwise healthy-looking people. An entertaining and informative
account of all the deception and self-deception involved in creating and maintaining these beliefs is given
by James R andi, The Faith Healers, Prometheus Books, 1989. A professional magician, he presents many
similar topics in several of his other books. See also his www.randi.org website for more details.
** The word ‘movement’ is rather modern; it was imported into English from the old French and became
popular only at the end of the eighteenth century. It is never used by Shakespeare.
1 why should we care about motion? 23

F I G U R E 9 Transport, growth
and transformation (© Philip
Plisson).

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 10 One of the most difﬁcult
volitional movements known, performed
by Alexander Tsukanov, the ﬁrst man able
to do this: jumping from one ultimate
wheel to another (© Moscow State Circus).

erties as passive movement in non-living systems. A distinction between the two types
of motion is thus unnecessary. Of course, from the emotional viewpoint, the differences
Ref. 20 are important; for example, grace can only be ascribed to volitional movements.
Since passive and volitional motion have the same properties, through the study of
motion of non-living objects we can learn something about the human condition. This
is most evident when touching the topics of determinism, causality, probability, infinity,
time, love and death, to name but a few of the themes we will encounter during our
adventure.
In the nineteenth and twentieth centuries other classically held beliefs about mo-
tion fell by the wayside. Extensive observations showed that all transformations and
all growth phenomena, including behaviour change and evolution, are also examples of
transport. In other words, over 2 000 years of studies have shown that the ancient classi-
24 1 why should we care about motion?

fication of observations was useless:

⊳ All change is transport.

And

⊳ Transport and motion are the same.

In the middle of the twentieth century the study of motion culminated in the exper-
imental confirmation of an even more specific idea, previously articulated in ancient
Greece:

⊳ Every type of change is due to the motion of particles.

It takes time and work to reach this conclusion, which appears only when we relentlessly
pursue higher and higher precision in the description of nature. The first five parts of

Motion Mountain – The Adventure of Physics
Challenge 7 s this adventure retrace the path to this result. (Do you agree with it?)
The last decade of the twentieth century again completely changed the description of
motion: the particle idea turns out to be limited and wrong. This recent result, reached
through a combination of careful observation and deduction, will be explored in the last
part of our adventure. But we still have some way to go before we reach that result, just
below the summit of our journey.
In summary, history has shown that classifying the various types of motion is not
productive. Only by trying to achieve maximum precision can we hope to arrive at the
fundamental properties of motion. Precision, not classification, is the path to follow. As

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ernest Rutherford said jokingly: ‘All science is either physics or stamp collecting.’
In order to achieve precision in our description of motion, we need to select specific
examples of motion and study them fully in detail. It is intuitively obvious that the most
precise description is achievable for the simplest possible examples. In everyday life, this
is the case for the motion of any non-living, solid and rigid body in our environment,
such as a stone thrown through the air. Indeed, like all humans, we learned to throw ob-
Ref. 21 jects long before we learned to walk. Throwing is one of the first physical experiments
we performed by ourselves. The importance of throwing is also seen from the terms de-
rived from it: in Latin, words like subject or ‘thrown below’, object or ‘thrown in front’,
and interjection or ‘thrown in between’; in Greek, the act of throwing led to terms like
symbol or ‘thrown together’, problem or ‘thrown forward’, emblem or ‘thrown into’, and
– last but not least – devil or ‘thrown through’. And indeed, during our early childhood,
by throwing stones, toys and other objects until our parents feared for every piece of the
household, we explored the perception and the properties of motion. We do the same
here.

“ ”
Die Welt ist unabhängig von meinem Willen.*
Ludwig Wittgenstein, Tractatus, 6.373

* ‘The world is independent of my will.’
1 why should we care about motion? 25

Perception, permanence and change

“
Only wimps specialize in the general case; real

”
scientists pursue examples.
Adapted by Michael Berry from a remark by
Beresford Parlett

Human beings enjoy perceiving. Perception starts before birth, and we continue enjoying
it for as long as we can. That is why television or videos, even when devoid of content,
are so successful. During our walk through the forest at the foot of Motion Mountain
we cannot avoid perceiving. Perception is first of all the ability to distinguish. We use
the basic mental act of distinguishing in almost every instant of life; for example, during
childhood we first learned to distinguish familiar from unfamiliar observations. This is
possible in combination with another basic ability, namely the capacity to memorize ex-
periences. Memory gives us the ability to experience, to talk and thus to explore nature.
Perceiving, classifying and memorizing together form learning. Without any one of these
three abilities, we could not study motion.
Children rapidly learn to distinguish permanence from variability. They learn to re-

Motion Mountain – The Adventure of Physics
cognize human faces, even though a face never looks exactly the same each time it is seen.
From recognition of faces, children extend recognition to all other observations. Recog-
nition works pretty well in everyday life; it is nice to recognize friends, even at night, and
even after many beers (not a challenge). The act of recognition thus always uses a form
of generalization. When we observe, we always have some general idea in our mind. Let
us specify the main ones.
Sitting on the grass in a clearing of the forest at the foot of Motion Mountain, surroun-
ded by the trees and the silence typical of such places, a feeling of calmness and tranquil-
lity envelops us. We are thinking about the essence of perception. Suddenly, something

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
moves in the bushes; immediately our eyes turn and our attention focuses. The nerve
cells that detect motion are part of the most ancient part of our brain, shared with birds
Ref. 22 and reptiles: the brain stem. Then the cortex, or modern brain, takes over to analyse the
type of motion and to identify its origin. Watching the motion across our field of vision,
we observe two invariant entities: the fixed landscape and the moving animal. After we
recognize the animal as a deer, we relax again.
How did we distinguish, in case of Figure 11, between landscape and deer? Perception
involves several processes in the eye and in the brain. An essential part for these pro-
cesses is motion, as is best deduced from the flip film shown in the lower left corners
Ref. 23 of these pages. Each image shows only a rectangle filled with a mathematically random
pattern. But when the pages are scanned in rapid succession, you discern a shape – a
square – moving against a fixed background. At any given instant, the square cannot
be distinguished from the background; there is no visible object at any given instant of
time. Nevertheless it is easy to perceive its motion.* Perception experiments such as this
one have been performed in many variations. Such experiments showed that detecting
a moving square against a random background is nothing special to humans; flies have
the same ability, as do, in fact, all animals that have eyes.

* The human eye is rather good at detecting motion. For example, the eye can detect motion of a point of
light even if the change of angle is smaller than that which can be distinguished in a fixed image. Details of
Ref. 11 this and similar topics for the other senses are the domain of perception research.
26 1 why should we care about motion?

F I G U R E 11 How do we distinguish a deer
from its environment? (© Tony Rodgers).

The flip film in the lower left corner, like many similar experiments, illustrates two

Motion Mountain – The Adventure of Physics
central attributes of motion. First, motion is perceived only if an object can be distin-
guished from a background or environment. Many motion illusions focus on this point.*
Second, motion is required to define both the object and the environment, and to dis-
tinguish them from each other. In fact, the concept of space is – among others – an
abstraction of the idea of background. The background is extended; the moving entity is
localized. Does this seem boring? It is not; just wait for a second.
We call a localized entity of investigation that can change or move a physical system –
or simply a system. A system is a recognizable, thus permanent part of nature. Systems
can be objects – also called ‘physical bodies’ – or radiation. Therefore, images, which are
made of radiation, are aspects of physical systems, but not themselves physical systems.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 8 s These connections are summarized in Table 2. Now, are holes physical systems?
In other words, we call the set of localized aspects that remain invariant or permanent
during motion, such as size, shape, colour etc., taken together, a (physical) object or a
(physical) body. We will tighten the definition shortly, to distinguish objects from images.
We note that to specify permanent moving objects, we need to distinguish them from
the environment. In other words, right from the start, we experience motion as a relative
process; it is perceived in relation and in opposition to the environment.
The conceptual distinction between localized, isolable objects and the extended envir-
Challenge 9 s onment is important. True, it has the appearance of a circular definition. (Do you agree?)
Page 438 Indeed, this issue will keep us busy later on. On the other hand, we are so used to our
ability to isolate local systems from the environment that we take it for granted. How-
ever, as we will discover later on in our walk, this distinction turns out to be logically
Vol. VI, page 85 and experimentally impossible!** The reason for this impossibility will turn out to be
fascinating. To discover the impossibility, we note, as a first step, that apart from mov-

* The topic of motion perception is full of interesting aspects. An excellent introduction is chapter 6 of the
beautiful text by Donald D. Hoffman, Visual Intelligence – How We Create What We See, W.W. Norton
& Co., 1998. His collection of basic motion illusions can be experienced and explored on the associated
www.cogsci.uci.edu/~ddhoff website.
** Contrary to what is often read in popular literature, the distinction is possible in quantum theory. It
becomes impossible only when quantum theory is unified with general relativity.
1 why should we care about motion? 27

TA B L E 2 Family tree of the basic physical concepts.

motion
the basic type of change: a system changing position relative to a background

parts/systems relations background
permanent measurable
bounded produce boundaries unbounded
have shapes produce shapes extended

objects radiation states interactions phase space space-time
impenetrable penetrable global local composed simple

The corresponding aspects:

mass intensity instant source dimension curvature

Motion Mountain – The Adventure of Physics
size colour position domain distance topology
charge image momentum strength volume distance
spin appearance energy direction subspaces area
etc. etc. etc. etc. etc. etc.

world – nature – universe – cosmos
the collection of all parts, relations and backgrounds

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
ing entities and the permanent background, we also need to describe their relations. The
necessary concepts are summarized in Table 2.

“
Wisdom is one thing: to understand the thought

”
which steers all things through all things.
Ref. 24 Heraclitus of Ephesus

Does the world need states?

“
Das Feste, das Bestehende und der Gegenstand
sind Eins. Der Gegenstand ist das Feste,
Bestehende; die Konfiguration ist das

”
Wechselnde, Unbeständige.*
Ludwig Wittgenstein, Tractatus, 2.027 – 2.0271

What distinguishes the various patterns in the lower left corners of this text? In everyday
life we would say: the situation or configuration of the involved entities. The situation
somehow describes all those aspects that can differ from case to case. It is customary to
call the list of all variable aspects of a set of objects their (physical) state of motion, or
simply their state. How is the state characterized?

* ‘The fixed, the existent and the object are one. The object is the fixed, the existent; the configuration is the
changing, the variable.’
28 1 why should we care about motion?

The configurations in the lower left corners differ first of all in time. Time is what
makes opposites possible: a child is in a house and the same child is outside the house.
Time describes and resolves this type of contradiction. But the state not only distin-
guishes situations in time: the state contains all those aspects of a system – i.e., of a group
of objects – that set it apart from all similar systems. Two similar objects can differ, at
each instant of time, in their
— position,
— velocity,
— orientation, or
— angular velocity.
These properties determine the state and pinpoint the individuality of a physical system
among exact copies of itself. Equivalently, the state describes the relation of an object or
a system with respect to its environment. Or, again, equivalently:

⊳ The state describes all aspects of a system that depend on the observer.

Motion Mountain – The Adventure of Physics
The definition of state is not boring at all – just ponder this: Does the universe have a
Challenge 10 s state? And: is the list of state properties just given complete?
In addition, physical systems are described by their permanent, intrinsic properties.
Some examples are
— mass,
— shape,
— colour,
— composition.
Intrinsic properties do not depend on the observer and are independent of the state of

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the system. They are permanent – at least for a certain time interval. Intrinsic properties
also allow to distinguish physical systems from each other. And again, we can ask: What
is the complete list of intrinsic properties in nature? And does the universe have intrinsic
Challenge 11 s properties?
The various aspects of objects and of their states are called (physical) observables. We
will refine this rough, preliminary definition in the following.
Describing nature as a collection of permanent entities and changing states is the
starting point of the study of motion. Every observation of motion requires the distinc-
tion of permanent, intrinsic properties – describing the objects that move – and changing
states – describing the way the objects move. Without this distinction, there is no motion.
Without this distinction, there is not even a way to talk about motion.
Using the terms just introduced, we can say

⊳ Motion is the change of state of permanent objects.

The exact separation between those aspects belonging to the object, the permanent in-
trinsic properties, and those belonging to the state, the varying state properties, depends
on the precision of observation. For example, the length of a piece of wood is not per-
manent; wood shrinks and bends with time, due to processes at the molecular level. To
be precise, the length of a piece of wood is not an aspect of the object, but an aspect of
its state. Precise observations thus shift the distinction between the object and its state;
1 why should we care about motion? 29

the distinction itself does not disappear – at least not in the first five volumes of our
adventure.
At the end of the twentieth century, neuroscience discovered that the distinction
between changing states and permanent objects is not only made by scientists and en-
gineers. Also nature makes the distinction. In fact, nature has hard-wired the distinction
into the brain! Using the output signals from the visual cortex that processes what the
eyes observe, the adjacent parts on the upper side of the human brain – the dorsal stream
– process the state of the objects that are seen, such their distance and motion, whereas
the adjacent parts on the lower side of the human brain – the ventral stream – process
intrinsic properties, such as shape, colours and patterns.
In summary, states are indeed required for the description of motion. So are perman-
ent, intrinsic properties. In order to proceed and to achieve a complete description of
motion, we thus need a complete description of the possible states and a complete de-
scription of the intrinsic properties of objects. The first approach that attempts this is
called Galilean physics; it starts by specifying our everyday environment and the motion
in it as precisely as possible.

Motion Mountain – The Adventure of Physics
Galilean physics in six interesting statements
The study of everyday motion, Galilean physics, is already worthwhile in itself: we will
uncover many results that are in contrast with our usual experience. For example, if we
recall our own past, we all have experienced how important, delightful or unwelcome
surprises can be. Nevertheless, the study of everyday motion shows that there are no sur-
prises in nature. Motion, and thus the world, is predictable or deterministic.
The main surprise of our exploration of motion is that there are no surprises in nature.
Nature is predictable. In fact, we will uncover six aspects of the predictability of everyday

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
motion:
1. Continuity. We know that eyes, cameras and measurement apparatus have a finite
resolution. All have a smallest distance they can observe. We know that clocks have a
smallest time they can measure. Despite these limitations, in everyday life all move-
ments, their states, as well as space and time themselves, are continuous.
2. Conservation. We all observe that people, music and many other things in motion
stop moving after a while. The study of motion yields the opposite result: motion
never stops. In fact, three aspects of motion do not change, but are conserved: mo-
mentum, angular momentum and energy are conserved, separately, in all examples
of motion. No exception to these three types of conservation has ever been observed.
(In contrast, mass is often, but not always conserved.) In addition, we will discover
that conservation implies that motion and its properties are the same at all places and
at all times: motion is universal.
3. Relativity. We all know that motion differs from rest. Despite this experience, careful
study shows that there is no intrinsic difference between the two. Motion and rest
depend on the observer. Motion is relative. And so is rest. This is the first step towards
understanding the theory of relativity.
4. Reversibility. We all observe that many processes happen only in one direction. For
example, spilled milk never returns into the container by itself. Despite such obser-
30 1 why should we care about motion?

F I G U R E 12 A block and tackle and a differential pulley (left) and a farmer (right).

Motion Mountain – The Adventure of Physics
vations, the study of motion will show us that all everyday motion is reversible. Phys-
icists call this the invariance of everyday motion under motion reversal. Sloppily, but
incorrectly, one sometimes speaks of ‘time reversal’.
5. Mirror invariance. Most of us find scissors difficult to handle with the left hand, have
difficulties to write with the other hand, and have grown with a heart on the left side.
Despite such observations, our exploration will show that everyday motion is mirror-
invariant (or parity-invariant). Mirror processes are always possible in everyday life.
6. Change minimization. We all are astonished by the many observations that the world

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
offers: colours, shapes, sounds, growth, disasters, happiness, friendship, love. The
variation, beauty and complexity of nature is amazing. We will confirm that all ob-
servations are due to motion. And despite the appearance of complexity, all motion
is simple. Our study will show that all observations can be summarized in a simple
way: Nature is lazy. All motion happens in a way that minimizes change. Change can
be measured, using a quantity called ‘action’, and nature keeps it to a minimum. Situ-
ations – or states, as physicists like to say – evolve by minimizing change. Nature is
lazy.
These six aspects are essential in understanding motion in sport, in music, in animals, in
machines or among the stars. This first volume of our adventure will be an exploration of
such movements. In particular, we will confirm, against all appearances of the contrary,
the mentioned six key properties in all cases of everyday motion.

Curiosities and fun challenges ab ou t motion*
In contrast to most animals, sedentary creatures, like plants or sea anemones, have no
legs and cannot move much; for their self-defence, they developed poisons. Examples of
such plants are the stinging nettle, the tobacco plant, digitalis, belladonna and poppy;

* Sections entitled ‘curiosities’ are collections of topics and problems that allow one to check and to expand
the usage of concepts already introduced.
1 why should we care about motion? 31

poisons include caffeine, nicotine, and curare. Poisons such as these are at the basis of
most medicines. Therefore, most medicines exist essentially because plants have no legs.
∗∗
A man climbs a mountain from 9 a.m. to 1 p.m. He sleeps on the top and comes down
the next day, taking again from 9 a.m. to 1 p.m. for the descent. Is there a place on the
Challenge 12 s path that he passes at the same time on the two days?
∗∗
Every time a soap bubble bursts, the motion of the surface during the burst is the same,
Challenge 13 s even though it is too fast to be seen by the naked eye. Can you imagine the details?
∗∗
Challenge 14 s Is the motion of a ghost an example of motion?
∗∗

Motion Mountain – The Adventure of Physics
Challenge 15 s Can something stop moving? How would you show it?
∗∗
Challenge 16 s Does a body moving forever in a straight line show that nature or space is infinite?
∗∗
What is the length of rope one has to pull in order to lift a mass by a height ℎ with a block
Challenge 17 s and tackle with four wheels, as shown on the left of Figure 12? Does the farmer on the
right of the figure do something sensible?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In the past, block and tackles were important in many machines. Two particularly
Ref. 25 useful versions are the differential block and tackle, also called a differential pulley, which
is easy to make, and the Spanish burton, which has the biggest effect with the smallest
Challenge 18 e number of wheels. There is even a so-called fool’s tackle. Enjoy their exploration.
All these devices are examples of the golden rule of mechanics: what you gain in force,
you loose in displacement. Or, equivalently: force times displacement – also called (phys-
ical) work – remains unchanged, whatever mechanical device you may use. This is one
example of conservation that is observed in everyday motion.
∗∗
Challenge 19 s Can the universe move?
∗∗
To talk about precision with precision, we need to measure precision itself. How would
Challenge 20 s you do that?
∗∗
Challenge 21 s Would we observe motion if we had no memory?
∗∗
Challenge 22 s What is the lowest speed you have observed? Is there a lowest speed in nature?
32 1 why should we care about motion?

ball

v
block
perfectly flat table

F I G U R E 13 What happens? F I G U R E 14 What is the speed of the rollers? Are
other roller shapes possible?

∗∗
According to legend, Sissa ben Dahir, the Indian inventor of the game of chaturanga or
chess, demanded from King Shirham the following reward for his invention: he wanted

Motion Mountain – The Adventure of Physics
one grain of wheat for the first square, two for the second, four for the third, eight for
the fourth, and so on. How much time would all the wheat fields of the world take to
Challenge 23 s produce the necessary grains?
∗∗
When a burning candle is moved, the flame lags behind the candle. How does the flame
Challenge 24 s behave if the candle is inside a glass, still burning, and the glass is accelerated?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
A good way to make money is to build motion detectors. A motion detector is a small
box with a few wires. The box produces an electrical signal whenever the box moves.
What types of motion detectors can you imagine? How cheap can you make such a box?
Challenge 25 d How precise?
∗∗
A perfectly frictionless and spherical ball lies near the edge of a perfectly flat and hori-
Challenge 26 d zontal table, as shown in Figure 13. What happens? In what time scale?
∗∗
You step into a closed box without windows. The box is moved by outside forces un-
Challenge 27 s known to you. Can you determine how you are moving from inside the box?
∗∗
When a block is rolled over the floor over a set of cylinders, as shown in Figure 14, how
Challenge 28 s are the speed of the block and that of the cylinders related?
∗∗
Ref. 18 Do you dislike formulae? If you do, use the following three-minute method to change
Challenge 29 s the situation. It is worth trying it, as it will make you enjoy this book much more. Life is
short; as much of it as possible, like reading this text, should be a pleasure.
1 why should we care about motion? 33

1. Close your eyes and recall an experience that was absolutely marvellous, a situation
when you felt excited, curious and positive.
2. Open your eyes for a second or two and look at page 280 – or any other page that
contains many formulae.
3. Then close your eyes again and return to your marvellous experience.
4. Repeat the observation of the formulae and the visualization of your memory – steps
2 and 3 – three more times.
Then leave the memory, look around yourself to get back into the here and now, and test
yourself. Look again at page 280. How do you feel about formulae now?
∗∗
In the sixteenth century, Niccolò Tartaglia* proposed the following problem. Three
young couples want to cross a river. Only a small boat that can carry two people is avail-
able. The men are extremely jealous, and would never leave their brides with another
Challenge 30 s man. How many journeys across the river are necessary?

Motion Mountain – The Adventure of Physics
∗∗
Cylinders can be used to roll a flat object over the floor, as shown in Figure 14. The cylin-
ders keep the object plane always at the same distance from the floor. What cross-sections
other than circular, so-called curves of constant width, can a cylinder have to realize the
Challenge 31 s same feat? How many examples can you find? Are objects different than cylinders pos-
sible?
∗∗
Hanging pictures on a wall is not easy. First puzzle: what is the best way to hang a picture

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
on one nail? The method must allow you to move the picture in horizontal position after
Challenge 32 s the nail is in the wall, in the case that the weight is not equally distributed. Second puzzle:
Ref. 26 Can you hang a picture on a wall – this time with a long rope – over two nails in such a
Challenge 33 s way that pulling either nail makes the picture fall? And with three nails? And 𝑛 nails?

First summary on motion
Motion, the change of position of physical systems, is the most fundamental observation
in nature. Everyday motion is predictable and deterministic. Predictability is reflected in
six aspects of motion: continuity, conservation, reversibility, mirror-invariance, relativity
and minimization. Some of these aspects are valid for all motion, and some are valid only
Challenge 34 d for everyday motion. Which ones, and why? We explore this now.

* Niccolò Fontana Tartaglia (1499–1557), was an important Renaissance mathematician.
Chapter 2

F ROM MOT ION M E A SU R E M E N T TO
C ON T I N U I T Y

“
Physic ist wahrlich das eigentliche Studium des

”
Menschen.**
Georg Christoph Lichtenberg

T
he simplest description of motion is the one we all, like cats or monkeys, use
hroughout our everyday life: only one thing can be at a given spot at a given time.

Motion Mountain – The Adventure of Physics
his general description can be separated into three assumptions: matter is impen-
etrable and moves, time is made of instants, and space is made of points. Without these
Challenge 35 s three assumptions (do you agree with them?) it is not even possible to define velocity.
We thus need points embedded in continuous space and time to talk about motion. This
description of nature is called Galilean physics, or sometimes Newtonian physics.
Galileo Galilei (1564–1642), Tuscan professor of mathematics, was the central founder
of modern physics. He became famous for advocating the importance of observations as
checks of statements about nature. By requiring and performing these checks through-
out his life, he was led to continuously increase the accuracy in the description of mo-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
tion. For example, Galileo studied motion by measuring change of position with a self-
constructed stopwatch. Galileo’s experimental aim was to measure all that is measurable
about motion. His approach changed the speculative description of ancient Greece into
the experimental physics of Renaissance Italy.***
After Galileo, the English alchemist, occultist, theologian, physicist and politician
Isaac Newton (1643–1727) continued to explore with vigour the idea that different types
of motion have the same properties, and he made important steps in constructing the
concepts necessary to demonstrate this idea.****
Above all, the explorations and books by Galileo popularized the fundamental exper-
imental statements on the properties of speed, space and time.
** ‘Physics truly is the proper study of man.’ Georg Christoph Lichtenberg (b. 1742 Ober-Ramstadt,
d. 1799 Göttingen) was an important physicist and essayist.
*** The best and most informative book on the life of Galileo and his times is by Pietro Redondi (see the
section on page 335). Galileo was born in the year the pencil was invented. Before his time, it was impossible
to do paper and pencil calculations. For the curious, the www.mpiwg-berlin.mpg.de website allows you to
read an original manuscript by Galileo.
**** Newton was born a year after Galileo died. For most of his life Newton searched for the philosopher’s
stone. Newton’s hobby, as head of the English mint, was to supervise personally the hanging of counterfeit-
Ref. 27 ers. About Newton’s lifelong infatuation with alchemy, see the books by Dobbs. A misogynist throughout
his life, Newton believed himself to be chosen by god; he took his Latin name, Isaacus Neuutonus, and
formed the anagram Jeova sanctus unus. About Newton and his importance for classical mechanics, see the
Ref. 28 text by Clifford Truesdell.
2 from motion measurement to continuity 35

F I G U R E 15 Galileo Galilei (1564–1642).

Motion Mountain – The Adventure of Physics
F I G U R E 16 Some speed measurement devices: an anemometer, a tachymeter for inline skates, a sports
radar gun and a Pitot–Prandtl tube in an aeroplane (© Fachhochschule Koblenz, Silva, Tracer, Wikimedia).

What is velo cit y?

“ ”
There is nothing else like it.
Jochen Rindt*

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Velocity fascinates. To physicists, not only car races are interesting, but any moving en-
tity is. Therefore, physicists first measure as many examples as possible. A selection of
measured speed values is given in Table 3. The units and prefixes used are explained in
Page 453 detail in Appendix B. Some speed measurement devices are shown in Figure 16.
Everyday life teaches us a lot about motion: objects can overtake each other, and they
can move in different directions. We also observe that velocities can be added or changed
smoothly. The precise list of these properties, as given in Table 4, is summarized by math-
ematicians in a special term; they say that velocities form a Euclidean vector space.** More
Page 81 details about this strange term will be given shortly. For now we just note that in describ-
ing nature, mathematical concepts offer the most accurate vehicle.
When velocity is assumed to be a Euclidean vector, it is called Galilean velocity. Ve-
locity is a profound concept. For example, velocity does not need space and time meas-
urements to be defined. Are you able to find a means of measuring velocities without
* Jochen Rindt (1942–1970), famous Austrian Formula One racing car driver, speaking about speed.
** It is named after Euclid, or Eukleides, the great Greek mathematician who lived in Alexandria around
300 bce. Euclid wrote a monumental treatise of geometry, the Στοιχεῖα or Elements, which is one of the
milestones of human thought. The text presents the whole knowledge of geometry of that time. For the first
time, Euclid introduces two approaches that are now in common use: all statements are deduced from a
small number of basic axioms and for every statement a proof is given. The book, still in print today, has
been the reference geometry text for over 2000 years. On the web, it can be found at aleph0.clarku.edu/
~djoyce/java/elements/elements.html.
36 2 from motion measurement to continuity

TA B L E 3 Some measured velocity values.

O b s e r va t i o n Ve l o c i t y

Growth of deep sea manganese crust 80 am/s
Can you find something slower? Challenge 36 s
Stalagmite growth 0.3 pm/s
Lichen growth down to 7 pm/s
Typical motion of continents 10 mm/a = 0.3 nm/s
Human growth during childhood, hair growth 4 nm/s
Tree growth up to 30 nm/s
Electron drift in metal wire 1 μm/s
Sperm motion 60 to 160 μm/s
Speed of light at Sun’s centre Ref. 29 1 mm/s
Ketchup motion 1 mm/s
Slowest speed of light measured in matter on Earth Ref. 30 0.3 m/s

Motion Mountain – The Adventure of Physics
Speed of snowflakes 0.5 m/s to 1.5 m/s
Signal speed in human nerve cells Ref. 31 0.5 m/s to 120 m/s
Wind speed at 1 and 12 Beaufort (light air and hurricane) < 1.5 m/s, > 33 m/s
Speed of rain drops, depending on radius 2 m/s to 8 m/s
Fastest swimming fish, sailfish (Istiophorus platypterus) 22 m/s
2009 Speed sailing record over 500 m (by trimaran Hydroptère) 26.4 m/s
Fastest running animal, cheetah (Acinonyx jubatus) 30 m/s
Speed of air in throat when sneezing 42 m/s
Fastest throw: a cricket ball thrown with baseball technique while running 50 m/s

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Freely falling human, depending on clothing 50 to 90 m/s
Fastest bird, diving Falco peregrinus 60 m/s
Fastest badminton smash 70 m/s
Average speed of oxygen molecule in air at room temperature 280 m/s
Speed of sound in dry air at sea level and standard temperature 330 m/s
Speed of the equator 434 m/s
Cracking whip’s end 750 m/s
Speed of a rifle bullet 1 km/s
Speed of crack propagation in breaking silicon 5 km/s
Highest macroscopic speed achieved by man – the Helios II satellite 70.2 km/s
Speed of Earth through universe 370 km/s
Average speed (and peak speed) of lightning tip 600 km/s (50 Mm/s)
Highest macroscopic speed measured in our galaxy Ref. 32 0.97 ⋅ 108 m/s
Speed of electrons inside a colour TV tube 1 ⋅ 108 m/s
Speed of radio messages in space 299 792 458 m/s
Highest ever measured group velocity of light 10 ⋅ 108 m/s
Speed of light spot from a lighthouse when passing over the Moon 2 ⋅ 109 m/s
Highest proper velocity ever achieved for electrons by man 7 ⋅ 1013 m/s
Highest possible velocity for a light spot or a shadow no limit
2 from motion measurement to continuity 37

TA B L E 4 Properties of everyday – or Galilean – velocity.

Ve l o c i t i e s Physical M at h e m at i c a l Definition
can propert y name
Be distinguished distinguishability element of set Vol. III, page 285
Change gradually continuum real vector space Page 80, Vol. V,
page 364
Point somewhere direction vector space, dimensionality Page 80
Be compared measurability metricity Vol. IV, page 236
Be added additivity vector space Page 80
Have defined angles direction Euclidean vector space Page 81
Exceed any limit infinity unboundedness Vol. III, page 286

Challenge 37 d measuring space and time? If so, you probably want to skip to the next volume, jump-
ing 2000 years of enquiries. If you cannot do so, consider this: whenever we measure a

Motion Mountain – The Adventure of Physics
quantity we assume that everybody is able to do so, and that everybody will get the same
result. In other words, we define measurement as a comparison with a standard. We thus
implicitly assume that such a standard exists, i.e., that an example of a ‘perfect’ velocity
can be found. Historically, the study of motion did not investigate this question first, be-
cause for many centuries nobody could find such a standard velocity. You are thus in
good company.
How is velocity measured in everyday life? Animals and people estimate their velo-
city in two ways: by estimating the frequency of their own movements, such as their
steps, or by using their eyes, ears, sense of touch or sense of vibration to deduce how

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
their own position changes with respect to the environment. But several animals have
additional capabilities: certain snakes can determine speeds with their infrared-sensing
organs, others with their magnetic field sensing organs. Still other animals emit sounds
that create echoes in order to measure speeds to high precision. Other animals use the
stars to navigate. A similar range of solutions is used by technical devices. Table 5 gives
an overview.
Velocity is not always an easy subject. Physicists like to say, provokingly, that what
cannot be measured does not exist. Can you measure your own velocity in empty inter-
Challenge 38 s stellar space?
Velocity is of interest to both engineers and evolution scientist. In general, self-
propelled systems are faster the larger they are. As an example, Figure 17 shows how
this applies to the cruise speed of flying things. In general, cruise speed scales with the
sixth root of the weight, as shown by the trend line drawn in the graph. (Can you find
Challenge 39 d out why?) By the way, similar allometric scaling relations hold for many other properties
of moving systems, as we will see later on.
Some researchers have specialized in the study of the lowest velocities found in nature:
Ref. 33 they are called geologists. Do not miss the opportunity to walk across a landscape while
Challenge 40 e listening to one of them.
Velocity is a profound subject for an additional reason: we will discover that all its
seven properties of Table 4 are only approximate; none is actually correct. Improved ex-
periments will uncover exceptions for every property of Galilean velocity. The failure of
38 2 from motion measurement to continuity

1 101 102 103 104

Airbus 380
wing load W/A [N/m2] Boeing 747
DC10
Concorde
106 Boeing 727
Boeing 737
Fokker F-28 F-14
Fokker F-27 MIG 23
F-16
105
Learjet 31
Beechcraft King Air
Beechcraft Bonanza Beechcraft Baron
Piper Warrior
104
Schleicher ASW33B
Schleicher ASK23
Ultralight
Quicksilver B
103 human-powered plane Skysurfer

Motion Mountain – The Adventure of Physics
Pteranodon
102 griffon vulture (Gyps fulvus) wandering albatross (Diomedea exulans)
white-tailed eagle (Haliaeetus albicilla) whooper swan (Cygnus cygnus)
white stork (Ciconia ciconia) graylag goose (Anser anser)
black-backed gull (Larus marinus) cormorant (Phalacrocorax carbo)
pheasant (Phasianus colchicus)
herring gull (Larus argentatus) wild duck (Anas platyrhynchos)
10
peregrine falcon (Falco peregrinus)
carrion craw (Corvus corone) coot (Fulica atra)
weight W [N]

barn owl (Tyto alba) moorhen (Gallinula chloropus)
black headed gull (Larus ridibundus)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
common tern (Sterna hirundo)
1 blackbird (Turdus merula)
starling (Sturnus vulgaris)
common swift (Apus Apus) ortolan bunting (Emberiza hortulana)
sky lark (Alauda arvensis) house sparrow (Passer domesticus)
barn swallow (Hirundo rustica)
European Robin (Erithacus rubecula) great tit (Parus major)
house martin (Delichon urbica)
10-1 winter wren (Troglodytes troglodytes)
canary (Serinus canaria)
goldcrest (Regulus Regulus) hummingbird (Trochilidae)
privet hawkmoth (Sphinx ligustri) stag betle (Lucanus cervus)
blue underwing (Catocala fraxini) sawyer beetle (Prionus coriarius)
10-2 yellow-striped dragonfly(S. flaveolum) cockchafer (Melolontha melolontha)
eyed hawk-moth (S. ocellata) small stag beetle (Dorcus parallelopipedus)
swallowtail (P. machaon) june bug (Amphimallon solstitialis)
green dragonfly (Anax junius)
garden bumble bee (Bombus hortorum)
large white (P. brassicae)
10-3 common wasp (Vespa vulgaris)
ant lion (Myrmeleo
honey bee (Apis mellifera)
formicarius)
blowfly (Calliphora vicina)
small white (P. rapae)
crane fly (Tipulidae)
scorpionfly (Panorpidae)
damsel fly house fly (Musca domestica)
10-4 (Coenagrionidae)
midge (Chironomidae)

gnat (Culicidae)
mosquito (Culicidae)
10-5
fruit fly (Drosophila melanogaster)
1 2 3 5 7 10 20 30 50 70 100 200
cruise speed at sea level v [m/s]

F I G U R E 17 How wing load and sea-level cruise speed scales with weight in ﬂying objects, compared
with the general trend line (after a graph © Henk Tennekes).
2 from motion measurement to continuity 39

TA B L E 5 Speed measurement devices in biological and engineered systems.

Measurement Device Range

Own running speed in insects, leg beat frequency measured 0 to 33 m/s
mammals and humans with internal clock
Own car speed tachymeter attached to 0 to 150 m/s
wheels
Predators and hunters measuring prey vision system 0 to 30 m/s
speed
Police measuring car speed radar or laser gun 0 to 90 m/s
Bat measuring own and prey speed at doppler sonar 0 to 20 m/s
night
Sliding door measuring speed of doppler radar 0 to 3 m/s
approaching people
Own swimming speed in fish and friction and deformation of 0 to 30 m/s
humans skin

Motion Mountain – The Adventure of Physics
Own swimming speed in dolphins and sonar to sea floor 0 to 20 m/s
ships
Diving speed in fish, animals, divers pressure change 0 to 5 m/s
and submarines
Water predators and fishing boats sonar 0 to 20 m/s
measuring prey speed
Own speed relative to Earth in insects often none (grasshoppers) n.a.
Own speed relative to Earth in birds visual system 0 to 60 m/s
Own speed relative to Earth in radio goniometry, radar 0 to 8000 m/s

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
aeroplanes or rockets
Own speed relative to air in insects filiform hair deflection, 0 to 60 m/s
and birds feather deflection
Own speed relative to air in aeroplanes Pitot–Prandtl tube 0 to 340 m/s
Wind speed measurement in thermal, rotating or 0 to 80 m/s
meteorological stations ultrasound anemometers
Swallows measuring prey speed visual system 0 to 20 m/s
Bats measuring prey speed sonar 0 to 20 m/s
Macroscopic motion on Earth Global Positioning System, 0 to 100 m/s
Galileo, Glonass
Pilots measuring target speed radar 0 to 1000 m/s
Motion of stars optical Doppler effect 0 to 1000 km/s
Motion of star jets optical Doppler effect 0 to 200 Mm/s

the last three properties of Table 4 will lead us to special and general relativity, the failure
of the middle two to quantum theory and the failure of the first two properties to the uni-
fied description of nature. But for now, we’ll stick with Galilean velocity, and continue
with another Galilean concept derived from it: time.
40 2 from motion measurement to continuity

F I G U R E 18 A typical path followed by a stone thrown through the air – a parabola – with photographs

Motion Mountain – The Adventure of Physics
(blurred and stroboscopic) of a table tennis ball rebounding on a table (centre) and a stroboscopic
photograph of a water droplet rebounding on a strongly hydrophobic surface (right, © Andrew
Davidhazy, Max Groenendijk).

“
Without the concepts place, void and time,
change cannot be. [...] It is therefore clear [...]
that their investigation has to be carried out, by

”
studying each of them separately.
Aristotle* Physics, Book III, part 1.

“ ”
Time is an accident of motion.
Theophrastus**

“
Time does not exist in itself, but only through
the perceived objects, from which the concepts

”
of past, of present and of future ensue.
Lucretius,*** De rerum natura, lib. 1, v. 460 ss.

In their first years of life, children spend a lot of time throwing objects around. The term
‘object’ is a Latin word meaning ‘that which has been thrown in front.’ Developmental
Ref. 21 psychology has shown experimentally that from this very experience children extract
the concepts of time and space. Adult physicists do the same when studying motion at
university.
When we throw a stone through the air, we can define a sequence of observations.
Figure 18 illustrates how. Our memory and our senses give us this ability. The sense of

* Aristotle (b. 384/3 Stageira, d. 322 bce Euboea), important Greek philosopher and scientist, founder of
the Peripatetic school located at the Lyceum, a gymnasium dedicated to Apollo Lyceus.
** Theophrastus of Eresos (c. 371 – c. 287) was a revered Lesbian philosopher, and successor of Aristoteles
at the Lyceum.
*** Titus Lucretius Carus (c. 95 to c. 55 bce), Roman scholar and poet.
2 from motion measurement to continuity 41

TA B L E 6 Selected time measurements.

O b s e r va t i o n Time

Shortest measurable time 10−44 s
Shortest time ever measured 10 ys
Time for light to cross a typical atom 0.1 to 10 as
Shortest laser light pulse produced so far 200 as
Period of caesium ground state hyperfine transition 108.782 775 707 78 ps
Beat of wings of fruit fly 1 ms
Period of pulsar (rotating neutron star) PSR 1913+16 0.059 029 995 271(2) s
Human ‘instant’ 20 ms
Shortest lifetime of living being 0.3 d
Average length of day 400 million years ago 79 200 s
Average length of day today 86 400.002(1) s
From birth to your 1000 million seconds anniversary 31.7 a

Motion Mountain – The Adventure of Physics
Age of oldest living tree 4600 a
Use of human language 0.2 Ma
Age of Himalayas 35 to 55 Ma
Age of oldest rocks, found in Isua Belt, Greenland 3.8 Ga
and in Porpoise Cove, Hudson Bay
Age of Earth 4.6 Ga
Age of oldest stars 13.8 Ga
Age of most protons in your body 13.8 Ga
Lifetime of tantalum nucleus 180𝑚 Ta 1015 a

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Lifetime of bismuth 209 Bi nucleus 1.9(2) ⋅ 1019 a

hearing registers the various sounds during the rise, the fall and the landing of the stone.
Our eyes track the location of the stone from one point to the next. All observations have
their place in a sequence, with some observations preceding them, some observations
simultaneous to them, and still others succeeding them. We say that observations are
perceived to happen at various instants – also called ‘points in time’ – and we call the
sequence of all instants time.
An observation that is considered the smallest part of a sequence, i.e., not itself a
sequence, is called an event. Events are central to the definition of time; in particular,
Challenge 41 s starting or stopping a stopwatch are events. (But do events really exist? Keep this question
in the back of your head as we move on.)
Sequential phenomena have an additional property known as stretch, extension or
duration. Some measured values are given in Table 6.* Duration expresses the idea that
sequences take time. We say that a sequence takes time to express that other sequences
can take place in parallel with it.
How exactly is the concept of time, including sequence and duration, deduced from
observations? Many people have looked into this question: astronomers, physicists,

* A year is abbreviated a (Latin ‘annus’).
42 2 from motion measurement to continuity

watchmakers, psychologists and philosophers. All find:

⊳ Time is deduced by comparing motions.

This is even the case for children and animals. Beginning at a very young age, they
Ref. 21 develop the concept of ‘time’ from the comparison of motions in their surroundings.
Grown-ups take as a standard the motion of the Sun and call the resulting type of time
local time. From the Moon they deduce a lunar calendar. If they take a particular village
clock on a European island they call it the universal time coordinate (UTC), once known
as ‘Greenwich mean time.’*Astronomers use the movements of the stars and call the res-
ult ephemeris time (or one of its successors). An observer who uses his personal watch
calls the reading his proper time; it is often used in the theory of relativity.
Not every movement is a good standard for time. In the year 2000, an Earth rotation
Page 456 did not take 86 400 seconds any more, as it did in the year 1900, but 86 400.002 seconds.
Can you deduce in which year your birthday will have shifted by a whole day from the
Challenge 43 s time predicted with 86 400 seconds?

Motion Mountain – The Adventure of Physics
All methods for the definition of time are thus based on comparisons of motions.
In order to make the concept as precise and as useful as possible, a standard reference
motion is chosen, and with it a standard sequence and a standard duration is defined.
The device that performs this task is called a clock. We can thus answer the question of
the section title:

⊳ Time is what we read from a clock.

Note that all definitions of time used in the various branches of physics are equivalent to

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
this one; no ‘deeper’ or more fundamental definition is possible.** Note that the word
‘moment’ is indeed derived from the word ‘movement’. Language follows physics in this
case. Astonishingly, the definition of time just given is final; it will never be changed, not
even at the top of Motion Mountain. This is surprising at first sight, because many books
have been written on the nature of time. Instead, they should investigate the nature of
motion!

⊳ Every clock reminds us that in order to understand time, we need to under-
stand motion.

But this is the aim of our walk anyhow. We are thus set to discover all the secrets of time
as a side result of our adventure.
Time is not only an aspect of observations, it is also a facet of personal experience.
Even in our innermost private life, in our thoughts, feelings and dreams, we experience
sequences and durations. Children learn to relate this internal experience of time with

* Official UTC is used to determine the phase of the power grid, phone and internet companies’ bit streams
and the signal to the GPS system. The latter is used by many navigation systems around the world, especially
in ships, aeroplanes and mobile phones. For more information, see the www.gpsworld.com website. The
time-keeping infrastructure is also important for other parts of the modern economy. Can you spot the
Challenge 42 s most important ones?
Ref. 34 ** The oldest clocks are sundials. The science of making them is called gnomonics.
2 from motion measurement to continuity 43

TA B L E 7 Properties of Galilean time.

I ns ta nt s o f t i m e Physical M at h e m at i c a l Definition
propert y name

Can be distinguished distinguishability element of set Vol. III, page 285
Can be put in order sequence order Vol. V, page 364
Define duration measurability metricity Vol. IV, page 236
Can have vanishing duration continuity denseness, completeness Vol. V, page 364
Allow durations to be added additivity metricity Vol. IV, page 236
Don’t harbour surprises translation invariance homogeneity Page 238
Don’t end infinity unboundedness Vol. III, page 286
Are equal for all observers absoluteness uniqueness

external observations, and to make use of the sequential property of events in their ac-
tions. Studies of the origin of psychological time show that it coincides – apart from its

Motion Mountain – The Adventure of Physics
lack of accuracy – with clock time.* Every living human necessarily uses in his daily
life the concept of time as a combination of sequence and duration; this fact has been
checked in numerous investigations. For example, the term ‘when’ exists in all human
Ref. 36 languages.
Time is a concept necessary to distinguish between observations. In any sequence of
observations, we observe that events succeed each other smoothly, apparently without
end. In this context, ‘smoothly’ means that observations that are not too distant tend to
be not too different. Yet between two instants, as close as we can observe them, there is
always room for other events. Durations, or time intervals, measured by different people

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
with different clocks agree in everyday life; moreover, all observers agree on the order
of a sequence of events. Time is thus unique in everyday life. One also says that time is
absolute in everyday life.
Time is necessary to distinguish between observations. For this reason, all observing
devices that distinguish between observations, from brains to dictaphones and video
cameras, have internal clocks. In particular, all animal brains have internal clocks. These
brain clocks allow their users to distinguish between present, recent and past data and
observations.
When Galileo studied motion in the seventeenth century, there were as yet no stop-
watches. He thus had to build one himself, in order to measure times in the range
Challenge 44 s between a fraction and a few seconds. Can you imagine how he did it?
If we formulate with precision all the properties of time that we experience in our
daily life, we are lead to Table 7. This concept of time is called Galilean time. All its prop-
erties can be expressed simultaneously by describing time with the help of real numbers.
In fact, real numbers have been constructed by mathematicians to have exactly the same
Vol. III, page 295 properties as Galilean time, as explained in the chapter on the brain. In the case of Ga-

Vol. V, page 42 * The brain contains numerous clocks. The most precise clock for short time intervals, the internal interval
timer of the brain, is more accurate than often imagined, especially when trained. For time periods between
Ref. 35 a few tenths of a second, as necessary for music, and a few minutes, humans can achieve timing accuracies
of a few per cent.
44 2 from motion measurement to continuity

lilean time, every instant of time can be described by a real number, often abbreviated
𝑡. The duration of a sequence of events is then given by the difference between the time
values of the final and the starting event.
We will have quite some fun with Galilean time in this part of our adventure. However,
hundreds of years of close scrutiny have shown that every single property of Galilean time
listed in Table 7 is approximate, and none is strictly correct. This story is told in the rest
of our adventure.

Clo cks

“
The most valuable thing a man can spend is

”
time.
Theophrastus

A clock is a moving system whose position can be read.
There are many types of clocks: stopwatches, twelve-hour clocks, sundials, lunar
clocks, seasonal clocks, etc. A few are shown in Figure 19. Most of these clock types are

Motion Mountain – The Adventure of Physics
Ref. 37 also found in plants and animals, as shown in Table 8.
Ref. 38 Interestingly, there is a strict rule in the animal kingdom: large clocks go slow. How
this happens is shown in Figure 20, another example of an allometric scaling ‘law’.
A clock is a moving system whose position can be read. Of course, a precise clock
is a system moving as regularly as possible, with as little outside disturbance as pos-
sible. Clockmakers are experts in producing motion that is as regular as possible. We
Page 181 will discover some of their tricks below. We will also explore, later on, the limits for the
Vol. V, page 45 precision of clocks.
Is there a perfect clock in nature? Do clocks exist at all? We will continue to study
these questions throughout this work and eventually reach a surprising conclusion. At

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
this point, however, we state a simple intermediate result: since clocks do exist, somehow
Challenge 45 s there is in nature an intrinsic, natural and ideal way to measure time. Can you see it?
2 from motion measurement to continuity 45

light from the Sun

time read off :
11h00 CEST sub-solar poin t

sub-solar point
close to Mekk a

close to Mekka
Sun’s orbit sun's orbi t

on May 15th on May 15 th

mirror reflects
the sunlight
winter-spring
display screen winter-spring
display scree n

time scale ring
CEST

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 19 Different types of clocks: a high-tech sundial (size c. 30 cm), a naval pocket chronometer
(size c. 6 cm), a caesium atomic clock (size c. 4 m), a group of cyanobacteria and the Galilean satellites of
Jupiter (© Carlo Heller at www.heliosuhren.de, Anonymous, INMS, Wikimedia, NASA).
46 2 from motion measurement to continuity

TA B L E 8 Examples of biological rhythms and clocks.

Living being O s c i l l at i n g s ys t e m Period
Sand hopper (Talitrus saltator) knows in which direction to flee from circadian
the position of the Sun or Moon
Human (Homo sapiens) gamma waves in the brain 0.023 to 0.03 s
alpha waves in the brain 0.08 to 0.13 s
heart beat 0.3 to 1.5 s
delta waves in the brain 0.3 to 10 s
blood circulation 30 s
cellular circhoral rhythms 1 to 2 ks
rapid-eye-movement sleep period 5.4 ks
nasal cycle 4 to 9 ks
growth hormone cycle 11 ks
suprachiasmatic nucleus (SCN), 90 ks
circadian hormone concentration,

Motion Mountain – The Adventure of Physics
temperature, etc.; leads to jet lag
skin clock circadian
monthly period 2.4(4) Ms
built-in aging 3.2(3) Gs
Common fly (Musca domestica) wing beat 30 ms
Fruit fly (Drosophila wing beat for courting 34 ms
melanogaster)
Most insects (e.g. wasps, fruit winter approach detection (diapause) by yearly
flies) length of day measurement; triggers

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
metabolism changes
Algae (Acetabularia) Adenosinetriphosphate (ATP)
concentration
Moulds (e.g. Neurospora crassa) conidia formation circadian
Many flowering plants flower opening and closing circadian
Tobacco plant flower opening clock; triggered by annual
length of days, discovered in 1920 by
Garner and Allard
Arabidopsis circumnutation circadian
growth a few hours
Telegraph plant (Desmodium side leaf rotation 200 s
gyrans)
Forsythia europaea, F. suspensa, Flower petal oscillation, discovered by 5.1 ks
F. viridissima, F. spectabilis Van Gooch in 2002
2 from motion measurement to continuity 47

Maximum lifespan of wild birds
+7

+6
Reproductive maturity

Growth-time in birds
+5
Gestation time (max 100 cycles per lifetime)

Motion Mountain – The Adventure of Physics
Log10 of time/min

Metabolism of fat, 0.1% of body mass
(max 1 000 000 cycles per lifetime)
+2
Sleep cycle

Insulin clearance of body plasma volume
+1
(max 3 000 000 cycles per lifetime)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
0
Circulation of blood volume
(max 30 000 000 cycles per lifetime)

-1
Respiratory cycles
(max 200 000 000 cycles per lifetime)
Gut contraction
-2 (max 300 000 000 cycles per lifetime)

Cardiac cycle
(max 1 000 000 000 cycles per lifetime)
-3
Fast muscle contraction
(max 120 000 000 000 cycles per lifetime)
-4
0.001 0.01 0.1 1 10 100 1000
Body mass/kg

F I G U R E 20 How biological rhythms scale with size in mammals: all scale more or less with a quarter
power of the mass (after data from the EMBO and Enrique Morgado).
48 2 from motion measurement to continuity

Why d o clo cks go clo ckwise?

“ ”
Challenge 46 s What time is it at the North Pole now?

Most rotational motions in our society, such as athletic races, horse, bicycle or ice skat-
ing races, turn anticlockwise.* Mathematicians call this the positive rotation sense. Every
supermarket leads its guests anticlockwise through the hall. Why? Most people are right-
handed, and the right hand has more freedom at the outside of a circle. Therefore thou-
sands of years ago chariot races in stadia went anticlockwise. As a result, all stadium races
still do so to this day, and that is why runners move anticlockwise. For the same reason,
helical stairs in castles are built in such a way that defending right-handers, usually from
above, have that hand on the outside.
On the other hand, the clock imitates the shadow of sundials; obviously, this is true
on the northern hemisphere only, and only for sundials on the ground, which were the
most common ones. (The old trick to determine south by pointing the hour hand of
a horizontal watch to the Sun and halving the angle between it and the direction of 12

Motion Mountain – The Adventure of Physics
o’clock does not work on the southern hemisphere – but there you can determine north
in this way.) So every clock implicitly continues to state on which hemisphere it was
invented. In addition, it also tells us that sundials on walls came in use much later than
those on the floor.

Does time flow?

“
Wir können keinen Vorgang mit dem ‘Ablauf
der Zeit’ vergleichen – diesen gibt es nicht –,
sondern nur mit einem anderen Vorgang (etwa

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
”
dem Gang des Chronometers).**
Ludwig Wittgenstein, Tractatus, 6.3611

“ ”
Si le temps est un fleuve, quel est son lit?***

The expression ‘the flow of time’ is often used to convey that in nature change follows
after change, in a steady and continuous manner. But though the hands of a clock ‘flow’,
time itself does not. Time is a concept introduced specially to describe the flow of events
around us; it does not itself flow, it describes flow. Time does not advance. Time is neither
linear nor cyclic. The idea that time flows is as hindering to understanding nature as is
Vol. III, page 90 the idea that mirrors exchange right and left.
The misleading use of the incorrect expression ‘flow of time’ was propagated first
Ref. 39 by some flawed Greek thinkers and then again by Newton. And it still continues. Ar-
istotle, careful to think logically, pointed out its misconception, and many did so after
him. Nevertheless, expressions such as ‘time reversal’, the ‘irreversibility of time’, and
the much-abused ‘time’s arrow’ are still common. Just read a popular science magazine

* Notable exceptions are most, but not all, Formula 1 races.
** ‘We cannot compare any process with ‘the passage of time’ – there is no such thing – but only with
another process (say, with the working of a chronometer).’
*** ‘If time is a river, what is his bed?’
2 from motion measurement to continuity 49

Challenge 47 e chosen at random. The fact is: time cannot be reversed, only motion can, or more pre-
cisely, only velocities of objects; time has no arrow, only motion has; it is not the flow
of time that humans are unable to stop, but the motion of all the objects in nature. In-
Ref. 40 credibly, there are even books written by respected physicists that study different types
of ‘time’s arrows’ and compare them with each other. Predictably, no tangible or new
result is extracted.

⊳ Time does not flow. Only bodies flow.

Time has no direction. Motion has. For the same reason, colloquial expressions such as
‘the start (or end) of time’ should be avoided. A motion expert translates them straight
away into ‘the start (or end) of motion’.

What is space?

“
The introduction of numbers as coordinates [...]

”
is an act of violence [...].

Motion Mountain – The Adventure of Physics
Hermann Weyl, Philosophie der Mathematik
und Naturwissenschaft.*

Whenever we distinguish two objects from each other, such as two stars, we first of
all distinguish their positions. We distinguish positions with our senses of sight, touch,
proprioception and hearing. Position is therefore an important aspect of the physical
state of an object. A position is taken by only one object at a time. Positions are limited.
The set of all available positions, called (physical) space, acts as both a container and a
background.
Closely related to space and position is size, the set of positions an object occupies.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Small objects occupy only subsets of the positions occupied by large ones. We will discuss
Page 52 size in more detail shortly.
How do we deduce space from observations? During childhood, humans (and most
higher animals) learn to bring together the various perceptions of space, namely the
visual, the tactile, the auditory, the kinaesthetic, the vestibular etc., into one self-
consistent set of experiences and description. The result of this learning process is a
certain concept of space in the brain. Indeed, the question ‘where?’ can be asked and
answered in all languages of the world. Being more precise, adults derive space from dis-
tance measurements. The concepts of length, area, volume, angle and solid angle are all
deduced with their help. Geometers, surveyors, architects, astronomers, carpet salesmen
and producers of metre sticks base their trade on distance measurements.

⊳ Space is formed from all the position and distance relations between objects
using metre sticks.

Humans developed metre sticks to specify distances, positions and sizes as accurately as
possible.
* Hermann Weyl (1885–1955) was one of the most important mathematicians of his time, as well as an
important theoretical physicist. He was one of the last universalists in both fields, a contributor to quantum
theory and relativity, father of the term ‘gauge’ theory, and author of many popular texts.
50 2 from motion measurement to continuity

F I G U R E 21 Two
proofs that space has
three dimensions: the
vestibular labyrinth in
the inner ear of
mammals (here a
human) with three
canals and a knot
(© Northwestern
University).

Metre sticks work well only if they are straight. But when humans lived in the jungle,
there were no straight objects around them. No straight rulers, no straight tools, noth-

Motion Mountain – The Adventure of Physics
Challenge 48 s ing. Today, a cityscape is essentially a collection of straight lines. Can you describe how
humans achieved this?
Once humans came out of the jungle with their newly built metre sticks, they collec-
ted a wealth of results. The main ones are listed in Table 9; they are easily confirmed by
personal experience. In particular, objects can take positions in an apparently continuous
manner: there indeed are more positions than can be counted.* Size is captured by de-
fining the distance between various positions, called length, or by using the field of view
an object takes when touched, called its surface area. Length and area can be measured
with the help of a metre stick. (Selected measurement results are given in Table 10; some

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
length measurement devices are shown in Figure 23.) The length of objects is independ-
ent of the person measuring it, of the position of the objects and of their orientation. In
daily life the sum of angles in any triangle is equal to two right angles. There are no limits
to distances, lengths and thus to space.
Experience shows us that space has three dimensions; we can define sequences of
positions in precisely three independent ways. Indeed, the inner ear of (practically) all
vertebrates has three semicircular canals that sense the body’s acceleration in the three
dimensions of space, as shown in Figure 21.** Similarly, each human eye is moved by
Challenge 49 s three pairs of muscles. (Why three?) Another proof that space has three dimensions is
provided by shoelaces: if space had more than three dimensions, shoelaces would not
be useful, because knots exist only in three-dimensional space. But why does space have
three dimensions? This is one of the most difficult question of physics. We leave it open
for the time being.
It is often said that thinking in four dimensions is impossible. That is wrong. Just try.
Challenge 50 s For example, can you confirm that in four dimensions knots are impossible?
Like time intervals, length intervals can be described most precisely with the help

* For a definition of uncountability, see page 288 in Volume III.
** Note that saying that space has three dimensions implies that space is continuous; the mathematician
and philosopher Luitzen Brouwer (b. 1881 Overschie, d. 1966 Blaricum) showed that dimensionality is only
a useful concept for continuous sets.
2 from motion measurement to continuity 51

TA B L E 9 Properties of Galilean space.

Points, or Physical M at h e m at i c a l Defini-
p o s i t i o n s i n s pa c e propert y name tion
Can be distinguished distinguishability element of set Vol. III, page 285
Can be lined up if on one line sequence order Vol. V, page 364
Can form shapes shape topology Vol. V, page 363
Lie along three independent possibility of knots 3-dimensionality Page 81, Vol. IV,
directions page 235
Can have vanishing distance continuity denseness, Vol. V, page 364
completeness
Define distances measurability metricity Vol. IV, page 236
Allow adding translations additivity metricity Vol. IV, page 236
Define angles scalar product Euclidean space Page 81
Don’t harbour surprises translation invariance homogeneity
Can beat any limit infinity unboundedness Vol. III, page 286

Motion Mountain – The Adventure of Physics
Defined for all observers absoluteness uniqueness Page 52

of real numbers. In order to simplify communication, standard units are used, so that
everybody uses the same numbers for the same length. Units allow us to explore the
general properties of Galilean space experimentally: space, the container of objects, is
continuous, three-dimensional, isotropic, homogeneous, infinite, Euclidean and unique
– or ‘absolute’. In mathematics, a structure or mathematical concept with all the prop-
erties just mentioned is called a three-dimensional Euclidean space. Its elements, (math-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
ematical) points, are described by three real parameters. They are usually written as

(𝑥, 𝑦, 𝑧) (1)

and are called coordinates. They specify and order the location of a point in space. (For
Page 81 the precise definition of Euclidean spaces, see below.)
What is described here in just half a page actually took 2000 years to be worked out,
mainly because the concepts of ‘real number’ and ‘coordinate’ had to be discovered
first. The first person to describe points of space in this way was the famous mathem-
atician and philosopher René Descartes*, after whom the coordinates of expression (1)
are named Cartesian.
Like time, space is a necessary concept to describe the world. Indeed, space is auto-
matically introduced when we describe situations with many objects. For example, when
many spheres lie on a billiard table, we cannot avoid using space to describe the relations
between them. There is no way to avoid using spatial concepts when talking about nature.
Even though we need space to talk about nature, it is still interesting to ask why this
is possible. For example, since many length measurement methods do exist – some are

* René Descartes or Cartesius (b. 1596 La Haye, d. 1650 Stockholm), mathematician and philosopher, au-
thor of the famous statement ‘je pense, donc je suis’, which he translated into ‘cogito ergo sum’ – I think
therefore I am. In his view, this is the only statement one can be sure of.
52 2 from motion measurement to continuity

F I G U R E 22 René Descartes (1596 –1650).

listed in Table 11 – and since they all yield consistent results, there must be a natural or
Challenge 51 s ideal way to measure distances, sizes and straightness. Can you find it?
As in the case of time, each of the properties of space just listed has to be checked. And
again, careful observations will show that each property is an approximation. In simpler
and more drastic words, all of them are wrong. This confirms Weyl’s statement at the
beginning of this section. In fact, his statement about the violence connected with the

Motion Mountain – The Adventure of Physics
introduction of numbers is told by every forest in the world. The rest of our adventure
will show this.

“ ”
Μέτρον ἄριστον.*
Cleobulus

Are space and time absolu te or relative?
In everyday life, the concepts of Galilean space and time include two opposing aspects;
the contrast has coloured every discussion for several centuries. On the one hand, space

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
and time express something invariant and permanent; they both act like big containers
for all the objects and events found in nature. Seen this way, space and time have an ex-
istence of their own. In this sense one can say that they are fundamental or absolute. On
the other hand, space and time are tools of description that allow us to talk about rela-
tions between objects. In this view, they do not have any meaning when separated from
objects, and only result from the relations between objects; they are derived, relational
Challenge 52 e or relative. Which of these viewpoints do you prefer? The results of physics have altern-
ately favoured one viewpoint or the other. We will repeat this alternation throughout our
Ref. 41 adventure, until we find the solution. And obviously, it will turn out to be a third option.

Size – why length and area exist, bu t volume d oes not
A central aspect of objects is their size. As a small child, under school age, every human
learns how to use the properties of size and space in their actions. As adults seeking
precision, with the definition of distance as the difference between coordinates allows us
to define length in a reliable way. It took hundreds of years to discover that this is not the
case. Several investigations in physics and mathematics led to complications.
The physical issues started with an astonishingly simple question asked by Lewis

* ‘Measure is the best (thing).’ Cleobulus (Κλεοβουλος) of Lindos, (c. 620–550 BCE ) was another of the
proverbial seven sages.
2 from motion measurement to continuity 53

TA B L E 10 Some measured distance values.

O b s e r va t i o n D i s ta nce

Galaxy Compton wavelength 10−85 m (calculated only)
Planck length, the shortest measurable length 10−35 m
Proton diameter 1 fm
Electron Compton wavelength 2.426 310 215(18) pm
Smallest air oscillation detectable by human ear 11 pm
Hydrogen atom size 30 pm
Size of small bacterium 0.2 μm
Wavelength of visible light 0.4 to 0.8 μm
Radius of sharp razor blade 5 μm
Point: diameter of smallest object visible with naked eye 20 μm
Diameter of human hair (thin to thick) 30 to 80 μm
Record diameter of hailstone 20 cm

Motion Mountain – The Adventure of Physics
Total length of DNA in each human cell 2m
Longest human throw with any object, using a boomerang 427 m
Highest human-built structure, Burj Khalifa 828 m
Largest living thing, the fungus Armillaria ostoyae 3 km
Largest spider webs in Mexico c. 5 km
Length of Earth’s Equator 40 075 014.8(6) m
Total length of human blood vessels (rough estimate) 4𝑡𝑜16 ⋅ 104 km
Total length of human nerve cells (rough estimate) 1.5𝑡𝑜8 ⋅ 105 km
Average distance to Sun 149 597 870 691(30) m

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Light year 9.5 Pm
Distance to typical star at night 10 Em
Size of galaxy 1 Zm
Distance to Andromeda galaxy 28 Zm
Most distant visible object 125 Ym

Richardson:* How long is the western coastline of Britain?
Following the coastline on a map using an odometer, a device shown in Figure 24,
Richardson found that the length 𝑙 of the coastline depends on the scale 𝑠 (say 1 : 10 000
or 1 : 500 000) of the map used:
𝑙 = 𝑙0 𝑠0.25 (2)

(Richardson found other exponentials for other coasts.) The number 𝑙0 is the length at
scale 1 : 1. The main result is that the larger the map, the longer the coastline. What would
happen if the scale of the map were increased even beyond the size of the original? The
length would increase beyond all bounds. Can a coastline really have infinite length?
Yes, it can. In fact, mathematicians have described many such curves; nowadays, they

* Lewis Fray Richardson (1881–1953), English physicist and psychologist.
54 2 from motion measurement to continuity

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 23 Three mechanical (a vernier caliper, a micrometer screw, a moustache) and three optical
(the eyes, a laser meter, a light curtain) length and distance measurement devices (© www.
medien-werkstatt.de, Naples Zoo, Keyence, and Leica Geosystems).

are called fractals. An infinite number of them exist, and Figure 25 shows one example.*

* Most of these curves are self-similar, i.e., they follow scaling ‘laws’ similar to the above-mentioned. The
term ‘fractal’ is due to the mathematician Benoît Mandelbrot and refers to a strange property: in a certain
sense, they have a non-integral number 𝐷 of dimensions, despite being one-dimensional by construction.
Mandelbrot saw that the non-integer dimension was related to the exponent 𝑒 of Richardson by 𝐷 = 1 + 𝑒,
Ref. 42 thus giving 𝐷 = 1.25 in the example above. The number 𝐷 varies from case to case. Measurements yield
a value 𝐷 = 1.14 for the land frontier of Portugal, 𝐷 = 1.13 for the Australian coast and 𝐷 = 1.02 for the
South African coast.
2 from motion measurement to continuity 55

TA B L E 11 Length measurement devices in biological and engineered systems.

Measurement Device Range

Humans
Measurement of body shape, e.g. finger muscle sensors 0.3 mm to 2 m
distance, eye position, teeth distance
Measurement of object distance stereoscopic vision 1 to 100 m
Measurement of object distance sound echo effect 0.1 to 1000 m
Animals
Measurement of hole size moustache up to 0.5 m
Measurement of walking distance by desert ants step counter up to 100 m
Measurement of flight distance by honey bees eye up to 3 km
Measurement of swimming distance by sharks magnetic field map up to 1000 km
Measurement of prey distance by snakes infrared sensor up to 2 m
Measurement of prey distance by bats, dolphins, sonar up to 100 m

Motion Mountain – The Adventure of Physics
and hump whales
Measurement of prey distance by raptors vision 0.1 to 1000 m
Machines
Measurement of object distance by laser light reflection 0.1 m to 400 Mm
Measurement of object distance by radar radio echo 0.1 to 50 km
Measurement of object length interferometer 0.5 μm to 50 km
Measurement of star, galaxy or quasar distance intensity decay up to 125 Ym
Measurement of particle size accelerator down to 10−18 m

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 24 A curvimeter or odometer
(photograph © Frank Müller).

n=1 n=2 n=3 n=∞

F I G U R E 25 An example of a fractal: a self-similar curve of inﬁnite length (far right), and its construction.

Challenge 53 e Can you construct another?
Length has other strange properties. The mathematician Giuseppe Vitali was the first
to discover that it is possible to cut a line segment of length 1 into pieces that can be reas-
56 2 from motion measurement to continuity

sembled – merely by shifting them in the direction of the segment – into a line segment
of length 2. Are you able to find such a division using the hint that it is only possible
Challenge 54 d using infinitely many pieces?
To sum up

⊳ Length exists. But length is well defined only for lines that are straight or
nicely curved, but not for intricate lines, or for lines that can be cut into
infinitely many pieces.

We therefore avoid fractals and other strangely shaped curves in the following, and we
take special care when we talk about infinitely small segments. These are the central as-
sumptions in the first five volumes of this adventure, and we should never forget them!
We will come back to these assumptions in the last part of our adventure.
In fact, all these problems pale when compared with the following problem. Com-
monly, area and volume are defined using length. You think that it is easy? You’re wrong,
as well as being a victim of prejudices spread by schools around the world. To define area

Motion Mountain – The Adventure of Physics
and volume with precision, their definitions must have two properties: the values must
be additive, i.e., for finite and infinite sets of objects, the total area and volume must be
the sum of the areas and volumes of each element of the set; and the values must be ri-
gid, i.e., if we cut an area or a volume into pieces and then rearrange the pieces, the value
must remain the same. Are such definitions possible? In other words, do such concepts
of volume and area exist?
For areas in a plane, we proceed in the following standard way: we define the area 𝐴
of a rectangle of sides 𝑎 and 𝑏 as 𝐴 = 𝑎𝑏; since any polygon can be rearranged into a
Challenge 55 s rectangle with a finite number of straight cuts, we can then define an area value for all

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
polygons. Subsequently, we can define the area for nicely curved shapes as the limit of
Page 251 the sum of infinitely many polygons. This method is called integration; it is introduced
in detail in the section on physical action.
However, integration does not allow us to define area for arbitrarily bounded regions.
Challenge 56 s (Can you imagine such a region?) For a complete definition, more sophisticated tools are
needed. They were discovered in 1923 by the famous mathematician Stefan Banach.* He
proved that one can indeed define an area for any set of points whatsoever, even if the
border is not nicely curved but extremely complicated, such as the fractal curve previ-
ously mentioned. Today this generalized concept of area, technically a ‘finitely additive
isometrically invariant measure,’ is called a Banach measure in his honour. Mathem-
aticians sum up this discussion by saying that since in two dimensions there is a Banach
measure, there is a way to define the concept of area – an additive and rigid measure –
for any set of points whatsoever.** In short,

⊳ Area exists. Area is well defined for plane and other nicely behaved surfaces,

* Stefan Banach (b. 1892 Krakow, d. 1945 Lvov), important mathematician.
** Actually, this is true only for sets on the plane. For curved surfaces, such as the surface of a sphere, there
are complications that will not be discussed here. In addition, the problems mentioned in the definition of
length of fractals also reappear for area if the surface to be measured is not flat. A typical example is the
area of the human lung: depending on the level of details examined, the area values vary from a few up to
over a hundred square metres.
2 from motion measurement to continuity 57

dihedral
angle

F I G U R E 26 A polyhedron with one of
its dihedral angles (© Luca Gastaldi).

Motion Mountain – The Adventure of Physics
but not for intricate shapes.

What is the situation in three dimensions, i.e., for volume? We can start in the same
way as for area, by defining the volume 𝑉 of a rectangular polyhedron with sides 𝑎, 𝑏, 𝑐 as
𝑉 = 𝑎𝑏𝑐. But then we encounter a first problem: a general polyhedron cannot be cut into
a cube by straight cuts! The limitation was discovered in 1900 and 1902 by Max Dehn.*
He found that the possibility depends on the values of the edge angles, or dihedral angles,
as the mathematicians call them. (They are defined in Figure 26.) If one ascribes to every

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
edge of a general polyhedron a number given by its length 𝑙 times a special function
𝑔(𝛼) of its dihedral angle 𝛼, then Dehn found that the sum of all the numbers for all
the edges of a solid does not change under dissection, provided that the function fulfils
𝑔(𝛼 + 𝛽) = 𝑔(𝛼) + 𝑔(𝛽) and 𝑔(π) = 0. An example of such a strange function 𝑔 is the
one assigning the value 0 to any rational multiple of π and the value 1 to a basis set of
irrational multiples of π. The values for all other dihedral angles of the polyhedron can
then be constructed by combination of rational multiples of these basis angles. Using this
Challenge 57 s function, you may then deduce for yourself that a cube cannot be dissected into a regular
tetrahedron because their respective Dehn invariants are different.**
Despite the problems with Dehn invariants, a rigid and additive concept of volume
for polyhedra does exist, since for all polyhedra and, in general, for all ‘nicely curved’
shapes, the volume can be defined with the help of integration.
Now let us consider general shapes and general cuts in three dimensions, not just the
‘nice’ ones mentioned so far. We then stumble on the famous Banach–Tarski theorem
Ref. 43 (or paradox). In 1924, Stefan Banach and Alfred Tarski*** proved that it is possible to

* Max Dehn (b. 1878 Hamburg, d. 1952 Black Mountain), mathematician, student of David Hilbert.
** This is also told in the beautiful book by M. Aigler & G. M. Z iegler, Proofs from the Book, Springer
Verlag, 1999. The title is due to the famous habit of the great mathematician Paul Erdős to imagine that all
beautiful mathematical proofs can be assembled in the ‘book of proofs’.
*** Alfred Tarski (b. 1902 Warsaw, d. 1983 Berkeley), influential mathematician.
58 2 from motion measurement to continuity

F I G U R E 27 Straight lines found in nature: cerussite (picture width approx. 3 mm, © Stephan Wolfsried)
and selenite (picture width approx. 15 m, © Arch. Speleoresearch & Films/La Venta at www.laventa.it and
www.naica.com.mx).

cut one sphere into five pieces that can be recombined to give two spheres, each the size
of the original. This counter-intuitive result is the Banach–Tarski theorem. Even worse,
another version of the theorem states: take any two sets not extending to infinity and

Motion Mountain – The Adventure of Physics
containing a solid sphere each; then it is always possible to dissect one into the other
with a finite number of cuts. In particular it is possible to dissect a pea into the Earth, or
vice versa. Size does not count!* In short, volume is thus not a useful concept at all!
The Banach–Tarski theorem raises two questions: first, can the result be applied to
gold or bread? That would solve many problems. Second, can it be applied to empty
Challenge 58 s space? In other words, are matter and empty space continuous? Both topics will be ex-
plored later in our walk; each issue will have its own, special consequences. For the mo-
ment, we eliminate this troubling issue by restricting our interest – again – to smoothly
curved shapes (and cutting knives). With this restriction, volumes of matter and of empty

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
space do behave nicely: they are additive and rigid, and show no paradoxes.** Indeed,
the cuts required for the Banach–Tarski paradox are not smooth; it is not possible to
perform them with an everyday knife, as they require (infinitely many) infinitely sharp
bends performed with an infinitely sharp knife. Such a knife does not exist. Nevertheless,
we keep in the back of our mind that the size of an object or of a piece of empty space is
a tricky quantity – and that we need to be careful whenever we talk about it.
In summary,

⊳ Volume only exists as an approximation. Volume is well-defined only for
regions with smooth surfaces. Volume does not exist in general, when in-
finitely sharp cuts are allowed.

We avoid strangely shaped volumes, surfaces and curves in the following, and we take
special care when we talk about infinitely small entities. We can talk about length, area
and volume only with this restriction. This avoidance is a central assumption in the first
* The proof of the result does not need much mathematics; it is explained beautifully by Ian Stewart in
Paradox of the spheres, New Scientist, 14 January 1995, pp. 28–31. The proof is based on the axiom of choice,
Vol. III, page 286 which is presented later on. The Banach–Tarski paradox also exists in four dimensions, as it does in any
Ref. 44 higher dimension. More mathematical detail can be found in the beautiful book by Stan Wagon.
** Mathematicians say that a so-called Lebesgue measure is sufficient in physics. This countably additive
isometrically invariant measure provides the most general way to define a volume.
2 from motion measurement to continuity 59

Motion Mountain – The Adventure of Physics
F I G U R E 28 A photograph of the Earth – seen from the direction of the Sun (NASA).

five volumes of this adventure. Again: we should never forget these restrictions, even
though they are not an issue in everyday life. We will come back to the assumptions at

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the end of our adventure.

What is straight?
When you see a solid object with a straight edge, it is a 99 %-safe bet that it is man-
made. Of course, there are exceptions, as shown in Figure 27.* The largest crystals ever
Ref. 46 found are 18 m in length. But in general, the contrast between the objects seen in a city
– buildings, furniture, cars, electricity poles, boxes, books – and the objects seen in a
forest – trees, plants, stones, clouds – is evident: in the forest no object is straight or flat,
whereas in the city most objects are.
Page 479 Any forest teaches us the origin of straightness; it presents tall tree trunks and rays of
daylight entering from above through the leaves. For this reason we call a line straight if
it touches either a plumb-line or a light ray along its whole length. In fact, the two defin-
Challenge 60 s itions are equivalent. Can you confirm this? Can you find another definition? Obviously,
we call a surface flat if for any chosen orientation and position the surface touches a
plumb-line or a light ray along its whole extension.

Challenge 59 ny * Why do crystals have straight edges? Another example of straight lines in nature, unrelated to atomic
Page 429 structures, is the well-known Irish geological formation called the Giant’s Causeway. Other candidates that
might come to mind, such as certain bacteria which have (almost) square or (almost) triangular shapes are
Ref. 45 not counter-examples, as the shapes are only approximate.
60 2 from motion measurement to continuity

Motion Mountain – The Adventure of Physics
F I G U R E 29 A model illustrating the hollow Earth theory, showing how day and night appear
(© Helmut Diehl).

In summary, the concept of straightness – and thus also of flatness – is defined with

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the help of bodies or radiation. In fact, all spatial concepts, like all temporal concepts,
require motion for their definition.

A hollow E arth?
Space and straightness pose subtle challenges. Some strange people maintain that all hu-
mans live on the inside of a sphere; they call this the hollow Earth model. They claim that
Ref. 47 the Moon, the Sun and the stars are all near the centre of the hollow sphere, as illustrated
in Figure 29. They also explain that light follows curved paths in the sky and that when
conventional physicists talk about a distance 𝑟 from the centre of the Earth, the real hol-
Challenge 61 s low Earth distance is 𝑟he = 𝑅2Earth /𝑟. Can you show that this model is wrong? Roman
Sexl* used to ask this question to his students and fellow physicists.
The answer is simple: if you think you have an argument to show that the hollow
Earth model is wrong, you are mistaken! There is no way of showing that such a view is
wrong. It is possible to explain the horizon, the appearance of day and night, as well as the
Challenge 62 e satellite photographs of the round Earth, such as Figure 28. To explain what happened
during a flight to the Moon is also fun. A consistent hollow Earth view is fully equivalent
to the usual picture of an infinitely extended space. We will come back to this problem
Vol. II, page 285 in the section on general relativity.
* Roman Sexl, (1939–1986), important Austrian physicist, author of several influential textbooks on gravit-
ation and relativity.
2 from motion measurement to continuity 61

Curiosities and fun challenges ab ou t everyday space and time
How does one measure the speed of a gun bullet with a stopwatch, in a space of 1 m3 ,
Challenge 63 s without electronics? Hint: the same method can also be used to measure the speed of
light.
∗∗
For a striking and interactive way to zoom through all length scales in nature, from the
Planck length to the size of the universe, see the website www.htwins.net/scale2/.
∗∗
Challenge 64 s What is faster: an arrow or a motorbike?
∗∗
Challenge 65 s Why are manholes always round?
∗∗

Motion Mountain – The Adventure of Physics
Apart from the speed of light, another speed is important in nature: the speed of ear
Ref. 48 growth. It was determined in published studies which show that on average, for old men,
age 𝑡 and ear circumference 𝑒 are related by 𝑒 = 𝑡0.51 mm/a + 88.1 mm. In the units of
the expression, ‘a’, from Latin ‘annus’, is the international abbreviation for ‘year’.
∗∗
Can you show to a child that the sum of the angles in a triangle equals two right angles?
Challenge 66 e What about a triangle on a sphere or on a saddle?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Do you own a glass whose height is larger than its maximum circumference?
∗∗
A gardener wants to plant nine trees in such a way that they form ten straight lines with
Challenge 67 e three trees each. How does he do it?
∗∗
How fast does the grim reaper walk? This question is the title of a publication in the
Challenge 68 d British Medial Journal from the year 2011. Can you imagine how it is answered?
∗∗
Time measurements require periodic phenomena. Tree rings are traces of the seasons.
Glaciers also have such traces, the ogives. Similar traces are found in teeth. Do you know
more examples?
∗∗
A man wants to know how many stairs he would have to climb if the escalator in front of
him, which is running upwards, were standing still. He walks up the escalator and counts
60 stairs; walking down the same escalator with the same speed he counts 90 stairs. What
62 2 from motion measurement to continuity

F I G U R E 30 At what height is a conical glass half full?

Challenge 69 s is the answer?
∗∗

Motion Mountain – The Adventure of Physics
You have two hourglasses: one needs 4 minutes and one needs 3 minutes. How can you
Challenge 70 e use them to determine when 5 minutes are over?
∗∗
You have two fuses of different length that each take one minute to burn. You are not
allowed to bend them nor to use a ruler. How do you determine that 45 s have gone by?
Challenge 71 e Now the tougher puzzle: How do you determine that 10 s have gone by with a single fuse?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
You have three wine containers: a full one of 8 litres, an empty one of 5 litres, and another
Challenge 72 e empty one of 3 litres. How can you use them to divide the wine evenly into two?
∗∗
Challenge 73 s How can you make a hole in a postcard that allows you to step through it?
∗∗
What fraction of the height of a conical glass, shown in Figure 30, must be filled to make
Challenge 74 s the glass half full?
∗∗
Challenge 75 s How many pencils are needed to draw a line as long as the Equator of the Earth?
∗∗
Challenge 76 e Can you place five equal coins so that each one touches the other four? Is the stacking of
two layers of three coins, each layer in a triangle, a solution for six coins? Why?
What is the smallest number of coins that can be laid flat on a table so that every coin
Challenge 77 e is touching exactly three other coins?
∗∗
Challenge 78 e Can you find three crossing points on a chessboard that lie on an equilateral triangle?
2 from motion measurement to continuity 63

rubber band

F I G U R E 31 Can
the snail reach
the horse once it
starts galloping
away?

∗∗
The following bear puzzle is well known: A hunter leaves his home, walks 10 km to the
South and 10 km to the West, shoots a bear, walks 10 km to the North, and is back home.

Motion Mountain – The Adventure of Physics
What colour is the bear? You probably know the answer straight away. Now comes the
harder question, useful for winning money in bets. The house could be on several addi-
tional spots on the Earth; where are these less obvious spots from which a man can have
exactly the same trip (forget the bear now) that was just described and be at home again?
Challenge 79 s

∗∗
Imagine a rubber band that is attached to a wall on one end and is attached to a horse at
the other end, as shown in Figure 31. On the rubber band, near the wall, there is a snail.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Both the snail and the horse start moving, with typical speeds – with the rubber being
Challenge 80 s infinitely stretchable. Can the snail reach the horse?
∗∗
For a mathematician, 1 km is the same as 1000 m. For a physicist the two are different!
Indeed, for a physicist, 1 km is a measurement lying between 0.5 km and 1.5 km, whereas
1000 m is a measurement between 999.5 m and 1000.5 m. So be careful when you write
down measurement values. The professional way is to write, for example, 1000(8) m to
mean 1000 ± 8 m, i.e., a value that lies between 992 and 1008 m with a probability of
Page 459 68.3 %.
∗∗
Imagine a black spot on a white surface. What is the colour of the line separating the spot
Challenge 81 s from the background? This question is often called Peirce’s puzzle.
∗∗
Also bread is an (approximate) fractal, though an irregular one. The fractal dimension
Challenge 82 s of bread is around 2.7. Try to measure it!
∗∗
Challenge 83 e How do you find the centre of a beer mat using paper and pencil?
64 2 from motion measurement to continuity

∗∗
How often in 24 hours do the hour and minute hands of a clock lie on top of each other?
Challenge 84 s For clocks that also have a second hand, how often do all three hands lie on top of each
other?
∗∗
Challenge 85 s How often in 24 hours do the hour and minute hands of a clock form a right angle?
∗∗
How many times in twelve hours can the two hands of a clock be exchanged with the
Challenge 86 s result that the new situation shows a valid time? What happens for clocks that also have
a third hand for seconds?
∗∗
Challenge 87 s How many minutes does the Earth rotate in one minute?

Motion Mountain – The Adventure of Physics
∗∗
What is the highest speed achieved by throwing (with and without a racket)? What was
Challenge 88 s the projectile used?
∗∗
A rope is put around the Earth, on the Equator, as tightly as possible. The rope is
Challenge 89 s then lengthened by 1 m. Can a mouse slip under the rope? The original, tight rope is
lengthened by 1 mm. Can a child slip under the rope?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Jack was rowing his boat on a river. When he was under a bridge, he dropped a ball into
the river. Jack continued to row in the same direction for 10 minutes after he dropped
the ball. He then turned around and rowed back. When he reached the ball, the ball had
Challenge 90 s floated 600 m from the bridge. How fast was the river flowing?
∗∗
Adam and Bert are brothers. Adam is 18 years old. Bert is twice as old as at the time
Challenge 91 e when Adam was the age that Bert is now. How old is Bert?
∗∗
‘Where am I?’ is a common question; ‘When am I?’ is almost never asked, not even in
Challenge 92 s other languages. Why?
∗∗
Challenge 93 s Is there a smallest time interval in nature? A smallest distance?
∗∗
Given that you know what straightness is, how would you characterize or define the
Challenge 94 s curvature of a curved line using numbers? And that of a surface?
2 from motion measurement to continuity 65

1 5 10

F I G U R E 32 A 9-to-10 vernier/nonius/clavius and a 19-to-20 version (in fact, a 38-to-40 version) in a
caliper (© www.medien-werkstatt.de).

∗∗
Challenge 95 s What is the speed of your eyelid?
∗∗

Motion Mountain – The Adventure of Physics
The surface area of the human body is about 400 m2 . Can you say where this large num-
Challenge 96 s ber comes from?
∗∗
How does a vernier work? It is called nonius in other languages. The first name is derived
from a French military engineer* who did not invent it, the second is a play of words
on the Latinized name of the Portuguese inventor of a more elaborate device** and the
Latin word for ‘nine’. In fact, the device as we know it today – shown in Figure 32 –
was designed around 1600 by Christophonius Clavius,*** the same astronomer whose

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
studies were the basis of the Gregorian calendar reform of 1582. Are you able to design
a vernier/nonius/clavius that, instead of increasing the precision tenfold, does so by an
Challenge 97 s arbitrary factor? Is there a limit to the attainable precision?
∗∗
Page 55 Fractals in three dimensions bear many surprises. Let us generalize Figure 25 to three
dimensions. Take a regular tetrahedron; then glue on every one of its triangular faces a
smaller regular tetrahedron, so that the surface of the body is again made up of many
equal regular triangles. Repeat the process, glueing still smaller tetrahedrons to these
new (more numerous) triangular surfaces. What is the shape of the final fractal, after an
Challenge 98 s infinite number of steps?
∗∗
Motoring poses many mathematical problems. A central one is the following parallel
parking challenge: what is the shortest distance 𝑑 from the car in front necessary to leave
Challenge 99 s a parking spot without using reverse gear? (Assume that you know the geometry of your

* Pierre Vernier (1580–1637), French military officer interested in cartography.
** Pedro Nuñes or Peter Nonnius (1502–1578), Portuguese mathematician and cartographer.
*** Christophonius Clavius or Schlüssel (1537–1612), Bavarian astronomer, one of the main astronomers of
his time.
66 2 from motion measurement to continuity

𝑑 𝑏

𝑤

𝐿
F I G U R E 33 Leaving a
parking space.

TA B L E 12 The exponential notation: how to write small and large numbers.

Number Exponential Number Exponential
n o tat i o n n o tat i o n

1 100

Motion Mountain – The Adventure of Physics
0.1 10−1 10 101
0.2 2 ⋅ 10−1 20 2 ⋅ 101
0.0324 3.24 ⋅ 10−2 32.4 3.24 ⋅ 101
0.01 10−2 100 102
0.001 10−3 1000 103
0.000 1 10−4 10 000 104
0.000 056 5.6 ⋅ 10−5 56 000 5.6 ⋅ 104
0.000 01 10−5 etc. 100 000 105 etc.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
car, as shown in Figure 33, and its smallest outer turning radius 𝑅, which is known for
every car.) Next question: what is the smallest gap required when you are allowed to
Challenge 100 s manoeuvre back and forward as often as you like? Now a problem to which no solution
seems to be available in the literature: How does the gap depend on the number, 𝑛, of
Challenge 101 s times you use reverse gear? (The author had offered 50 euro for the first well-explained
solution; the winning solution by Daniel Hawkins is now found in the appendix.)
∗∗
Scientists use a special way to write large and small numbers, explained in Table 12.
∗∗
Ref. 49 In 1996 the smallest experimentally probed distance was 10−19 m, achieved between
quarks at Fermilab. (To savour the distance value, write it down without the exponent.)
Challenge 102 s What does this measurement mean for the continuity of space?
∗∗
Zeno, the Greek philosopher, discussed in detail what happens to a moving object at a
given instant of time. To discuss with him, you decide to build the fastest possible shutter
for a photographic camera that you can imagine. You have all the money you want. What
2 from motion measurement to continuity 67

𝑎
𝑟 𝐴
𝛼 Ω
𝑟
𝑎
𝛼= 𝑟
WF

F I G U R E 34 The deﬁnition of plane and solid angles.

is the shortest shutter time you would achieve?

Motion Mountain – The Adventure of Physics
Challenge 103 s

∗∗
Can you prove Pythagoras’ theorem by geometrical means alone, without using
Challenge 104 s coordinates? (There are more than 30 possibilities.)
∗∗
Page 59 Why are most planets and moons, including ours, (almost) spherical (see, for example,
Challenge 105 s Figure 28)?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
A rubber band connects the tips of the two hands of a clock. What is the path followed
Challenge 106 s by the mid-point of the band?
∗∗
There are two important quantities connected to angles. As shown in Figure 34, what is
usually called a (plane) angle is defined as the ratio between the lengths of the arc and
the radius. A right angle is π/2 radian (or π/2 rad) or 90°.
The solid angle is the ratio between area and the square of the radius. An eighth of a
sphere is π/2 steradian or π/2 sr. (Mathematicians, of course, would simply leave out the
steradian unit.) As a result, a small solid angle shaped like a cone and the angle of the
Challenge 107 s cone tip are different. Can you find the relationship?
∗∗
The definition of angle helps to determine the size of a firework display. Measure the time
𝑇, in seconds, between the moment that you see the rocket explode in the sky and the
moment you hear the explosion, measure the (plane) angle 𝛼 – pronounced ‘alpha’ – of
the ball formed by the firework with your hand. The diameter 𝐷 is

6m
𝐷≈ 𝑇𝛼 . (3)
𝑠°
68 2 from motion measurement to continuity

cosec cot cot
c
se
c co
cos se

tan tan
sin sin
angle angle
sec cos

circle of radius 1 circle of radius 1

F I G U R E 35 Two equivalent deﬁnitions of the sine, cosine, tangent, cotangent, secant and cosecant of
an angle.

Motion Mountain – The Adventure of Physics
sky yks

earth htrae

F I G U R E 36 How the apparent size of the Moon and the Sun changes during a day.

Challenge 108 e Why? For more information about fireworks, see the cc.oulu.fi/~kempmp website. By
the way, the angular distance between the knuckles of an extended fist are about 3°, 2°
and 3°, the size of an extended hand 20°. Can you determine the other angles related to
Challenge 109 s your hand?
∗∗
Using angles, the sine, cosine, tangent, cotangent, secant and cosecant can be defined, as
shown in Figure 35. You should remember this from secondary school. Can you confirm
Challenge 110 e that sin 15° = (√6 − √2 )/4, sin 18° = (−1 + √5 )/4, sin 36° = √10 − 2√5 /4, sin 54° =
2 from motion measurement to continuity 69

F I G U R E 37 How the size of the
Moon actually changes during
its orbit (© Anthony
Ayiomamitis).

Motion Mountain – The Adventure of Physics
(1 + √5 )/4 and that sin 72° = √10 + 2√5 /4? Can you show also that

sin 𝑥 𝑥 𝑥 𝑥
= cos cos cos ... (4)
𝑥 2 4 8

Challenge 111 e is correct?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Measuring angular size with the eye only is tricky. For example, can you say whether the
Moon is larger or smaller than the nail of your thumb at the end of your extended arm?
Challenge 112 e Angular size is not an intuitive quantity; it requires measurement instruments.
A famous example, shown in Figure 36, illustrates the difficulty of estimating angles.
Both the Sun and the Moon seem larger when they are on the horizon. In ancient times,
Ptolemy explained this so-called Moon illusion by an unconscious apparent distance
Ref. 50 change induced by the human brain. Indeed, the Moon illusion disappears when you
look at the Moon through your legs. In fact, the Moon is even further away from the
observer when it is just above the horizon, and thus its image is smaller than it was a few
Challenge 113 s hours earlier, when it was high in the sky. Can you confirm this?
The Moon’s angular size changes even more due to another effect: the orbit of the
Moon round the Earth is elliptical. An example of the consequence is shown in Figure 37.
∗∗
Galileo also made mistakes. In his famous book, the Dialogues, he says that the curve
formed by a thin chain hanging between two nails is a parabola, i.e., the curve defined
Challenge 114 d by 𝑦 = 𝑥2 . That is not correct. What is the correct curve? You can observe the shape
(approximately) in the shape of suspension bridges.
∗∗
Draw three circles, of different sizes, that touch each other, as shown in Figure 38. Now
70 2 from motion measurement to continuity

O
C
B

F I G U R E 38 A famous puzzle: how are the F I G U R E 39 What is the area ABC,
four radii related? given the other three areas and three
right angles at O?

Motion Mountain – The Adventure of Physics
wall
blue ladder
of length b
ladder of red ladder
length l of length r
wall wall
height
h?
F I G U R E 40
square Two ladder

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
box of puzzles: a
side b height h
moderately
difﬁcult
(left) and a
difﬁcult one
distance d ?
(right).

draw a fourth circle in the space between, touching the outer three. What simple relation
Challenge 115 s do the inverse radii of the four circles obey?
∗∗
Take a tetrahedron OABC whose triangular sides OAB, OBC and OAC are rectangular in
O, as shown in Figure 39. In other words, the edges OA, OB and OC are all perpendicular
to each other. In the tetrahedron, the areas of the triangles OAB, OBC and OAC are
Challenge 116 s respectively 8, 4 and 1. What is the area of triangle ABC?
∗∗
Ref. 51 There are many puzzles about ladders. Two are illustrated in Figure 40. If a 5 m ladder
is put against a wall in such a way that it just touches a box with 1 m height and depth,
Challenge 117 s how high does the ladder reach? If two ladders are put against two facing walls, and if
2 from motion measurement to continuity 71

F I G U R E 41 Anticrepuscular rays - where is
the Sun in this situation? (© Peggy Peterson)

the lengths of the ladders and the height of the crossing point are known, how distant
Challenge 118 d are the walls?

Motion Mountain – The Adventure of Physics
∗∗
With two rulers, you can add and subtract numbers by lying them side by side. Are you
Challenge 119 s able to design rulers that allow you to multiply and divide in the same manner?
∗∗
How many days would a year have if the Earth turned the other way with the same ro-
Challenge 120 s tation frequency?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 121 s The Sun is hidden in the spectacular situation shown in Figure 41 Where is it?
∗∗
A slightly different, but equally fascinating situation – and useful for getting used to per-
spective drawing – appears when you have a lighthouse in your back. Can you draw the
Challenge 122 e rays you see in the sky up to the horizon?
∗∗
Two cylinders of equal radius intersect at a right angle. What is the value of the intersec-
Challenge 123 s tion volume? (First make a drawing.)
∗∗
Two sides of a hollow cube with side length 1 dm are removed, to yield a tunnel with
square opening. Is it true that a cube with edge length 1.06 dm can be made to pass
Challenge 124 s through the hollow cube with side length 1 dm?
∗∗
Ref. 52 Could a two-dimensional universe exist? Alexander Dewdney imagined such a universe
in great detail and wrote a well-known book about it. He describes houses, the trans-
portation system, digestion, reproduction and much more. Can you explain why a two-
72 2 from motion measurement to continuity

F I G U R E 42 Ideal conﬁgurations of ropes made of two, three and four strands. In the ideal
conﬁguration, the speciﬁc pitch angle relative to the equatorial plane – 39.4°, 42.8° and 43.8°,
respectively – leads to zero-twist structures. In these ideal conﬁgurations, the rope will neither rotate in

Motion Mountain – The Adventure of Physics
one nor in the other direction under vertical strain (© Jakob Bohr).

Challenge 125 d dimensional universe is impossible?
∗∗
Ropes are wonderful structures. They are flexible, they are helically woven, but despite
this, they do not unwind or twist, they are almost inextensible, and their geometry de-
Ref. 53 pends little on the material used in making them. What is the origin of all these proper-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
ties?
Laying rope is an old craft; it is based on a purely geometric result: among all possible
helices of 𝑛 strands of given length laid around a central structure of fixed radius, there
is one helix for which the number of turns is maximal. For purely geometric reasons,
ropes with that specific number of turns and the corresponding inner radius have the
mentioned properties that make ropes so useful. The geometries of ideal ropes made of
two, three and four strands are shown in Figure 42.
∗∗
Challenge 126 s Some researchers are investigating whether time could be two-dimensional. Can this be?
∗∗
Other researchers are investigating whether space could have more than three dimen-
Challenge 127 s sions. Can this be?
∗∗
One way to compare speeds of animals and machines is to measure them in ‘body lengths
per second’. The click beetle achieves a value of around 2000 during its jump phase,
certain Archaea (bacteria-like) cells a value of 500, and certain hummingbirds 380. These
are the record-holders so far. Cars, aeroplanes, cheetahs, falcons, crabs, and all other
Ref. 54 motorized systems are much slower.
2 from motion measurement to continuity 73

F I G U R E 43 An open research problem: What
is the ropelength of a tight knot? (© Piotr
Pieranski, from Ref. 55)

∗∗
Challenge 128 e Why is the cross section of a tube usually circular?
∗∗
What are the dimensions of an open rectangular water tank that contains 1 m3 of water
Challenge 129 e and uses the smallest amount of wall material?

Motion Mountain – The Adventure of Physics
∗∗
Draw a square consisting of four equally long connecting line segments hinged at the
vertices. Such a structure may be freely deformed into a rhombus if some force is applied.
How many additional line interlinks of the same length must be supplemented to avoid
this freedom and to prevent the square from being deformed? The extra line interlinks
must be in the same plane as the square and each one may only be pegged to others at
Challenge 130 s the endpoints.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Area measurements can be difficult. In 2014 it became clear that the area of the gastro-
intestinal tract of adult health humans is between 30 and 40 m2 . For many years, the
mistaken estimate for the area was between 180 and 300 m2 .
∗∗
If you never explored plane geometry, do it once in your life. An excellent introduc-
tion is Claudi Alsina & Ro ger B. Nelsen, Icons of Mathematics: An Exploration
of Twenty Key Images, MAA Press, 2011. This is a wonderful book with many simple and
surprising facts about geometry that are never told in school or university. You will enjoy
it.
∗∗
Triangles are full of surprises. Together, Leonhard Euler, Charles Julien Brianchon and
Jean Victor Poncelet discovered that in any triangle, nine points lie on the same circle:
the midpoints of the sides, the feet of the altitude lines, and the midpoints of the altitude
segments connecting each vertex to the orthocenter. Euler also discovered that in every
triangle, the orthocenter, the centroid, the circumcenter and the center of the nine-point-
circle lie on the same line, now called the Euler line.
For the most recent recearch results on plane triangles, see the wonderful Encyclopedia
of Triangle Centers, available at faculty.evansville.edu/ck6/encyclopedia/ETC.html.
74 2 from motion measurement to continuity

∗∗
Here is a simple challenge on length that nobody has solved yet. Take a piece of ideal
rope: of constant radius, ideally flexible, and completely slippery. Tie a tight knot into it,
as shown in Figure 43. By how much did the two ends of the rope come closer together?
Challenge 131 r

Summary ab ou t everyday space and time
Motion defines speed, time and length. Observations of everyday life and precision ex-
periments are conveniently and precisely described by describing velocity as a vector,
space as three-dimensional set of points, and time as a one-dimensional real line, also
made of points. These three definitions form the everyday, or Galilean, description of our
environment.
Modelling velocity, time and space as continuous quantities is precise and convenient.
The modelling works during most of the adventures that follows. However, this common
model of space and time cannot be confirmed by experiment. For example, no experi-
ment can check distances larger than 1025 m or smaller than 10−25 m; the continuum

Motion Mountain – The Adventure of Physics
model is likely to be incorrect at smaller scales. We will find out in the last part of our
adventure that this is indeed the case.

HOW TO DE S C R I BE MOT ION
– K I N E M AT IC S

“
La filosofia è scritta in questo grandissimo libro
che continuamente ci sta aperto innanzi agli
occhi (io dico l’universo) ... Egli è scritto in

”
lingua matematica.**
Galileo Galilei, Il saggiatore VI.

Motion Mountain – The Adventure of Physics
E
xperiments show that the properties of motion, time and space are
xtracted from the environment both by children and animals. This
xtraction has been confirmed for cats, dogs, rats, mice, ants and fish, among
others. They all find the same results.
First of all, motion is change of position with time. This description is illustrated by rap-
idly flipping the lower left corners of this book, starting at page 242. Each page simulates
an instant of time, and the only change that takes place during motion is in the position
of the object, say a stone, represented by the dark spot. The other variations from one
picture to the next, which are due to the imperfections of printing techniques, can be
taken to simulate the inevitable measurement errors.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Stating that ‘motion is the change of position with time’ is neither an explanation nor
a definition, since both the concepts of time and position are deduced from motion itself.
It is only a description of motion. Still, the statement is useful, because it allows for high
precision, as we will find out by exploring gravitation and electrodynamics. After all, pre-
cision is our guiding principle during this promenade. Therefore the detailed description
of changes in position has a special name: it is called kinematics.
The idea of change of positions implies that the object can be followed during its mo-
tion. This is not obvious; in the section on quantum theory we will find examples where
this is impossible. But in everyday life, objects can always be tracked. The set of all pos-
itions taken by an object over time forms its path or trajectory. The origin of this concept
Ref. 56 is evident when one watches fireworks or again the flip film in the lower left corners
starting at page 242.
In everyday life, animals and humans agree on the Euclidean properties of velocity,
space and time. In particular, this implies that a trajectory can be described by specify-
ing three numbers, three coordinates (𝑥, 𝑦, 𝑧) – one for each dimension – as continuous

** Science is written in this huge book that is continuously open before our eyes (I mean the universe) ... It
is written in mathematical language.
76 3 how to describe motion – kinematics

collision

F I G U R E 44 Two ways to test that the time of free fall does not depend on horizontal velocity.

Vol. III, page 289 functions of time 𝑡. (Functions are defined in detail later on.) This is usually written as

Motion Mountain – The Adventure of Physics
𝑥 = 𝑥(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) . (5)

For example, already Galileo found, using stopwatch and ruler, that the height 𝑧 of any
thrown or falling stone changes as

𝑧(𝑡) = 𝑧0 + 𝑣𝑧0 (𝑡 − 𝑡0 ) − 12 𝑔 (𝑡 − 𝑡0 )2 (6)

where 𝑡0 is the time the fall starts, 𝑧0 is the initial height, 𝑣𝑧0 is the initial velocity in the
vertical direction and 𝑔 = 9.8 m/s2 is a constant that is found to be the same, within

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
about one part in 300, for all falling bodies on all points of the surface of the Earth.
Ref. 57 Where do the value 9.8 m/s2 and its slight variations come from? A preliminary answer
will be given shortly, but the complete elucidation will occupy us during the larger part
of this hike.
The special case with no initial velocity is of great interest. Like a few people before
him, Galileo made it clear that 𝑔 is the same for all bodies, if air resistance can be neg-
Page 202 lected. He had many arguments for this conclusion; can you find one? And of course,
his famous experiment at the leaning tower in Pisa confirmed the statement. (It is a false
Ref. 58 urban legend that Galileo never performed the experiment. He did it.)
Equation (6) therefore allows us to determine the depth of a well, given the time a
Challenge 132 s stone takes to reach its bottom. The equation also gives the speed 𝑣 with which one hits
the ground after jumping from a tree, namely

𝑣 = √2𝑔ℎ . (7)

A height of 3 m yields a velocity of 27 km/h. The velocity is thus proportional only to the
square root of the height. Does this mean that one’s strong fear of falling results from an
Challenge 133 s overestimation of its actual effects?
Galileo was the first to state an important result about free fall: the motions in the
horizontal and vertical directions are independent. He showed that the time it takes for
3 how to describe motion – kinematics 77

space-time configuration hodograph phase space
diagrams space graph

𝑧 𝑧 𝑣𝑧 𝑚𝑣𝑧

𝑡 𝑥 v𝑥 𝑧
𝑥 𝑚𝑣𝑥

𝑡 𝑥

Motion Mountain – The Adventure of Physics
F I G U R E 45 Various types of graphs describing the same path of a thrown stone.

a cannon ball that is shot exactly horizontally to fall is independent of the strength of the
gunpowder, as shown in Figure 44. Many great thinkers did not agree with this statement
Ref. 59 even after his death: in 1658 the Academia del Cimento even organized an experiment
to check this assertion, by comparing the flying cannon ball with one that simply fell
Challenge 134 s vertically. Can you imagine how they checked the simultaneity? Figure 44 shows how
you can check this at home. In this experiment, whatever the spring load of the cannon,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the two bodies will always collide in mid-air (if the table is high enough), thus proving
the assertion.
In other words, a flying cannon ball is not accelerated in the horizontal direction.
Its horizontal motion is simply unchanging – as long as air resistance is negligible. By
extending the description of equation (6) with the two expressions for the horizontal
coordinates 𝑥 and 𝑦, namely

𝑥(𝑡) = 𝑥0 + 𝑣x0 (𝑡 − 𝑡0 )
𝑦(𝑡) = 𝑦0 + 𝑣y0 (𝑡 − 𝑡0 ) , (8)

a complete description for the path followed by thrown stones results. A path of this shape
Page 40 is called a parabola; it is shown in Figures 18, 44 and 45. (A parabolic shape is also used
Challenge 135 s for light reflectors inside pocket lamps or car headlights. Can you show why?)
Ref. 60 Physicists enjoy generalizing the idea of a path. As Figure 45 shows, a path is a trace
left in a diagram by a moving object. Depending on what diagram is used, these paths
have different names. Space-time diagrams are useful to make the theory of relativity ac-
cessible. The configuration space is spanned by the coordinates of all particles of a system.
For many particles, it has a high number of dimensions and plays an important role in
Page 415 self-organization. The difference between chaos and order can be described as a differ-
ence in the properties of paths in configuration space. Hodographs, the paths in ‘velocity
78 3 how to describe motion – kinematics

space’, are used in weather forecasting. The phase space diagram is also called state space
diagram. It plays an essential role in thermodynamics.

Throwing , jumping and sho oting
The kinematic description of motion is useful for answering a whole range of questions.
∗∗
What is the upper limit for the long jump? The running peak speed world record
Ref. 61 in 2019 was over 12.5 m/s ≈ 45 km/h by Usain Bolt, and the 1997 women’s record
Ref. 62 was 11 m/s ≈ 40 km/h. However, male long jumpers never run much faster than about
9.5 m/s. How much extra jump distance could they achieve if they could run full speed?
How could they achieve that? In addition, long jumpers take off at angles of about 20°,
Ref. 63 as they are not able to achieve a higher angle at the speed they are running. How much
Challenge 136 s would they gain if they could achieve 45°? Is 45° the optimal angle?
∗∗

Motion Mountain – The Adventure of Physics
Why was basketball player Dirk Nowitzki so successful? His trainer Holger Geschwind-
ner explained him that a throw is most stable against mistakes when it falls into the
basket at around 47 degrees from the horizontal. He further told Nowitzki that the ball
flies in a plane, and that therefore the arms should also move in that plane only. And he
explained that when the ball leaves the hand, it should roll over the last two fingers like
a train moves on rails. Using these criteria to check and to improve Nowitzki’s throws,
he made him into one of the best basket ball throwers in the world.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
What do the athletes Usain Bolt and Michael Johnson, the last two world record holders
on the 200 m race at time of this writing, have in common? They were tall, athletic, and
had many fast twitch fibres in the muscles. These properties made them good sprinters.
A last difference made them world-class sprinters: they had a flattened spine, with almost
no S-shape. This abnormal condition saves them a little bit of time at every step, because
their spine is not as flexible as in usual people. This allows them to excel at short-distance
races.
∗∗
Athletes continuously improve speed records. Racing horses do not. Why? For racing
horses, breathing rhythm is related to gait; for humans, it is not. As a result, racing horses
cannot change or improve their technique, and the speed of racing horses is essentially
the same since it is measured.
∗∗
What is the highest height achieved by a human throw of any object? What is the longest
Challenge 137 s distance achieved by a human throw? How would you clarify the rules? Compare the
results with the record distance with a crossbow, 1, 871.8 m, achieved in 1988 by Harry
Drake, the record distance with a footbow, 1854.4 m, achieved in 1971 also by Harry
Drake, and with a hand-held bow, 1, 222.0 m, achieved in 1987 by Don Brown.
3 how to describe motion – kinematics 79

F I G U R E 46 Three superimposed
images of a frass pellet shot away by a
caterpillar inside a rolled-up leaf
(© Stanley Caveney).

Motion Mountain – The Adventure of Physics
∗∗
Challenge 138 s How can the speed of falling rain be measured using an umbrella? The answer is import-
ant: the same method can also be used to measure the speed of light, as we will find out
Vol. II, page 17 later. (Can you guess how?)
∗∗
When a dancer jumps in the air, how many times can he or she rotate around his or her
Challenge 139 s vertical axis before arriving back on earth?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Numerous species of moth and butterfly caterpillars shoot away their frass – to put it
Ref. 64 more crudely: their faeces – so that its smell does not help predators to locate them. Stan-
ley Caveney and his team took photographs of this process. Figure 46 shows a caterpillar
(yellow) of the skipper Calpodes ethlius inside a rolled up green leaf caught in the act.
Given that the record distance observed is 1.5 m (though by another species, Epargyreus
Challenge 140 s clarus), what is the ejection speed? How do caterpillars achieve it?
∗∗
What is the horizontal distance one can reach by throwing a stone, given the speed and
Challenge 141 s the angle from the horizontal at which it is thrown?
∗∗
Challenge 142 s What is the maximum numbers of balls that could be juggled at the same time?
∗∗
Is it true that rain drops would kill if it weren’t for the air resistance of the atmosphere?
Challenge 143 s What about hail?
∗∗
80 3 how to describe motion – kinematics

0.001 0.01 0.1 1 10

antilope
cat leopard tiger
20 W/kg
lesser dog human horse
1
galago
Height of jump [m]

locusts and
grasshoppers
0.1
fleas
standing jumps running jumps

0.01 elephant

0.001 0.01 0.1 1 10
Length of animal [m]

Motion Mountain – The Adventure of Physics
F I G U R E 47 The height achieved by jumping land animals.

Challenge 144 s Are bullets, fired into the air from a gun, dangerous when they fall back down?
∗∗
Police finds a dead human body at the bottom of cliff with a height of 30 m, at a distance
Challenge 145 s of 12 m from the cliff. Was it suicide or murder?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Ref. 65 All land animals, regardless of their size, achieve jumping heights of at most 2.2 m, as
Challenge 146 s shown in Figure 47. The explanation of this fact takes only two lines. Can you find it?
The last two issues arise because the equation (6) describing free fall does not hold in all
cases. For example, leaves or potato crisps do not follow it. As Galileo already knew, this
is a consequence of air resistance; we will discuss it shortly. Because of air resistance, the
path of a stone is not a parabola.
In fact, there are other situations where the path of a falling stone is not a parabola,
Challenge 147 s even without air resistance. Can you find one?

Enjoying vectors
Physical quantities with a defined direction, such as speed, are described with three num-
bers, or three components, and are called vectors. Learning to calculate with such multi-
component quantities is an important ability for many sciences. Here is a summary.
Vectors can be pictured by small arrows. Note that vectors do not have specified points
at which they start: two arrows with same direction and the same length are the same
vector, even if they start at different points in space. Since vectors behave like arrows,
vectors can be added and they can be multiplied by numbers. For example, stretching an
arrow 𝑎 = (𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧 ) by a number 𝑐 corresponds, in component notation, to the vector
𝑐𝑎 = (𝑐𝑎𝑥 , 𝑐𝑎𝑦 , 𝑐𝑎𝑧 ).
3 how to describe motion – kinematics 81

In precise, mathematical language, a vector is an element of a set, called vector space,
in which the following properties hold for all vectors 𝑎 and 𝑏 and for all numbers 𝑐 and 𝑑:

𝑐(𝑎 + 𝑏) = 𝑐𝑎 + 𝑐𝑏 , (𝑐 + 𝑑)𝑎 = 𝑐𝑎 + 𝑑𝑎 , (𝑐𝑑)𝑎 = 𝑐(𝑑𝑎) and 1𝑎 = 𝑎 . (9)

Examples of vector spaces are the set of all positions of an object, or the set of all its
Challenge 148 s possible velocities. Does the set of all rotations form a vector space?
All vector spaces allow defining a unique null vector and a unique negative vector for
Challenge 149 e each vector.
In most vector spaces of importance when describing nature the concept of length –
specifying the ‘magnitude’ – of a vector can be introduced. This is done via an inter-
mediate step, namely the introduction of the scalar product of two vectors. The product
is called ‘scalar’ because its result is a scalar; a scalar is a number that is the same for
all observers; for example, it is the same for observers with different orientations.* The
scalar product between two vectors 𝑎 and 𝑏 is a number that satisfies

Motion Mountain – The Adventure of Physics
𝑎𝑎 ⩾ 0 ,
𝑎𝑏 = 𝑏𝑎 ,
(𝑎 + 𝑎󸀠 )𝑏 = 𝑎𝑏 + 𝑎󸀠 𝑏 , (10)
󸀠 󸀠
𝑎(𝑏 + 𝑏 ) = 𝑎𝑏 + 𝑎𝑏 and
(𝑐𝑎)𝑏 = 𝑎(𝑐𝑏) = 𝑐(𝑎𝑏) .

This definition of a scalar product is not unique; however, there is a standard scalar
product. In Cartesian coordinate notation, the standard scalar product is given by

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
𝑎𝑏 = 𝑎𝑥 𝑏𝑥 + 𝑎𝑦 𝑏𝑦 + 𝑎𝑧 𝑏𝑧 . (11)

If the scalar product of two vectors vanishes the two vectors are orthogonal, at a right
Challenge 150 e angle to each other. (Show it!) Note that one can write either 𝑎𝑏 or 𝑎 ⋅ 𝑏 with a central
dot.
The length or magnitude or norm of a vector can then be defined as the square root of
the scalar product of a vector with itself: 𝑎 = √𝑎𝑎 . Often, and also in this text, lengths
are written in italic letters, whereas vectors are written in bold letters. The magnitude is
often written as 𝑎 = √𝑎2 . A vector space with a scalar product is called an Euclidean
vector space.
The scalar product is especially useful for specifying directions. Indeed, the scalar
product between two vectors encodes the angle between them. Can you deduce this im-
Challenge 151 s portant relation?

* We mention that in mathematics, a scalar is a number; in physics, a scalar is an invariant number, i.e.,
a number that is the same for all observers. Likewise, in mathematics, a vector is an element of a vector
space; in physics, a vector is an invariant element of a vector space, i.e., a quantity whose coordinates, when
observed by different observers, change like the components of velocity.
82 3 how to describe motion – kinematics

𝑦 derivative slope: d𝑦/d𝑡

secant slope: Δ𝑦/Δ𝑡

Δ𝑡

Δ𝑦
𝑡
F I G U R E 48 The derivative in a point as the limit of secants.

What is rest? What is velo cit y?
In the Galilean description of nature, motion and rest are opposites. In other words, a
body is at rest when its position, i.e., its coordinates, do not change with time. In other

Motion Mountain – The Adventure of Physics
words, (Galilean) rest is defined as

𝑥(𝑡) = const . (12)

We recall that 𝑥(𝑡) is the abbreviation for the three coordinates (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)). Later
we will see that this definition of rest, contrary to first impressions, is not much use and
will have to be expanded. Nevertheless, any definition of rest implies that non-resting
objects can be distinguished by comparing the rapidity of their displacement. Thus we
can define the velocity 𝑣 of an object as the change of its position 𝑥 with time 𝑡. This is
usually written as

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
d𝑥
𝑣= . (13)
d𝑡
In this expression, valid for each coordinate separately, d/d𝑡 means ‘change with time’.
We can thus say that velocity is the derivative of position with respect to time. The speed 𝑣
is the name given to the magnitude of the velocity 𝑣. Thus we have 𝑣 = √𝑣𝑣 . Derivatives
are written as fractions in order to remind the reader that they are derived from the idea
of slope. The expression

d𝑠 Δ𝑠
is meant as an abbreviation of lim , (14)
d𝑡 Δ𝑡→0 Δ𝑡

a shorthand for saying that the derivative at a point is the limit of the secant slopes in the
neighbourhood of the point, as shown in Figure 48. This definition implies the working
Challenge 152 e rules

d(𝑠 + 𝑟) d𝑠 d𝑟 d(𝑐𝑠) d𝑠 d d𝑠 d2 𝑠 d(𝑠𝑟) d𝑠 d𝑟
= + , =𝑐 , = , = 𝑟 + 𝑠 , (15)
d𝑡 d𝑡 d𝑡 d𝑡 d𝑡 d𝑡 d𝑡 d𝑡2 d𝑡 d𝑡 d𝑡
𝑐 being any number. This is all one ever needs to know about derivatives in physics.
Quantities such as d𝑡 and d𝑠, sometimes useful by themselves, are called differentials.
3 how to describe motion – kinematics 83

F I G U R E 49 Gottfried Wilhelm Leibniz (1646–1716).

These concepts are due to Gottfried Wilhelm Leibniz.* Derivatives lie at the basis of all
calculations based on the continuity of space and time. Leibniz was the person who made
it possible to describe and use velocity in physical formulae and, in particular, to use the
idea of velocity at a given point in time or space for calculations.
The definition of velocity assumes that it makes sense to take the limit Δ𝑡 → 0. In
other words, it is assumed that infinitely small time intervals do exist in nature. The

Motion Mountain – The Adventure of Physics
definition of velocity with derivatives is possible only because both space and time are
described by sets which are continuous, or in mathematical language, connected and com-
plete. In the rest of our walk we shall not forget that from the beginning of classical phys-
ics, infinities are present in its description of nature. The infinitely small is part of our
definition of velocity. Indeed, differential calculus can be defined as the study of infin-
ity and its uses. We thus discover that the appearance of infinity does not automatically
render a description impossible or imprecise. In order to remain precise, physicists use
only the smallest two of the various possible types of infinities. Their precise definition
Vol. III, page 288 and an overview of other types are introduced later on.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The appearance of infinity in the usual description of motion was first criticized in
Ref. 66 his famous ironical arguments by Zeno of Elea (around 445 b ce), a disciple of Parmen-
ides. In his so-called third argument, Zeno explains that since at every instant a given
object occupies a part of space corresponding to its size, the notion of velocity at a given
instant makes no sense; he provokingly concludes that therefore motion does not ex-
ist. Nowadays we would not call this an argument against the existence of motion, but
against its usual description, in particular against the use of infinitely divisible space and
Challenge 153 e time. (Do you agree?) Nevertheless, the description criticized by Zeno actually works
quite well in everyday life. The reason is simple but deep: in daily life, changes are indeed
continuous.
Large changes in nature are made up of many small changes. This property of nature is
not obvious. For example, we note that we have (again) tacitly assumed that the path of an
object is not a fractal or some other badly behaved entity. In everyday life this is correct;
in other domains of nature it is not. The doubts of Zeno will be partly rehabilitated later
Vol. VI, page 65 in our walk, and increasingly so the more we proceed. The rehabilitation is only partial,
as the final solution will be different from that which he envisaged; on the other hand,
* Gottfried Wilhelm Leibniz (b. 1646 Leipzig, d. 1716 Hannover), lawyer, physicist, mathematician, philo-
sopher, diplomat and historian. He was one of the great minds of mankind; he invented the differential
calculus (before Newton) and published many influential and successful books in the various fields he ex-
plored, among them De arte combinatoria, Hypothesis physica nova, Discours de métaphysique, Nouveaux
essais sur l’entendement humain, the Théodicée and the Monadologia.
84 3 how to describe motion – kinematics

TA B L E 13 Some measured acceleration values.

O b s e r va t i o n A c c e l e r at i o n

What is the lowest you can find? Challenge 154 s
Back-acceleration of the galaxy M82 by its ejected jet 10 f m/s2
Acceleration of a young star by an ejected jet 10 pm/s2
Fathoumi Acceleration of the Sun in its orbit around the Milky Way 0.2 nm/s2
Deceleration of the Pioneer satellites, due to heat radiation imbalance 0.8 nm/s2
Centrifugal acceleration at Equator due to Earth’s rotation 33 mm/s2
Electron acceleration in household electricity wire due to alternating 50 mm/s2
current
Acceleration of fast underground train 1.3 m/s2
Gravitational acceleration on the Moon 1.6 m/s2
Minimum deceleration of a car, by law, on modern dry asphalt 5.5 m/s2
Gravitational acceleration on the Earth’s surface, depending on 9.8 ± 0.3 m/s2

Motion Mountain – The Adventure of Physics
location
Standard gravitational acceleration 9.806 65 m/s2
Highest acceleration for a car or motorbike with engine-driven wheels 15 m/s2
Space rockets at take-off 20 to 90 m/s2
Acceleration of cheetah 32 m/s2
Gravitational acceleration on Jupiter’s surface 25 m/s2
Flying fly (Musca domestica) c. 100 m/s2
Acceleration of thrown stone c. 120 m/s2
Acceleration that triggers air bags in cars 360 m/s2

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Fastest leg-powered acceleration (by the froghopper, Philaenus 4 km/s2
spumarius, an insect)
Tennis ball against wall 0.1 Mm/s2
Bullet acceleration in rifle 2 Mm/s2
Fastest centrifuges 0.1 Gm/s2
Acceleration of protons in large accelerator 90 Tm/s2
Acceleration of protons inside nucleus 1031 m/s2
Highest possible acceleration in nature 1052 m/s2

the doubts about the idea of ‘velocity at a point’ will turn out to be well-founded. For
the moment though, we have no choice: we continue with the basic assumption that in
nature changes happen smoothly.
Why is velocity necessary as a concept? Aiming for precision in the description of
motion, we need to find the complete list of aspects necessary to specify the state of an
object. The concept of velocity is obviously on this list.
3 how to describe motion – kinematics 85

Acceleration
Continuing along the same line, we call acceleration 𝑎 of a body the change of velocity 𝑣
with time, or
d𝑣 d2 𝑥
𝑎= = 2 . (16)
d𝑡 d𝑡
Acceleration is what we feel when the Earth trembles, an aeroplane takes off, or a bicycle
goes round a corner. More examples are given in Table 13. Acceleration is the time de-
rivative of velocity. Like velocity, acceleration has both a magnitude and a direction. In
short, acceleration, like velocity, is a vector quantity. As usual, this property is indicated
by the use of a bold letter for its abbreviation.
In a usual car, or on a motorbike, we can feel being accelerated. (These accelerations
are below 1𝑔 and are therefore harmless.) We feel acceleration because some part inside
us is moved against some other part: acceleration deforms us. Such a moving part can
be, for example, some small part inside our ear, our stomach inside the belly, or simply
our limbs against our trunk. All acceleration sensors, including those listed in Table 14

Motion Mountain – The Adventure of Physics
or those shown in Figure 50, whether biological or technical, work in this way.
Acceleration is felt. Our body is deformed and the sensors in our body detect it, for ex-
ample in amusement parks. Higher accelerations can have stronger effects. For example,
when accelerating a sitting person in the direction of the head at two or three times the
value of usual gravitational acceleration, eyes stop working and the sight is greyed out,
because the blood cannot reach the eye any more. Between 3 and 5𝑔 of continuous accele-
Ref. 67 ration, or 7 to 9𝑔 of short time acceleration, consciousness is lost, because the brain does
not receive enough blood, and blood may leak out of the feet or lower legs. High acce-
leration in the direction of the feet of a sitting person can lead to haemorrhagic strokes

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in the brain. The people most at risk are jet pilots; they have special clothes that send
compressed air onto the pilot’s bodies to avoid blood accumulating in the wrong places.
Challenge 155 s Can you think of a situation where you are accelerated but do not feel it?
Velocity is the time derivative of position. Acceleration is the second time derivative
of position. Higher derivatives than acceleration can also be defined, in the same manner.
Challenge 156 s They add little to the description of nature, because – as we will show shortly – neither
these higher derivatives nor even acceleration itself are useful for the description of the
state of motion of a system.

From objects to point particles

“
Wenn ich den Gegenstand kenne, so kenne ich
auch sämtliche Möglichkeiten seines

”
Vorkommens in Sachverhalten.*
Ludwig Wittgenstein, Tractatus, 2.0123

One aim of the study of motion is to find a complete and precise description of both
states and objects. With the help of the concept of space, the description of objects can be
refined considerably. In particular, we know from experience that all objects seen in daily
Challenge 157 e life have an important property: they can be divided into parts. Often this observation is

* ‘If I know an object, then I also know all the possibilities of its occurrence in atomic facts.’
86 3 how to describe motion – kinematics

TA B L E 14 Some acceleration sensors.

Measurement Sensor Range

Direction of gravity in plants statoliths in cells 0 to 10 m/s2
(roots, trunk, branches, leaves)
Direction and value of the utricle and saccule in the inner 0 to 20 m/s2
accelerations in mammals ear (detecting linear accelerations),
and the membranes in each
semicircular canal (detecting
rotational accelerations)
Direction and value of acceleration piezoelectric sensors 0 to 20 m/s2
in modern step counters for hikers
Direction and value of acceleration airbag sensor using piezoelectric 0 to 2000 m/s2
in car crashes ceramics

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 50 Three accelerometers: a one-axis piezoelectric airbag sensor, a three-axis capacitive
accelerometer, and the utricle and saccule near the three semicircular canals inside the human ear
(© Bosch, Rieker Electronics, Northwestern University).

expressed by saying that all objects, or bodies, have two properties. First, they are made
out of matter,* defined as that aspect of an object responsible for its impenetrability,
i.e., the property preventing two objects from being in the same place. Secondly, bodies
Ref. 68 * Matter is a word derived from the Latin ‘materia’, which originally meant ‘wood’ and was derived via
intermediate steps from ‘mater’, meaning ‘mother’.
3 how to describe motion – kinematics 87

α γ
Betelgeuse
Bellatrix

ε δ Mintaka
ζ Alnilam
Alnitak

β
κ
Rigel
Saiph
F I G U R E 51 Orion in natural colours (© Matthew Spinelli) and Betelgeuse (ESA, NASA).

have a certain form or shape, defined as the precise way in which this impenetrability is

Motion Mountain – The Adventure of Physics
distributed in space.
In order to describe motion as accurately as possible, it is convenient to start with
those bodies that are as simple as possible. In general, the smaller a body, the simpler
it is. A body that is so small that its parts no longer need to be taken into account is
called a particle. (The older term corpuscle has fallen out of fashion.) Particles are thus
idealized small stones. The extreme case, a particle whose size is negligible compared
with the dimensions of its motion, so that its position is described completely by a single
triplet of coordinates, is called a point particle or a point mass or a mass point. In equation
(6), the stone was assumed to be such a point particle.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Do point-like objects, i.e., objects smaller than anything one can measure, exist in
daily life? Yes and no. The most notable examples are the stars. At present, angular sizes
as small as 2 μrad can be measured, a limit given by the fluctuations of the air in the
atmosphere. In space, such as for the Hubble telescope orbiting the Earth, the angular
limit is due to the diameter of the telescope and is of the order of 10 nrad. Practically
all stars seen from Earth are smaller than that, and are thus effectively ‘point-like’, even
when seen with the most powerful telescopes.
As an exception to the general rule, the size of a few large or nearby stars, mostly of
red giant type, can be measured with special instruments.* Betelgeuse, the higher of the
two shoulders of Orion shown in Figure 51, Mira in Cetus, Antares in Scorpio, Aldebaran
in Taurus and Sirius in Canis Major are examples of stars whose size has been measured;
Ref. 69 they are all less than two thousand light years from Earth. For a comparison of dimen-
sions, see Figure 52. Of course, like the Sun, also all other stars have a finite size, but one
Challenge 158 s cannot prove this by measuring their dimension in photographs. (True?)

* The website stars.astro.illinois.edu/sow/sowlist.html gives an introduction to the different types of stars.
The www.astro.wisc.edu/~dolan/constellations website provides detailed and interesting information about
constellations.
For an overview of the planets, see the beautiful book by Kenneth R. L ang &
Charles A. Whitney, Vagabonds de l’espace – Exploration et découverte dans le système solaire,
Springer Verlag, 1993. Amazingly beautiful pictures of the stars can be found in David Malin, A View
of the Universe, Sky Publishing and Cambridge University Press, 1993.
88 3 how to describe motion – kinematics

Motion Mountain – The Adventure of Physics
F I G U R E 52 A comparison of star sizes (© Dave Jarvis).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 53 Regulus and Mars,
photographed with an exposure
time of 10 s on 4 June 2010 with
a wobbling camera, show the
difference between a point-like
star that twinkles and an
extended planet that does not
(© Jürgen Michelberger).

The difference between ‘point-like’ and finite-size sources can be seen with the naked
Challenge 159 e eye: at night, stars twinkle, but planets do not. (Check it!) A beautiful visualization is
shown in Figure 53. This effect is due to the turbulence of air. Turbulence has an effect on
the almost point-like stars because it deflects light rays by small amounts. On the other
hand, air turbulence is too weak to lead to the twinkling of sources of larger angular size,
such as planets or artificial satellites,* because the deflection is averaged out in this case.

* A satellite is an object circling a planet, like the Moon; an artificial satellite is a system put into orbit by
humans, like the Sputniks.
3 how to describe motion – kinematics 89

An object is point-like for the naked eye if its angular size is smaller than about
Challenge 160 s 2 󸀠 = 0.6 mrad. Can you estimate the size of a ‘point-like’ dust particle? By the way, an
object is invisible to the naked eye if it is point-like and if its luminosity, i.e., the intensity
of the light from the object reaching the eye, is below some critical value. Can you esti-
mate whether there are any man-made objects visible from the Moon, or from the space
Challenge 161 s shuttle?
The above definition of ‘point-like’ in everyday life is obviously misleading. Do
proper, real point particles exist? In fact, is it at all possible to show that a particle has a
vanishing size? In the same way, we need to ask and check whether points in space do
exist. Our walk will lead us to the astonishing result that all the answers to these ques-
Challenge 162 s tions are negative. Can you imagine why? Do not be disappointed if you find this issue
difficult; many brilliant minds have had the same problem.
However, many particles, such as electrons, quarks or photons are point-like for all
practical purposes. Once we know how to describe the motion of point particles, we can
also describe the motion of extended bodies, rigid or deformable: we assume that they
are made of parts. This is the same approach as describing the motion of an animal as a

Motion Mountain – The Adventure of Physics
whole by combining the motion of its various body parts. The simplest description, the
continuum approximation, describes extended bodies as an infinite collection of point
particles. It allows us to understand and to predict the motion of milk and honey, the
motion of the air in hurricanes and of perfume in rooms. The motion of fire and all
other gaseous bodies, the bending of bamboo in the wind, the shape changes of chewing
Ref. 70 gum, and the growth of plants and animals can also be described in this way.
All observations so far have confirmed that the motion of large bodies can be de-
scribed to full precision as the result of the motion of their parts. All machines that hu-
mans ever built are based on this idea. A description that is even more precise than the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Vol. IV, page 15 continuum approximation is given later on. Describing body motion with the motion
of body parts will guide us through the first five volumes of our mountain ascent; for
example, we will understand life in this way. Only in the final volume will we discover
that, at a fundamental scale, this decomposition is impossible.

Legs and wheels
The parts of a body determine its shape. Shape is an important aspect of bodies: among
other things, it tells us how to count them. In particular, living beings are always made of
a single body. This is not an empty statement: from this fact we can deduce that animals
cannot have large wheels or large propellers, but only legs, fins, or wings. Why?
Living beings have only one surface; simply put, they have only one piece of skin.
Vol. V, page 365 Mathematically speaking, animals are connected. This is often assumed to be obvious, and
Ref. 71 it is often mentioned that the blood supply, the nerves and the lymphatic connections to a
rotating part would get tangled up. However, this argument is not so simple, as Figure 54
shows. The figure proves that it is indeed possible to rotate a body continuously against
a second one, without tangling up the connections. Three dimensions of space allow
tethered rotation. Can you find an example for this kind of motion, often called tethered
Challenge 163 s rotation, in your own body? Are you able to see how many cables may be attached to the
Challenge 164 s rotating body of the figure without hindering the rotation?
Despite the possibility of animals having rotating parts, the method of Figure 54 or
90 3 how to describe motion – kinematics

F I G U R E 54 Tethered rotation: How an object can rotate continuously without tangling up the
connection to a second object.

Motion Mountain – The Adventure of Physics
F I G U R E 55 Tethered rotation:

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the continuous rotation of an
object attached to its
environment (QuickTime ﬁlm
© Jason Hise).

Figure 55 still cannot be used to make a practical wheel or propeller. Can you see why?
Challenge 165 s Therefore, evolution had no choice: it had to avoid animals with (large) parts rotating
around axles. That is the reason that propellers and wheels do not exist in nature. Of
course, this limitation does not rule out that living bodies move by rotation as a whole:
Ref. 72 tumbleweed, seeds from various trees, some insects, several spiders, certain other anim-
als, children and dancers occasionally move by rolling or rotating as a whole.
Large single bodies, and thus all large living beings, can thus only move through de-
Ref. 73 formation of their shape: therefore they are limited to walking, running, jumping, rolling,
gliding, crawling, flapping fins, or flapping wings. Moving a leg is a common way to de-
form a body.
Ref. 74 Extreme examples of leg use in nature are shown in Figure 56 and Figure 57. The most
extreme example of rolling spiders – there are several species – are Cebrennus villosus and
Ref. 75 live in the sand in Morocco. They use their legs to accelerate the rolling, they can steer
the rolling direction and can even roll uphill slopes of 30 % – a feat that humans are
3 how to describe motion – kinematics 91

Motion Mountain – The Adventure of Physics
50 μm

F I G U R E 56 Legs and ‘wheels’ in living beings: the red millipede Aphistogoniulus erythrocephalus (15 cm
body length), a gecko on a glass pane (15 cm body length), an amoeba Amoeba proteus (1 mm size), the
rolling shrimp Nannosquilla decemspinosa (2 cm body length, 1.5 rotations per second, up to 2 m, can
even roll slightly uphill slopes) and the rolling caterpillar Pleurotya ruralis (can only roll downhill, to
escape predators), (© David Parks, Marcel Berendsen, Antonio Guillén Oterino, Robert Full, John

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Brackenbury / Science Photo Library ).

F I G U R E 57 Two of the rare lifeforms that are able to roll uphill also on steep slopes: the desert spider
Cebrennus villosus and Homo sapiens (© Ingo Rechenberg, Karva Javi).
92 3 how to describe motion – kinematics

unable to perform. Films of the rolling motion can be found at www.bionik.tu-berlin.
Vol. V, page 281 de.* Walking on water is shown in Figure 127 on page 170; examples of wings are given
Vol. V, page 282 later on, as are the various types of deformations that allow swimming in water.
In contrast, systems of several bodies, such as bicycles, pedal boats or other machines,
can move without any change of shape of their components, thus enabling the use of
axles with wheels, propellers and other rotating devices.**
In short, whenever we observe a construction in which some part is turning continu-
ously (and without the ‘wiring’ of Figure 54) we know immediately that it is an artefact:
it is a machine, not a living being (but built by one). However, like so many statements
about living creatures, this one also has exceptions.
The distinction between one and two bodies is poorly defined if the whole system is
made of only a few molecules. This happens most clearly inside bacteria. Organisms such
as Escherichia coli, the well-known bacterium found in the human gut, or bacteria from
the Salmonella family, all swim using flagella. Flagella are thin filaments, similar to tiny
hairs that stick out of the cell membrane. In the 1970s it was shown that each flagellum,
made of one or a few long molecules with a diameter of a few tens of nanometres, does

Motion Mountain – The Adventure of Physics
Vol. V, page 282 in fact turn about its axis.
Bacteria are able to rotate their flagella in both clockwise and anticlockwise directions,
can achieve more than 1000 turns per second, and can turn all their flagella in perfect
Ref. 76 synchronization. These wheels are so tiny that they do not need a mechanical connec-
Ref. 77 tion; Figure 58 shows a number of motor models found in bacteria. The motion and the
construction of these amazing structures are shown in more details in the films Figure 59
and Figure 60.
In summary, wheels actually do exist in living beings, albeit only tiny ones. The growth
and motion of these wheels are wonders of nature. Macroscopic wheels in living beings

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
are not possible, though rolling motion is.

Curiosities and fun challenges ab ou t kinematics
Challenge 167 s What is the biggest wheel ever made?
∗∗
A football is shot, by a goalkeeper, with around 30 m/s. Use a video to calculate the dis-
tance it should fly and compare it with the distances realized in a soccer match. Where
Challenge 168 s does the difference come from?
∗∗
A train starts to travel at a constant speed of 10 m/s between two cities A and B, 36 km

* Rolling is also known for the Namibian wheel spiders of the Carparachne genus; films of their motion can
be found on the internet.
** Despite the disadvantage of not being able to use rotating parts and of being restricted to one piece
only, nature’s moving constructions, usually called animals, often outperform human-built machines. As
an example, compare the size of the smallest flying systems built by evolution with those built by humans.
(See, e.g., pixelito.reference.be.) There are two reasons for this discrepancy. First, nature’s systems have
integrated repair and maintenance systems. Second, nature can build large structures inside containers with
small openings. In fact, nature is very good at what people do when they build sailing ships inside glass
Challenge 166 s bottles. The human body is full of such examples; can you name a few?
3 how to describe motion – kinematics 93

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 58 Some types of ﬂagellar motors found in nature; the images are taken by cryotomography.
All yellow scale bars are 10 nm long (© S. Chen & al., EMBO Journal, Wiley & Sons).
94 3 how to describe motion – kinematics

F I G U R E 59
The rotational
motion of a
bacterial
ﬂagellum, and
its reversal

Motion Mountain – The Adventure of Physics
(QuickTime
ﬁlm © Osaka
University).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 60
The growth of
a bacterial
ﬂagellum,
showing the
molecular
assembly
(QuickTime
ﬁlm © Osaka
University).

apart. The train will take one hour for the journey. At the same time as the train, a fast
dove starts to fly from A to B, at 20 m/s. Being faster than the train, the dove arrives at
B first. The dove then flies back towards A; when it meets the train, it turns back again,
to city B. It goes on flying back and forward until the train reaches B. What distance did
Challenge 169 e the dove cover?
3 how to describe motion – kinematics 95

Motion Mountain – The Adventure of Physics
F I G U R E 61 Are comets, such as the beautiful comet McNaught seen in 2007, images or bodies? How
can you show it? (And why is the tail curved?) (© Robert McNaught)

F I G U R E 62 The parabola of safety around a cannon, shown in red. The highest points of all trajectories copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
form an ellipse, shown in blue. (© Theon)

∗∗
Figure 62 illustrates that around a cannon, there is a line outside which you cannot be
hit. Already in the 17th century, Evangelista Torricelli showed, without algebra, that the
Challenge 170 e line is a parabola, and called it the parabola of safety. Can you show this as well? Can you
confirm that the highest points of all trajectories lie on an ellipse? The parabola of safety
also appears in certain water fountains.
∗∗
Balance a pencil vertically (tip upwards!) on a piece of paper near the edge of a table.
Challenge 171 e How can you pull out the paper without letting the pencil fall?
96 3 how to describe motion – kinematics

F I G U R E 63 Observation of sonoluminescence with a simple set-up
that focuses ultrasound in water (© Detlef Lohse).

∗∗

Motion Mountain – The Adventure of Physics
Is a return flight by aeroplane – from a point A to B and back to A – faster if the wind
Challenge 172 e blows or if it does not?
∗∗
The level of acceleration that a human can survive depends on the duration over which
one is subjected to it. For a tenth of a second, 30 𝑔 = 300 m/s2 , as generated by an ejector
seat in an aeroplane, is acceptable. (It seems that the record acceleration a human has
survived is about 80 𝑔 = 800 m/s2 .) But as a rule of thumb it is said that accelerations of
15 𝑔 = 150 m/s2 or more are fatal.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
The highest microscopic accelerations are observed in particle collisions, where values up
to 1035 m/s2 are achieved. The highest macroscopic accelerations are probably found in
the collapsing interiors of supernovae, exploding stars which can be so bright as to be vis-
ible in the sky even during the daytime. A candidate on Earth is the interior of collapsing
bubbles in liquids, a process called cavitation. Cavitation often produces light, an effect
Ref. 78 discovered by Frenzel and Schultes in 1934 and called sonoluminescence. (See Figure 63.)
It appears most prominently when air bubbles in water are expanded and contracted by
underwater loudspeakers at around 30 kHz and allows precise measurements of bubble
motion. At a certain threshold intensity, the bubble radius changes at 1500 m/s in as little
Ref. 79 as a few μm, giving an acceleration of several 1011 m/s2 .
∗∗
Legs are easy to build. Nature has even produced a millipede, Illacme plenipes, that has
750 legs. The animal is 3 to 4 cm long and about 0.5 mm wide. This seems to be the record
so far. In contrast to its name, no millipede actually has a thousand legs.

Summary of kinematics
The description of everyday motion of mass points with three coordinates as
(𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) is simple, precise and complete. This description of paths is the basis
3 how to describe motion – kinematics 97

of kinematics. As a consequence, space is described as a three-dimensional Euclidean
space and velocity and acceleration as Euclidean vectors.
The description of motion with paths assumes that the motion of objects can be fol-
lowed along their paths. Therefore, the description often does not work for an important
case: the motion of images.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Chapter 4

F ROM OB J E C T S A N D I M AG E S TO
C ON SE RVAT ION

W
alking through a forest we observe two rather different types of motion:
e see the breeze move the leaves, and at the same time, on the ground,
Ref. 80 e see their shadows move. Shadows are a simple type of image. Both objects and
images are able to move; both change position over time. Running tigers, falling snow-
flakes, and material ejected by volcanoes, but also the shadow following our body, the

Motion Mountain – The Adventure of Physics
beam of light circling the tower of a lighthouse on a misty night, and the rainbow that
constantly keeps the same apparent distance from us are examples of motion.
Both objects and images differ from their environment in that they have boundaries
defining their size and shape. But everybody who has ever seen an animated cartoon
knows that images can move in more surprising ways than objects. Images can change
their size and shape, they can even change colour, a feat only a few objects are able to
perform.** Images can appear and disappear without a trace, multiply, interpenetrate, go
backwards in time and defy gravity or any other force. Images, even ordinary shadows,
can move faster than light. Images can float in space and keep the same distance from
approaching objects. Objects can do almost none of this. In general, the ‘laws of cartoon

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 82
physics’ are rather different from those in nature. In fact, the motion of images does not
seem to follow any rules, in contrast to the motion of objects. We feel the need for precise
criteria allowing the two cases to be distinguished.
Making a clear distinction between images and objects is performed using the same
method that children or animals use when they stand in front of a mirror for the first
time: they try to touch what they see. Indeed,

⊳ If we are able to touch what we see – or more precisely, if we are able to move
it with a collision – we call it an object, otherwise an image.***

** Excluding very slow changes such as the change of colour of leaves in the Autumn, in nature only certain
crystals, the octopus and other cephalopods, the chameleon and a few other animals achieve this. Of man-
made objects, television, computer displays, heated objects and certain lasers can do it. Do you know more
Challenge 173 s examples? An excellent source of information on the topic of colour is the book by K. Nassau, The Physics
and Chemistry of Colour – the fifteen causes of colour, J. Wiley & Sons, 1983. In the popular science domain,
the most beautiful book is the classic work by the Flemish astronomer Marcel G. J. Minnaert, Light
and Colour in the Outdoors, Springer, 1993, an updated version based on his wonderful book series, De
Ref. 81 natuurkunde van ‘t vrije veld, Thieme & Cie, 1937. Reading it is a must for all natural scientists. On the web,
there is also the – simpler, but excellent – webexhibits.org/causesofcolour website.
*** One could propose including the requirement that objects may be rotated; however, this requirement,
surprisingly, gives difficulties in the case of atoms, as explained on page 85 in Volume IV.
4 from objects and images to conservation 99

push

F I G U R E 64 In which direction does the bicycle turn?

Vol. IV, page 139 Images cannot be touched, but objects can. Images cannot hit each other, but objects can.
And as everybody knows, touching something means feeling that it resists movement.
Certain bodies, such as butterflies, pose little resistance and are moved with ease, others,
such as ships, resist more, and are moved with more difficulty.

⊳ The resistance to motion – more precisely, to change of motion – is called

Motion Mountain – The Adventure of Physics
inertia, and the difficulty with which a body can be moved is called its (in-
ertial) mass.

Images have neither inertia nor mass.
Summing up, for the description of motion we must distinguish bodies, which can
be touched and are impenetrable, from images, which cannot and are not. Everything
Challenge 174 s visible is either an object or an image; there is no third possibility. (Do you agree?) If
the object is so far away that it cannot be touched, such as a star or a comet, it can be
difficult to decide whether one is dealing with an image or an object; we will encounter

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
this difficulty repeatedly. For example, how would you show that comets – such as the
beautiful example of Figure 61 – are objects and not images, as Galileo (falsely) claimed?
Challenge 175 s
In the same way that objects are made of matter, images are made of radiation. Im-
Ref. 83 ages are the domain of shadow theatre, cinema, television, computer graphics, belief sys-
tems and drug experts. Photographs, motion pictures, ghosts, angels, dreams and many
hallucinations are images (sometimes coupled with brain malfunction). To understand
images, we need to study radiation (plus the eye and the brain). However, due to the
importance of objects – after all, we are objects ourselves – we study the latter first.

Motion and contact

“
Democritus affirms that there is only one type

”
of movement: That resulting from collision.
Ref. 84 Aetius, Opinions.

When a child rides a unicycle, she or he makes use of a general rule in our world: one
body acting on another puts it in motion. Indeed, in about six hours, anybody can learn
to ride and enjoy a unicycle. As in all of life’s pleasures, such as toys, animals, women,
machines, children, men, the sea, wind, cinema, juggling, rambling and loving, some-
thing pushes something else. Thus our first challenge is to describe the transfer of motion
due to contact – and to collisions – in more precise terms.
100 4 from objects and images to conservation

𝑣1 𝑣2

𝑣1 + Δ𝑣1 𝑣2 + Δ𝑣2

Motion Mountain – The Adventure of Physics
F I G U R E 65 Collisions deﬁne mass. F I G U R E 66 The standard kilogram (until 2019)
(© BIPM).

Contact is not the only way to put something into motion; a counter-example is an
apple falling from a tree or one magnet pulling another. Non-contact influences are more
fascinating: nothing is hidden, but nevertheless something mysterious happens. Contact

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
motion seems easier to grasp, and that is why one usually starts with it. However, despite
this choice, non-contact interactions cannot be avoided. Our choice to start with contact
will lead us to a similar experience to that of riding a bicycle. (See Figure 64.) If we ride
a bicycle at a sustained speed and try to turn left by pushing the right-hand steering bar,
we will turn right. By the way, this surprising effect, also known to motorbike riders,
obviously works only above a certain minimum speed. Can you determine what this
Challenge 176 s speed is? Be careful! Too strong a push will make you fall.
Something similar will happen to us as well; despite our choice of contact motion, the
rest of our walk will rapidly force us to study non-contact interactions.

What is mass?

“
Δός μοί (φησι) ποῦ στῶ καὶ κινῶ τὴν γῆν.

”
Da ubi consistam, et terram movebo.*
Archimedes

When we push something we are unfamiliar with, such as when we kick an object on
the street, we automatically pay attention to the same aspect that children explore when

* ‘Give me a place to stand, and I’ll move the Earth.’ Archimedes (c. 283–212), Greek scientist and engineer.
Ref. 85 This phrase is attributed to him by Pappus. Already Archimedes knew that the distinction used by lawyers
between movable and immovable objects made no sense.
4 from objects and images to conservation 101

F I G U R E 67 Antoine Lavoisier (1743 –1794) and his wife.

Motion Mountain – The Adventure of Physics
they stand in front of a mirror for the first time, or when they see a red laser spot for the
first time. They check whether the unknown entity can be pushed or caught, and they
pay attention to how the unknown object moves under their influence. All these are col-
lision experiments. The high-precision version of any collision experiment is illustrated
in Figure 65. Repeating such experiments with various pairs of objects, we find:

⊳ A fixed quantity 𝑚𝑖 can be ascribed to every object 𝑖, determined by the
relation
𝑚2 Δ𝑣
=− 1 (17)

where Δ𝑣 is the velocity change produced by the collision. The quantity 𝑚𝑖
is called the mass of the object 𝑖.

The more difficult it is to move an object, the higher the mass value. In order to have
mass values that are common to everybody, the mass value for one particular, selected
object has to be fixed in advance. Until 2019, there really was one such special object in the
world, shown in Figure 66; it was called the standard kilogram. It was kept with great care
in a glass container in Sèvres near Paris. Until 2019, the standard kilogram determined
the value of the mass of every other object in the world. The standard kilogram was
touched only once every few years because otherwise dust, humidity, or scratches would
change its mass. For example, the standard kilogram was not kept under vacuum, because
this would lead to outgassing and thus to changes in its mass. All the care did not avoid
the stability issues though, and in 2019, the kilogram unit has been redefined using the
fundamental constants 𝐺 (indirectly, via the caesium transition frequency), 𝑐 and ℏ that
are shown in Figure 1. Since that change, everybody can produce his or her own standard
kilogram in the laboratory – provided that sufficient care is used.
The mass thus measures the difficulty of getting something moving. High masses are
harder to move than low masses. Obviously, only objects have mass; images don’t. (By
Ref. 68 the way, the word ‘mass’ is derived, via Latin, from the Greek μαζα – bread – or the
102 4 from objects and images to conservation

F I G U R E 68 Christiaan Huygens (1629 –1695).

Hebrew ‘mazza’ – unleavened bread. That is quite a change in meaning.)
Experiments with everyday life objects also show that throughout any collision, the
sum of all masses is conserved:
∑ 𝑚𝑖 = const . (18)
𝑖

Motion Mountain – The Adventure of Physics
The principle of conservation of mass was first stated by Antoine-Laurent Lavoisier.*
Conservation of mass also implies that the mass of a composite system is the sum of the
mass of the components. In short, mass is also a measure for the quantity of matter.
In a famous experiment in the sixteenth century, for several weeks Santorio Santorio
(Sanctorius) (1561–1636), a friend of Galileo, lived with all his food and drink supply,
and also his toilet, on a large balance. He wanted to test mass conservation. How did the
Challenge 177 s measured weight change with time?
Various cult leaders pretended and still pretend that they can produce matter out of
nothing. This would be an example of non-conservation of mass. How can you show that
all such leaders are crooks?

Momentum and mass
The definition of mass can also be given in another way. We can ascribe a number 𝑚𝑖
to every object 𝑖 such that for collisions free of outside interference the following sum is
unchanged throughout the collision:

∑ 𝑚𝑖 𝑣𝑖 = const . (19)
𝑖

The product of the velocity 𝑣𝑖 and the mass 𝑚𝑖 is called the (linear) momentum of the
body. The sum, or total momentum of the system, is the same before and after the colli-
sion; momentum is a conserved quantity.

* Antoine-Laurent Lavoisier (b. 1743 Paris , d. 1794 Paris), chemist and genius. Lavoisier was the first to un-
derstand that combustion is a reaction with oxygen; he discovered the components of water and introduced
mass measurements into chemistry. A famous story about his character: When he was (unjustly) sentenced
to the guillotine during the French revolution, he decided to use the situation for a scientific experiment. He
announced that he would try to blink his eyes as frequently as possible after his head was cut off, in order to
show others how long it takes to lose consciousness. Lavoisier managed to blink eleven times. It is unclear
whether the story is true or not. It is known, however, that it could be true. Indeed, after a decapitation
Ref. 86 without pain or shock, a person can remain conscious for up to half a minute.
4 from objects and images to conservation 103

F I G U R E 69 Is this dangerous?

⊳ Momentum conservation defines mass.

The two conservation principles (18) and (19) were first stated in this way by the import-
ant physicist Christiaan Huygens:* Momentum and mass are conserved in the everyday

Motion Mountain – The Adventure of Physics
motion of objects. In particular, neither quantity can be defined for the motion of images.
Some typical momentum values are given in Table 15.
Momentum conservation implies that when a moving sphere hits a resting one of the
same mass and without loss of energy, a simple rule determines the angle between the
Challenge 179 s directions the two spheres take after the collision. Can you find this rule? It is particularly
useful when playing billiards. We will find out later that the rule is not valid for speeds
Vol. II, page 67 near that of light.
Another consequence of momentum conservation is shown in Figure 69: a man is
lying on a bed of nails with a large block of concrete on his stomach. Another man is

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
hitting the concrete with a heavy sledgehammer. As the impact is mostly absorbed by
Challenge 180 s the concrete, there is no pain and no danger – unless the concrete is missed. Why?
The above definition (17) of mass has been generalized by the physicist and philo-
sopher Ernst Mach** in such a way that it is valid even if the two objects interact without
contact, as long as they do so along the line connecting their positions.

⊳ The mass ratio between two bodies is defined as a negative inverse accelera-
tion ratio, thus as
𝑚2 𝑎
=− 1 , (20)
𝑚1 𝑎2

* Christiaan Huygens (b. 1629 ’s Gravenhage, d. 1695 Hofwyck) was one of the main physicists and math-
ematicians of his time. Huygens clarified the concepts of mechanics; he also was one of the first to show that
light is a wave. He wrote influential books on probability theory, clock mechanisms, optics and astronomy.
Among other achievements, Huygens showed that the Orion Nebula consists of stars, discovered Titan, the
moon of Saturn, and showed that the rings of Saturn consist of rock. (This is in contrast to Saturn itself,
whose density is lower than that of water.)
** Ernst Mach (1838 Chrlice–1916 Vaterstetten), Austrian physicist and philosopher. The mach unit for aero-
plane speed as a multiple of the speed of sound in air (about 0.3 km/s) is named after him. He also studied
the basis of mechanics. His thoughts about mass and inertia influenced the development of general relativ-
ity and led to Mach’s principle, which will appear later on. He was also proud to be the last scientist denying
– humorously, and against all evidence – the existence of atoms.
104 4 from objects and images to conservation

TA B L E 15 Some measured momentum values.

O b s e r va t i o n Momentum

Images 0
Momentum of a green photon 1.2 ⋅ 10−27 Ns
Average momentum of oxygen molecule in air 10−26 Ns
X-ray photon momentum 10−23 Ns
𝛾 photon momentum 10−17 Ns
Highest particle momentum in accelerators 1 fNs
Highest possible momentum of a single elementary 6.5 Ns
particle – the Planck momentum
Fast billiard ball 3 Ns
Flying rifle bullet 10 Ns
Box punch 15 to 50 Ns
Comfortably walking human 80 Ns

Motion Mountain – The Adventure of Physics
Lion paw strike c. 0.2 kNs
Whale tail blow c. 3 kNs
Car on highway 40 kNs
Impact of meteorite with 2 km diameter 100 TNs
Momentum of a galaxy in galaxy collision up to 1046 Ns

where 𝑎 is the acceleration of each body during the interaction.

This definition of mass has been explored in much detail in the physics community,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
mainly in the nineteenth century. A few points sum up the results:
— The definition of mass implies the conservation of total momentum ∑ 𝑚𝑣. Mo-
mentum conservation is not a separate principle. Conservation of momentum cannot
be checked experimentally, because mass is defined in such a way that the momentum
conservation holds.
— The definition of mass implies the equality of the products 𝑚1 𝑎1 and −𝑚2 𝑎2 . Such
products are called forces. The equality of acting and reacting forces is not a separate
principle; mass is defined in such a way that the principle holds.
— The definition of mass is independent of whether contact is involved or not, and
whether the accelerations are due to electricity, gravitation, or other interactions.*
Since the interaction does not enter the definition of mass, mass values defined with
the help of the electric, nuclear or gravitational interaction all agree, as long as mo-
mentum is conserved. All known interactions conserve momentum. For some un-
fortunate historical reasons, the mass value measured with the electric or nuclear in-
teractions is called the ‘inertial’ mass and the mass measured using gravity is called

* As mentioned above, only central forces obey the relation (20) used to define mass. Central forces act
Page 119 between the centre of mass of bodies. We give a precise definition later. However, since all fundamental
forces are central, this is not a restriction. There seems to be one notable exception: magnetism. Is the
Challenge 181 s definition of mass valid in this case?
4 from objects and images to conservation 105

TA B L E 16 Some measured mass values.

O b s e r va t i o n Mass

Probably lightest known object: neutrino c. 2 ⋅ 10−36 kg
Mass increase due to absorption of one green photon 4.1 ⋅ 10−36 kg
Lightest known charged object: electron 9.109 381 88(72) ⋅ 10−31 kg
Atom of argon 39.962 383 123(3) u = 66.359 1(1) yg
Lightest object ever weighed (a gold particle) 0.39 ag
Human at early age (fertilized egg) 10−8 g
Water adsorbed on to a kilogram metal weight 10−5 g
Planck mass 2.2 ⋅ 10−5 g
Fingerprint 10−4 g
Typical ant 10−4 g
Water droplet 1 mg
Honey bee, Apis mellifera 0.1 g

Motion Mountain – The Adventure of Physics
Euro coin 7.5 g
Blue whale, Balaenoptera musculus 180 Mg
Heaviest living things, such as the fungus Armillaria 106 kg
ostoyae or a large Sequoia Sequoiadendron giganteum
Heaviest train ever 99.7 ⋅ 106 kg
Largest ocean-going ship 400 ⋅ 106 kg
Largest object moved by man (Troll gas rig) 687.5 ⋅ 106 kg
Large antarctic iceberg 1015 kg
Water on Earth 1021 kg

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Earth’s mass 5.98 ⋅ 1024 kg
Solar mass 2.0 ⋅ 1030 kg
Our galaxy’s visible mass 3 ⋅ 1041 kg
Our galaxy’s estimated total mass 2 ⋅ 1042 kg
virgo supercluster 2 ⋅ 1046 kg
Total mass visible in the universe 1054 kg

the ‘gravitational’ mass. As it turns out, this artificial distinction makes no sense; this
becomes especially clear when we take an observation point that is far away from all
the bodies concerned.
— The definition of mass requires observers at rest or in inertial motion.
By measuring the masses of bodies around us we can explore the science and art of ex-
periments. An overview of mass measurement devices is given in Table 18 and Figure 71.
Some measurement results are listed in Table 16.
By measuring mass values around us we confirm the main properties of mass. First
of all, mass is additive in everyday life, as the mass of two bodies combined is equal to
the sum of the two separate masses. Furthermore, mass is continuous; it can seemingly
take any positive value. Finally, mass is conserved in everyday life; the mass of a system,
106 4 from objects and images to conservation

TA B L E 17 Properties of mass in everyday life.

Masses Physical M at h e m at i c a l Defini-
propert y name tion
Can be distinguished distinguishability element of set Vol. III, page 285
Can be ordered sequence order Vol. IV, page 224
Can be compared measurability metricity Vol. IV, page 236
Can change gradually continuity completeness Vol. V, page 364
Can be added quantity of matter additivity
Page 81
Beat any limit infinity unboundedness, openness Vol. III, page 286
Do not change conservation invariance 𝑚 = const
Do not disappear impenetrability positivity 𝑚⩾0

defined as the sum of the mass of all constituents, does not change over time if the system

Motion Mountain – The Adventure of Physics
is kept isolated from the rest of the world. Mass is not only conserved in collisions but
also during melting, evaporation, digestion and all other everyday processes.
All the properties of everyday mass are summarized in Table 17. Later we will find that
several of the properties are only approximate. High-precision experiments show devi-
ations.* However, the definition of mass remains unchanged throughout our adventure.
The definition of mass through momentum conservation implies that when an object
falls, the Earth is accelerated upwards by a tiny amount. If we could measure this tiny
amount, we could determine the mass of the Earth. Unfortunately, this measurement is
Challenge 182 s impossible. Can you find a better way to determine the mass of the Earth?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The definition of mass and momentum allows answering the question of Figure 70.
A brick hangs from the ceiling; a second thread hangs down from the brick, and you
Challenge 183 e can pull it. How can you tune your pulling method to make the upper thread break? The
lower one?
Summarizing Table 17, the mass of a body is thus most precisely described by a positive
real number, often abbreviated 𝑚 or 𝑀. This is a direct consequence of the impenetrabil-
ity of matter. Indeed, a negative (inertial) mass would mean that such a body would move
in the opposite direction of any applied force or acceleration. Such a body could not be
kept in a box; it would break through any wall trying to stop it. Strangely enough, neg-
ative mass bodies would still fall downwards in the field of a large positive mass (though
Challenge 184 e more slowly than an equivalent positive mass). Are you able to confirm this? However, a
small positive mass object would float away from a large negative-mass body, as you can
easily deduce by comparing the various accelerations involved. A positive and a negative
mass of the same value would remain at a constant distance and spontaneously accelerate
Challenge 185 e away along the line connecting the two masses. Note that both energy and momentum
Vol. II, page 72 are conserved in all these situations.** Negative-mass bodies have never been observed.
Vol. IV, page 192 Antimatter, which will be discussed later, also has positive mass.

* For example, in order to define mass we must be able to distinguish bodies. This seems a trivial require-
ment, but we discover that this is not always possible in nature.
** For more curiosities, see R. H. P rice, Negative mass can be positively amusing, American Journal of
4 from objects and images to conservation 107

ceiling

thread

brick

thread

hand

Motion Mountain – The Adventure of Physics
F I G U R E 70 Depending on the way you pull, either
the upper of the lower thread snaps. What are the
options?

TA B L E 18 Some mass sensors.

Measurement Sensor Range

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Precision scales balance, pendulum, or spring 1 pg to 103 kg
Particle collision speed below 1 mg
Sense of touch pressure sensitive cells 1 mg to 500 kg
Doppler effect on light reflected off interferometer 1 mg to 100 g
the object
Cosmonaut body mass spring frequency around 70 kg
measurement device
Truck scales hydraulic balance 103 to 60 ⋅ 103 kg
Ship weight water volume measurement up to 500 ⋅ 106 kg

Physics 61, pp. 216–217, 1993. Negative mass particles in a box would heat up a box made of positive mass
Page 110 while traversing its walls, and accelerating, i.e., losing energy, at the same time. They would allow one to
build a perpetuum mobile of the second kind, i.e., a device circumventing the second principle of thermo-
Challenge 186 e dynamics. Moreover, such a system would have no thermodynamic equilibrium, because its energy could
decrease forever. The more one thinks about negative mass, the more one finds strange properties contra-
Challenge 187 s dicting observations. By the way, what is the range of possible mass values for tachyons?
108 4 from objects and images to conservation

Motion Mountain – The Adventure of Physics
F I G U R E 71 Mass measurement devices: a vacuum balance used in 1890 by Dmitriy Ivanovich
Mendeleyev, a modern laboratory balance, a device to measure the mass of a cosmonaut in space and

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
a truck scales (© Thinktank Trust, Mettler-Toledo, NASA, Anonymous).

Is motion eternal? – C onservation of momentum

“
Every body continues in the state of rest or of
uniform motion in a straight line except in so

”
far as it doesn’t.
Arthur Eddington*

The product 𝑝 = 𝑚𝑣 of mass and velocity is called the momentum of a particle; it de-
scribes the tendency of an object to keep moving during collisions. The larger it is, the
harder it is to stop the object. Like velocity, momentum has a direction and a magnitude:
it is a vector. In French, momentum is called ‘quantity of motion’, a more appropriate
term. In the old days, the term ‘motion’ was used instead of ‘momentum’, for example
by Newton. The conservation of momentum, relation (19), therefore expresses the con-
servation of motion during interactions.
Momentum is an extensive quantity. That means that it can be said that it flows from
one body to the other, and that it can be accumulated in bodies, in the same way that
water flows and can be accumulated in containers. Imagining momentum as something

* Arthur Eddington (1882–1944), British astrophysicist.
4 from objects and images to conservation 109

cork

wine

wine
stone

Challenge 188 s F I G U R E 72 What happens in these four situations?

Ref. 87 that can be exchanged between bodies in collisions is always useful when thinking about
the description of moving objects.
Momentum is conserved. That explains the limitations you might experience when

Motion Mountain – The Adventure of Physics
being on a perfectly frictionless surface, such as ice or a polished, oil covered marble:
you cannot propel yourself forward by patting your own back. (Have you ever tried to
put a cat on such a marble surface? It is not even able to stand on its four legs. Neither
Challenge 189 s are humans. Can you imagine why?) Momentum conservation also answers the puzzles
of Figure 72.
The conservation of momentum and mass also means that teleportation (‘beam me
Challenge 190 s up’) is impossible in nature. Can you explain this to a non-physicist?
Momentum conservation implies that momentum can be imagined to be like an invis-
ible fluid. In an interaction, the invisible fluid is transferred from one object to another.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In such transfers, the amount of fluid is always constant.
Momentum conservation implies that motion never stops; it is only exchanged. On
the other hand, motion often ‘disappears’ in our environment, as in the case of a stone
dropped to the ground, or of a ball left rolling on grass. Moreover, in daily life we of-
ten observe the creation of motion, such as every time we open a hand. How do these
examples fit with the conservation of momentum?
It turns out that apparent momentum disappearance is due to the microscopic aspects
of the involved systems. A muscle only transforms one type of motion, namely that of the
electrons in certain chemical compounds* into another, the motion of the fingers. The
working of muscles is similar to that of a car engine transforming the motion of electrons
in the fuel into motion of the wheels. Both systems need fuel and get warm in the process.
We must also study the microscopic behaviour when a ball rolls on grass until it stops.
The apparent disappearance of motion is called friction. Studying the situation carefully,
we find that the grass and the ball heat up a little during this process. During friction,
visible motion is transformed into heat. A striking observation of this effect for a bicycle
Page 384 is shown below, in Figure 273. Later, when we discover the structure of matter, it will
become clear that heat is the disorganized motion of the microscopic constituents of
every material. When the microscopic constituents all move in the same direction, the
object as a whole moves; when they oscillate randomly, the object is at rest, but is warm.

Ref. 88 * The fuel of most processes in animals usually is adenosine triphosphate (ATP).
110 4 from objects and images to conservation

Heat is a form of motion. Friction thus only seems to be disappearance of motion; in fact
it is a transformation of ordered into unordered motion.
Page 395 Despite momentum conservation, macroscopic perpetual motion does not exist, since
friction cannot be completely eliminated.* Motion is eternal only at the microscopic
scale. In other words, the disappearance and also the spontaneous appearance of mo-
tion in everyday life is an illusion due to the limitations of our senses. For example, the
motion proper of every living being exists before its birth, and stays after its death. The
same happens with its energy. This result is probably the closest one can get to the idea
of everlasting life from evidence collected by observation. It is perhaps less than a co-
incidence that energy used to be called vis viva, or ‘living force’, by Leibniz and many
others.
Since motion is conserved, it has no origin. Therefore, at this stage of our walk we
cannot answer the fundamental questions: Why does motion exist? What is its origin?
The end of our adventure is nowhere near.

More conservation – energy

Motion Mountain – The Adventure of Physics
When collisions are studied in detail, a second conserved quantity turns up. Experiments
show that in the case of perfect, or elastic collisions – collisions without friction – the
following quantity, called the kinetic energy 𝑇 of the system, is also conserved:

𝑇 = ∑ 21 𝑚𝑖 𝑣𝑖2 = const . (21)
𝑖

Kinetic energy is the ability that a body has to induce change in bodies it hits. Kinetic
energy thus depends on the mass and on the square of the speed 𝑣 of a body. The full

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
name ‘kinetic energy’ was introduced by Gustave-Gaspard Coriolis.** Some measured
energy values are given in Table 19.

* Some funny examples of past attempts to built a perpetual motion machine are described in Stan-
islav Michel, Perpetuum mobile, VDI Verlag, 1976. Interestingly, the idea of eternal motion came
to Europe from India, via the Islamic world, around the year 1200, and became popular as it op-
posed the then standard view that all motion on Earth disappears over time. See also the web.archive.
org/web/20040812085618/http://www.geocities.com/mercutio78_99/pmm.html and the www.lhup.edu/
~dsimanek/museum/unwork.htm websites. The conceptual mistake made by eccentrics and used by crooks
is always the same: the hope of overcoming friction. (In fact, this applied only to the perpetual motion ma-
chines of the second kind; those of the first kind – which are even more in contrast with observation – even
try to generate energy from nothing.)
If the machine is well constructed, i.e., with little friction, it can take the little energy it needs for the
sustenance of its motion from very subtle environmental effects. For example, in the Victoria and Albert
Ref. 89 Museum in London one can admire a beautiful clock powered by the variations of air pressure over time.
Low friction means that motion takes a long time to stop. One immediately thinks of the motion of the
planets. In fact, there is friction between the Earth and the Sun. (Can you guess one of the mechanisms?)
Challenge 191 s But the value is so small that the Earth has already circled around the Sun for thousands of millions of years,
and will do so for quite some time more.
** Gustave-Gaspard Coriolis (b. 1792 Paris, d. 1843 Paris) was engineer and mathematician. He introduced
the modern concepts of ‘work’ and of ‘kinetic energy’, and explored the Coriolis effect discovered by
Page 138 Laplace. Coriolis also introduced the factor 1/2 in the kinetic energy 𝑇, in order that the relation d𝑇/d𝑣 = 𝑝
Challenge 192 s would be obeyed. (Why?)
4 from objects and images to conservation 111

The experiments and ideas mentioned so far can be summarized in the following
definition:

⊳ (Physical) energy is the measure of the ability to generate motion.

A body has a lot of energy if it has the ability to move many other bodies. Energy is a
number; energy, in contrast to momentum, has no direction. The total momentum of
two equal masses moving with opposite velocities is zero; but their total energy is not,
and it increases with velocity. Energy thus also measures motion, but in a different way
than momentum. Energy measures motion in a more global way.
An equivalent definition is the following:

⊳ Energy is the ability to perform work.

Here, the physical concept of work is just the precise version of what is meant by work in
everyday life. As usual, (physical) work is the product of force and distance in direction

Motion Mountain – The Adventure of Physics
of the force. In other words, work is the scalar product of force and distance. Physical
work is a quantity that describes the effort of pushing of an object along a distance. As a
result, in physics, work is a form of energy.
Another, equivalent definition of energy will become clear shortly:

⊳ Energy is what can be transformed into heat.

Energy is a word taken from ancient Greek; originally it was used to describe character,
and meant ‘intellectual or moral vigour’. It was taken into physics by Thomas Young

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
(1773–1829) in 1807 because its literal meaning is ‘force within’. (The letters 𝐸, 𝑊, 𝐴 and
several others are also used to denote energy.)
Both energy and momentum measure how systems change. Momentum tells how sys-
tems change over distance: momentum is action (or change) divided by distance. Mo-
mentum is needed to compare motion here and there.
Energy measures how systems change over time: energy is action (or change) divided
by time. Energy is needed to compare motion now and later.
Do not be surprised if you do not grasp the difference between momentum and energy
straight away: physicists took about a century to figure it out! So you are allowed to take
some time to get used to it. Indeed, for many decades, English physicists insisted on using
the same term for both concepts; this was due to Newton’s insistence that – no joke – the
existence of god implied that energy was the same as momentum. Leibniz, instead, knew
that energy increases with the square of the speed and proved Newton wrong. In 1722,
Willem Jacob ’s Gravesande even showed the difference between energy and momentum
Ref. 90 experimentally. He let metal balls of different masses fall into mud from different heights.
By comparing the size of the imprints he confirmed that Newton was wrong both with
his physical statements and his theological ones.
One way to explore the difference between energy and momentum is to think about
the following challenges. Which running man is more difficult to stop? One of mass 𝑚
running at speed 𝑣, or one with mass 𝑚/2 and speed 2𝑣, or one with mass 𝑚/2 and speed
Challenge 193 e √2 𝑣? You may want to ask a rugby-playing friend for confirmation.
112 4 from objects and images to conservation

TA B L E 19 Some measured energy values.

O b s e r va t i o n Energy

Average kinetic energy of oxygen molecule in air 6 zJ
Green photon energy 0.37 aJ
X-ray photon energy 1 fJ
𝛾 photon energy 1 pJ
Highest particle energy in accelerators 0.1 μJ
Kinetic energy of a flying mosquito 0.2 μJ
Comfortably walking human 20 J
Flying arrow 50 J
Right hook in boxing 50 J
Energy in torch battery 1 kJ
Energy in explosion of 1 g TNT 4.1 kJ
Energy of 1 kcal 4.18 kJ

Motion Mountain – The Adventure of Physics
Flying rifle bullet 10 kJ
One gram of fat 38 kJ
One gram of gasoline 44 kJ
Apple digestion 0.2 MJ
Car on highway 0.3 to 1 MJ
Highest laser pulse energy 1.8 MJ
Lightning flash up to 1 GJ
Planck energy 2.0 GJ
Small nuclear bomb (20 ktonne) 84 TJ

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Earthquake of magnitude 7 2 PJ
Largest nuclear bomb (50 Mtonne) 210 PJ
Impact of meteorite with 2 km diameter 1 EJ
Yearly machine energy use 420 EJ
Rotation energy of Earth 2 ⋅ 1029 J
Supernova explosion 1044 J
Gamma-ray burst up to 1047 J
Energy content 𝐸 = 𝑐2 𝑚 of Sun’s mass 1.8 ⋅ 1047 J
Energy content of Galaxy’s central black hole 4 ⋅ 1053 J

Another distinction between energy and momentum is illustrated by athletics: the real
long jump world record, almost 10 m, is still kept by an athlete who in the early twentieth
century ran with two weights in his hands, and then threw the weights behind him at the
Challenge 194 s moment he took off. Can you explain the feat?
When a car travelling at 100 m/s runs head-on into a parked car of the same kind and
Challenge 195 s make, which car receives the greatest damage? What changes if the parked car has its
brakes on?
To get a better feeling for energy, here is an additional aspect. The world consumption
of energy by human machines (coming from solar, geothermal, biomass, wind, nuclear,
4 from objects and images to conservation 113

F I G U R E 73 Robert Mayer (1814–1878).

hydro, gas, oil, coal, or animal sources) in the year 2000 was about 420 EJ,* for a world
Ref. 91 population of about 6000 million people. To see what this energy consumption means,
we translate it into a personal power consumption; we get about 2.2 kW. The watt W
is the unit of power, and is simply defined as 1 W = 1 J/s, reflecting the definition of
(physical) power as energy used per unit time. The precise wording is: power is energy
flowing per time through a defined closed surface. See Table 20 for some power values

Motion Mountain – The Adventure of Physics
found in nature, and Table 21 for some measurement devices.
As a working person can produce mechanical work of about 100 W, the average hu-
man energy consumption corresponds to about 22 humans working 24 hours a day. In
particular, if we look at the energy consumption in First World countries, the average
inhabitant there has machines working for him or her that are equivalent to several hun-
Challenge 196 s dred ‘servants’. Machines do a lot of good. Can you point out some of these machines?
Kinetic energy is thus not conserved in everyday life. For example, in non-elastic colli-
sions, such as that of a piece of chewing gum hitting a wall, kinetic energy is lost. Friction
destroys kinetic energy. At the same time, friction produces heat. It was one of the im-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
portant conceptual discoveries of physics that total energy is conserved if one includes
the discovery that heat is a form of energy. Friction is thus a process transforming kinetic
energy, i.e., the energy connected with the motion of a body, into heat. On a microscopic
scale, energy is always conserved.
Any example for the non-conservation of energy is only apparent. ** Indeed, without
energy conservation, the concept of time would not be definable! We will show this im-
portant connection shortly.
In summary, in addition to mass and momentum, everyday linear motion also con-
serves energy. To discover the last conserved quantity, we explore another type of motion:
rotation.

Page 453 * For the explanation of the abbreviation E, see Appendix B.
** In fact, the conservation of energy was stated in its full generality in public only in 1842, by Julius Robert
Mayer. He was a medical doctor by training, and the journal Annalen der Physik refused to publish his
paper, as it supposedly contained ‘fundamental errors’. What the editors called errors were in fact mostly
– but not only – contradictions of their prejudices. Later on, Helmholtz, Thomson-Kelvin, Joule and many
others acknowledged Mayer’s genius. However, the first to have stated energy conservation in its modern
form was the French physicist Sadi Carnot (1796–1832) in 1820. To him the issue was so clear that he did not
publish the result. In fact he went on and discovered the second ‘law’ of thermodynamics. Today, energy
conservation, also called the first ‘law’ of thermodynamics, is one of the pillars of physics, as it is valid in
all its domains.
114 4 from objects and images to conservation

TA B L E 20 Some measured power values.

O b s e r va t i o n Power

Radio signal from the Galileo space probe sending from Jupiter 10 zW
Power of flagellar motor in bacterium 0.1 pW
Power consumption of a typical cell 1 pW
sound power at the ear at hearing threshold 2.5 pW
CR-R laser, at 780 nm 40-80 mW
Sound output from a piano playing fortissimo 0.4 W
Dove (0.16 kg) basal metabolic rate 0.97 W
Rat (0.26 kg) basal metabolic rate 1.45 W
Pigeon (0.30 kg) basal metabolic rate 1.55 W
Hen (2.0 kg) basal metabolic rate 4.8 W
Incandescent light bulb light output 1 to 5 W
Dog (16 kg) basal metabolic rate 20 W

Motion Mountain – The Adventure of Physics
Sheep (45 kg) basal metabolic rate 50 W
Woman (60 kg) basal metabolic rate 68 W
Man (70 kg) basal metabolic rate 87 W
Incandescent light bulb electricity consumption 25 to 100 W
A human, during one work shift of eight hours 100 W
Cow (400 kg) basal metabolic rate 266 W
One horse, for one shift of eight hours 300 W
Steer (680 kg) basal metabolic rate 411 W
Eddy Merckx, the great bicycle athlete, during one hour 500 W

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Metric horse power power unit (75 kg ⋅ 9.81 m/s2 ⋅ 1 m/s) 735.5 W
British horse power power unit 745.7 W
Large motorbike 100 kW
Electrical power station output 0.1 to 6 GW
World’s electrical power production in 2000 Ref. 91 450 GW
Power used by the geodynamo 200 to 500 GW
Limit on wind energy production Ref. 92 18 to 68 TW
Input on Earth surface: Sun’s irradiation of Earth Ref. 93 0.17 EW
Input on Earth surface: thermal energy from inside of the Earth 32 TW
Input on Earth surface: power from tides (i.e., from Earth’s rotation) 3 TW
Input on Earth surface: power generated by man from fossil fuels 8 to 11 TW
Lost from Earth surface: power stored by plants’ photosynthesis 40 TW
World’s record laser power 1 PW
Output of Earth surface: sunlight reflected into space 0.06 EW
Output of Earth surface: power radiated into space at 287 K 0.11 EW
Peak power of the largest nuclear bomb 5 YW
Sun’s output 384.6 YW
Maximum power in nature, 𝑐5 /4𝐺 9.1 ⋅ 1051 W
4 from objects and images to conservation 115

TA B L E 21 Some power sensors.

Measurement Sensor Range

Heart beat as power meter deformation sensor and clock 75 to 2 000 W
Fitness power meter piezoelectric sensor 75 to 2 000 W
Electricity meter at home rotating aluminium disc 20 to 10 000 W
Power meter for car engine electromagnetic brake up to 1 MW
Laser power meter photoelectric effect in up to 10 GW
semiconductor
Calorimeter for chemical reactions temperature sensor up to 1 MW
Calorimeter for particles light detector up to a few μJ/ns

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 74 Some power measurement devices: a bicycle power meter, a laser power meter, and an
electrical power meter (© SRAM, Laser Components, Wikimedia).

The cross product, or vector product
The discussion of rotation is easiest if we introduce an additional way to multiply vectors.
This new product between two vectors 𝑎 and 𝑏 is called the cross product or vector product
𝑎 × 𝑏.
The result of the vector product is another vector; thus it differs from the scalar
product, whose result is a scalar, i.e., a number. The result of the vector product is that
vector
— that is orthogonal to both vectors to be multiplied,
— whose orientation is given by the right-hand rule, and
— whose length is given by the surface area of the parallelogram spanned by the two
vectors, i.e., by 𝑎𝑏 sin ∢(𝑎, 𝑏).
The definition implies that the cross product vanishes if and only if the vectors are par-
116 4 from objects and images to conservation

Challenge 197 e allel. From the definition you can also show that the vector product has the properties

𝑎 × 𝑏 = −𝑏 × 𝑎 , 𝑎 × (𝑏 + 𝑐) = 𝑎 × 𝑏 + 𝑎 × 𝑐 ,
𝜆𝑎 × 𝑏 = 𝜆(𝑎 × 𝑏) = 𝑎 × 𝜆𝑏 , 𝑎 × 𝑎 = 0 ,
𝑎(𝑏 × 𝑐) = 𝑏(𝑐 × 𝑎) = 𝑐(𝑎 × 𝑏) , 𝑎 × (𝑏 × 𝑐) = (𝑎𝑐)𝑏 − (𝑎𝑏)𝑐 ,
(𝑎 × 𝑏)(𝑐 × 𝑑) = 𝑎(𝑏 × (𝑐 × 𝑑)) = (𝑎𝑐)(𝑏𝑑) − (𝑏𝑐)(𝑎𝑑) ,
(𝑎 × 𝑏) × (𝑐 × 𝑑) = ((𝑎 × 𝑏)𝑑)𝑐 − ((𝑎 × 𝑏)𝑐)𝑑 ,
𝑎 × (𝑏 × 𝑐) + 𝑏 × (𝑐 × 𝑎) + 𝑐 × (𝑎 × 𝑏) = 0 . (22)

The vector product exists only in vector spaces with three dimensions. We will explore
Vol. IV, page 234 more details on this connection later on.
The vector product is useful to describe systems that rotate – and (thus) also systems
with magnetic forces. The motion of an orbiting body is always perpendicular both to
the axis and to the line that connects the body with the axis. In rotation, axis, radius and
velocity form a right-handed set of mutually orthogonal vectors. This connection lies at

Motion Mountain – The Adventure of Physics
the origin of the vector product.
Challenge 198 e Confirm that the best way to calculate the vector product 𝑎 × 𝑏 component by com-
ponent is given by the symbolic determinant

󵄨󵄨𝑒 𝑎 𝑏 󵄨󵄨 󵄨󵄨 + − + 󵄨󵄨
󵄨󵄨 𝑥 𝑥 𝑥 󵄨󵄨 󵄨󵄨 󵄨󵄨
󵄨 󵄨 󵄨󵄨 󵄨
𝑎 × 𝑏 = 󵄨󵄨󵄨𝑒𝑦 𝑎𝑦 𝑏𝑦 󵄨󵄨󵄨 or, sloppily 𝑎 × 𝑏 = 󵄨󵄨󵄨𝑎𝑥 𝑎𝑦 𝑎𝑧 󵄨󵄨󵄨󵄨 . (23)
󵄨󵄨 󵄨 󵄨󵄨 𝑏 𝑏 𝑏 󵄨󵄨
󵄨󵄨𝑒𝑧 𝑎𝑧 𝑏𝑧 󵄨󵄨󵄨 󵄨󵄨 𝑥 𝑦 𝑧 󵄨󵄨

These symbolic determinants are easy to remember and easy to perform, both with letters

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
and with numerical values. (Here, 𝑒𝑥 is the unit basis vector in the 𝑥 direction.) Written
out, the symbolic determinants are equivalent to the relation

𝑎 × 𝑏 = (𝑎𝑦 𝑏𝑧 − 𝑏𝑦 𝑎𝑧 , 𝑏𝑥 𝑎𝑧 − 𝑎𝑥 𝑏𝑧 , 𝑎𝑥 𝑏𝑦 − 𝑏𝑥 𝑎𝑦 ) (24)

which is harder to remember, though.
Challenge 199 e Show that the parallelepiped spanned by three arbitrary vectors 𝑎, 𝑏 and 𝑐 has the
volume 𝑉 = 𝑐 (𝑎 × 𝑏). Show that the pyramid or tetrahedron formed by the same three
Challenge 200 e vectors has one sixth of that volume.

Rotation and angular momentum
Rotation keeps us alive. Without the change of day and night, we would be either fried or
frozen to death, depending on our location on our planet. But rotation appears in many
other settings, as Table 22 shows. A short exploration of rotation is thus appropriate.
All objects have the ability to rotate. We saw before that a body is described by its
reluctance to move, which we called mass; similarly, a body also has a reluctance to turn.
This quantity is called its moment of inertia and is often abbreviated Θ – pronounced
‘theta’. The speed or rate of rotation is described by angular velocity, usually abbreviated
𝜔 – pronounced ‘omega’. A few values found in nature are given in Table 22.
The observables that describe rotation are similar to those describing linear motion,
4 from objects and images to conservation 117

TA B L E 22 Some measured rotation frequencies.

O b s e r va t i o n Angular velocity
𝜔 = 2π/𝑇
Galactic rotation 2π ⋅ 0.14 ⋅ 10−15 / s
6
= 2π /(220 ⋅ 10 a)
Average Sun rotation around its axis 2π ⋅3.8 ⋅ 10−7 / s = 2π / 30 d
Typical lighthouse 2π ⋅ 0.08/ s
Pirouetting ballet dancer 2π ⋅ 3/ s
Ship’s diesel engine 2π ⋅ 5/ s
Helicopter rotor 2π ⋅ 5.3/ s
Washing machine up to 2π ⋅ 20/ s
Bacterial flagella 2π ⋅ 100/ s
Fast CD recorder up to 2π ⋅ 458/ s
Racing car engine up to 2π ⋅ 600/ s
Fastest turbine built 2π ⋅ 103 / s

Motion Mountain – The Adventure of Physics
Fastest pulsars (rotating stars) up to at least 2π ⋅ 716/ s
Ultracentrifuge > 2π ⋅ 3 ⋅ 103 / s
Dental drill up to 2π ⋅ 13 ⋅ 103 / s
Technical record 2π ⋅ 333 ⋅ 103 / s
Proton rotation 2π ⋅ 1020 / s
Highest possible, Planck angular velocity 2π⋅ 1035 / s

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
as shown in Table 24. Like mass, the moment of inertia is defined in such a way that the
sum of angular momenta 𝐿 – the product of moment of inertia and angular velocity – is
conserved in systems that do not interact with the outside world:

∑ Θ𝑖 𝜔𝑖 = ∑ 𝐿 𝑖 = const . (25)
𝑖 𝑖

In the same way that the conservation of linear momentum defines mass, the conser-
vation of angular momentum defines the moment of inertia. Angular momentum is a
concept introduced in the 1730s and 1740s by Leonhard Euler and Daniel Bernoulli.
The moment of inertia can be related to the mass and shape of a body. If the body is
imagined to consist of small parts or mass elements, the resulting expression is

Θ = ∑ 𝑚𝑛 𝑟𝑛2 , (26)
𝑛

where 𝑟𝑛 is the distance from the mass element 𝑚𝑛 to the axis of rotation. Can you con-
Challenge 201 e firm the expression? Therefore, the moment of inertia of a body depends on the chosen
Challenge 202 s axis of rotation. Can you confirm that this is so for a brick?
In contrast to the case of mass, there is no conservation of the moment of inertia.
In fact, the value of the moment of inertia depends both on the direction and on the
118 4 from objects and images to conservation

TA B L E 23 Some measured angular momentum values.

O b s e r va t i o n Angular momentum

Smallest observed value in nature, ℏ/2, in elementary 0.53 ⋅ 10−34 Js
matter particles (fermions)
Spinning top 5 ⋅ 10−6 Js
CD (compact disc) playing c. 0.029 Js
Walking man (around body axis) c. 4 Js
Dancer in a pirouette 5 Js
Typical car wheel at 30 m/s 10 Js
Typical wind generator at 12 m/s (6 Beaufort) 104 Js
Earth’s atmosphere 1 to 2 ⋅ 1026 Js
Earth’s oceans 5 ⋅ 1024 Js
Earth around its axis 7.1 ⋅ 1033 Js
Moon around Earth 2.9 ⋅ 1034 Js

Motion Mountain – The Adventure of Physics
Earth around Sun 2.7 ⋅ 1040 Js
Sun around its axis 1.1 ⋅ 1042 Js
Jupiter around Sun 1.9 ⋅ 1043 Js
Solar System around Sun 3.2 ⋅ 1043 Js
Milky Way 1068 Js
All masses in the universe 0 (within measurement error)

TA B L E 24 Correspondence between linear and rotational motion.

Q ua nt i t y Linear motion R o tat i o na l

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
motion
State time 𝑡 time 𝑡
position 𝑥 angle 𝜑
momentum 𝑝 = 𝑚𝑣 angular momentum 𝐿 = Θ𝜔
energy 𝑚𝑣2 /2 energy Θ𝜔2 /2
Motion velocity 𝑣 angular velocity 𝜔
acceleration 𝑎 angular acceleration 𝛼
Reluctance to move mass 𝑚 moment of inertia Θ
Motion change force 𝑚𝑎 torque Θ𝛼

location of the axis used for its definition. For each axis direction, one distinguishes an
intrinsic moment of inertia, when the axis passes through the centre of mass of the body,
from an extrinsic moment of inertia, when it does not.* In the same way, we distinguish

* Extrinsic and intrinsic moment of inertia are related by

Θext = Θint + 𝑚𝑑2 , (27)

where 𝑑 is the distance between the centre of mass and the axis of extrinsic rotation. This relation is called
Challenge 203 s Steiner’s parallel axis theorem. Are you able to deduce it?
4 from objects and images to conservation 119

middle finger: "r x p"
2
𝐿 = 𝑟 × 𝑝 = Θ𝜔 = 𝑚𝑟 𝜔 fingers in
rotation
sense;
thumb
𝑟 shows
index: "p"
𝐴 angular
thumb: "r" momentum
𝑝 = 𝑚𝑣 = 𝑚𝜔 × 𝑟
F I G U R E 75 Angular momentum and other quantities for a point particle in circular motion, and the
two versions of the right-hand rule.

Motion Mountain – The Adventure of Physics
frictionless
axis

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 76 Can the ape reach the F I G U R E 77 How a snake turns itself around its axis.
banana?

intrinsic and extrinsic angular momenta. (By the way, the centre of mass of a body is that
imaginary point which moves straight during vertical fall, even if the body is rotating.
Challenge 204 s Can you find a way to determine its location for a specific body?)
We now define the rotational energy as

𝐿2
𝐸rot = 1
2
Θ 𝜔2 = . (28)
2Θ
The expression is similar to the expression for the kinetic energy of a particle. For rotating
objects with a fixed shape, rotational energy is conserved.
Can you guess how much larger the rotational energy of the Earth is compared with
Challenge 205 s the yearly electricity usage of humanity? In fact, if you could find a way to harness the
Earth’s rotational energy, you would become famous.
Every object that has an orientation also has an intrinsic angular momentum. (What
Challenge 206 s about a sphere?) Therefore, point particles do not have intrinsic angular momenta – at
120 4 from objects and images to conservation

least in classical physics. (This statement will change in quantum theory.) The extrinsic
angular momentum 𝐿 of a point particle is defined as

𝐿=𝑟×𝑝 (29)

where 𝑝 is the momentum of the particle and 𝑟 the position vector. The angular mo-
mentum thus points along the rotation axis, following the right-hand rule, as shown in
Figure 75. A few values observed in nature are given in Table 23. The definition implies
Challenge 207 e that the angular momentum can also be determined using the expression

2𝐴(𝑡)𝑚
𝐿= , (30)
𝑡
where 𝐴(𝑡) is the area swept by the position vector 𝑟 of the particle during time 𝑡. For
example, by determining the swept area with the help of his telescope, Johannes Kepler
discovered in the year 1609 that each planet orbiting the Sun has an angular momentum

Motion Mountain – The Adventure of Physics
value that is constant over time.
A physical body can rotate simultaneously about several axes. The film of Figure 108
Page 148 shows an example: The top rotates around its body axis and around the vertical at the
same time. A detailed exploration shows that the exact rotation of the top is given by
Challenge 208 e the vector sum of these two rotations. To find out, ‘freeze’ the changing rotation axis at
Page 161 a specific time. Rotations thus are a type of vectors.
As in the case of linear motion, rotational energy and angular momentum are not
always conserved in the macroscopic world: rotational energy can change due to fric-
tion, and angular momentum can change due to external forces (torques). But for closed
(undisturbed) systems, both angular momentum and rotational energy are always con-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
served. In particular, on a microscopic scale, most objects are undisturbed, so that con-
servation of rotational energy and angular momentum usually holds on microscopic
scales.
Angular momentum is conserved. This statement is valid for any axis of a physical
system, provided that external forces (torques) play no role. To make the point, Jean-Marc
Ref. 2 Lévy-Leblond poses the problem of Figure 76. Can the ape reach the banana without
leaving the plate, assuming that the plate on which the ape rests can turn around the axis
Challenge 209 s without any friction?
We note that many effects of rotation are the same as for acceleration: both accelera-
tion and rotation of a car pushed us in our seats. Therefore, many sensors for rotation are
Page 86 the same as the acceleration sensors we explored above. But a few sensors for rotation
Page 141 are fundamentally new. In particular, we will meet the gyroscope shortly.
On a frictionless surface, as approximated by smooth ice or by a marble floor covered
by a layer of oil, it is impossible to move forward. In order to move, we need to push
against something. Is this also the case for rotation?
Surprisingly, it is possible to turn even without pushing against something. You can
check this on a well-oiled rotating office chair: simply rotate an arm above the head. After
each turn of the hand, the orientation of the chair has changed by a small amount. In-
deed, conservation of angular momentum and of rotational energy do not prevent bodies
from changing their orientation. Cats learn this in their youth. After they have learned
4 from objects and images to conservation 121

ωr ω d = vp
P ωR

r ω R = vaxis

d
R

F I G U R E 78 The velocities and unit vectors F I G U R E 79 A simulated photograph of a
for a rolling wheel. rolling wheel with spokes.

Motion Mountain – The Adventure of Physics
the trick, if cats are dropped legs up, they can turn themselves in such a way that they
Ref. 94 always land feet first. Snakes also know how to rotate themselves, as Figure 77 shows.
Also humans have the ability: during the Olympic Games you can watch board divers
and gymnasts perform similar tricks. Rotation thus differs from translation in this im-
Challenge 210 d portant aspect. (Why?)

Rolling wheels
Rotation is an interesting phenomenon in many ways. A rolling wheel does not turn

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
around its axis, but around its point of contact. Let us show this.
A wheel of radius 𝑅 is rolling if the speed of the axis 𝑣axis is related to the angular
velocity 𝜔 by
𝑣
𝜔 = axis . (31)
𝑅
For any point P on the wheel, with distance 𝑟 from the axis, the velocity 𝑣P is the sum
of the motion of the axis and the motion around the axis. Figure 78 shows that 𝑣P is
orthogonal to 𝑑, the distance between the point P and the contact point of the wheel.
Challenge 211 e The figure also shows that the length ratio between 𝑣P and 𝑑 is the same as between 𝑣axis
and 𝑅. As a result, we can write
𝑣P = 𝜔 × 𝑑 , (32)

which shows that a rolling wheel does indeed rotate about its point of contact with the
ground.
Surprisingly, when a wheel rolls, some points on it move towards the wheel’s axis,
some stay at a fixed distance and others move away from it. Can you determine where
Challenge 212 s these various points are located? Together, they lead to an interesting pattern when a
Ref. 95 rolling wheel with spokes, such as a bicycle wheel, is photographed, as show in Figure 79.
Ref. 96 With these results you can tackle the following beautiful challenge. When a turning
bicycle wheel is deposed on a slippery surface, it will slip for a while, then slip and roll,
122 4 from objects and images to conservation

F I G U R E 80 The
measured motion of a
walking human (© Ray
McCoy).

Motion Mountain – The Adventure of Physics
and finally roll only. How does the final speed depend on the initial speed and on the
Challenge 213 d friction?

How d o we walk and run?

“ ”
Golf is a good walk spoiled.
The Allens

Why do we move our arms when walking or running? To save energy or to be graceful? In
fact, whenever a body movement is performed with as little energy as possible, it is both

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
natural and graceful. This correspondence can indeed be taken as the actual definition
of grace. The connection is common knowledge in the world of dance; it is also a central
Ref. 20 aspect of the methods used by actors to learn how to move their bodies as beautifully as
possible.
To convince yourself about the energy savings, try walking or running with your arms
fixed or moving in the opposite direction to usual: the effort required is considerably
higher. In fact, when a leg is moved, it produces a torque around the body axis which
has to be counterbalanced. The method using the least energy is the swinging of arms, as
depicted in Figure 80. Since the arms are lighter than the legs, they must move further
from the axis of the body, to compensate for the momentum; evolution has therefore
moved the attachment of the arms, the shoulders, farther apart than those of the legs,
the hips. Animals on two legs but without arms, such as penguins or pigeons, have more
difficulty walking; they have to move their whole torso with every step.
Ref. 97 Measurements show that all walking animals follow

𝑣max walking = (2.2 ± 0.2 m/s) √𝑙/m . (33)

Indeed, walking, the moving of one leg after the other, can be described as a concaten-
ation of (inverted) pendulum swings. The pendulum length is given by the leg length 𝑙.
The typical time scale of a pendulum is 𝑡 ∼ √𝑙/𝑔 . The maximum speed of walking then
4 from objects and images to conservation 123

becomes 𝑣 ∼ 𝑙/𝑡 ∼ √𝑔𝑙 , which is, up to a constant factor, the measured result.
Which muscles do most of the work when walking, the motion that experts call gait?
Ref. 98 In 1980, Serge Gracovetsky found that in human gait a large fraction of the power comes
from the muscles along the spine, not from those of the legs. (Indeed, people without
legs are also able to walk. However, a number of muscles in the legs must work in order
to walk normally.) When you take a step, the lumbar muscles straighten the spine; this
automatically makes it turn a bit to one side, so that the knee of the leg on that side
automatically comes forward. When the foot is moved, the lumbar muscles can relax,
and then straighten again for the next step. In fact, one can experience the increase in
Challenge 214 e tension in the back muscles when walking without moving the arms, thus confirming
where the human engine, the so-called spinal engine is located.
Human legs differ from those of apes in a fundamental aspect: humans are able to run.
In fact the whole human body has been optimized for running, an ability that no other
primate has. The human body has shed most of its hair to achieve better cooling, has
evolved the ability to run while keeping the head stable, has evolved the right length of
arms for proper balance when running, and even has a special ligament in the back that

Motion Mountain – The Adventure of Physics
works as a shock absorber while running. In other words, running is the most human of
all forms of motion.
Curiosities and fun challenges ab ou t mass, conservation and
rotation

“
It is a mathematical fact that the casting of this
pebble from my hand alters the centre of gravity

”
of the universe.
Thomas Carlyle,* Sartor Resartus III.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
A cup with water is placed on a weighing scale, as shown in Figure 81. How does the
Challenge 216 e mass result change if you let a piece of metal attached to a string hang into the water?
∗∗
Take ten coins of the same denomination. Put nine of them on a table and form a closed
loop with them of any shape you like, as shown in Figure 82. (The nine coins thus look
like a section of pearl necklace where the pearls touch each other.) Now take the tenth
coin and let it roll around the loop, thus without ever sliding it. How many turns does
Challenge 217 e this last coin make during one round?
∗∗
Conservation of momentum is best studied by playing and exploring billiards, snooker or
pool. The best introduction is provided by the trickshot films found across the internet.
Are you able to use momentum conservation to deduce ways for improving your billiards
Challenge 218 e game?
Another way to explore momentum conservation is to explore the ball-chain, or ball
collision pendulum, that was invented by Edme Mariotte. Decades later, Newton claimed
it as his, as he often did with other people’s results. Playing with the toy is fun – and
explaining its behaviour even more. Indeed, if you lift and let go three balls on one side,
Challenge 215 s * Thomas Carlyle (1797–1881), Scottish essayist. Do you agree with the quotation?
124 4 from objects and images to conservation

Motion Mountain – The Adventure of Physics
F I G U R E 81 How does the displayed weight value change when an object hangs into the water?

How many rotations?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 82 How many rotations does the tenth coin
perform in one round?

you will see three balls departing on the other side; for the explanation of this behaviour
the conservation of momentum and energy conservation are not sufficient, as you should
Challenge 219 d be able to find out. Are you able to build a high-precision ball-chain?
∗∗
Challenge 220 s There is a well-known way to experience 81 sunrises in just 80 days. How?
∗∗
4 from objects and images to conservation 125

F I G U R E 83 The ball-chain or cradle invented by
Mariotte allows exploring momentum
conservation, energy conservation, and the
difﬁculties of precision manufacturing (© www.
questacon.edu.au).

Motion Mountain – The Adventure of Physics
Walking is a source of many physics problems. When climbing a mountain, the most
Ref. 99 energy-effective way is not always to follow the steepest ascent; indeed, for steep slopes,
zig-zagging is more energy efficient. Why? And can you estimate the slope angle at which
Challenge 221 s this will happen?
∗∗
Asterix and his friends from the homonymous comic strip, fear only one thing: that the
Challenge 222 e sky might fall down. Is the sky an object? An image?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Death is a physical process and thus can be explored. In general, animals have a lifespan
𝑇 that scales with fourth root of their mass 𝑀. In other terms, 𝑇 = 𝑀1/4 . This is valid
from bacteria to insects to blue whales. Animals also have a power consumption per
mass, or metabolic rate per mass, that scales with the inverse fourth root. We conclude
that death occurs for all animals when a certain fixed energy consumption per mass has
been achieved. This is indeed the case; death occurs for most animals when they have
Ref. 100 consumed around 1 GJ/kg. (But quite a bit later for humans.) This surprisingly simple
result is valid, on average, for all known animals.
Note that the argument is only valid when different species are compared. The de-
pendence on mass is not valid when specimen of the same species are compared. (You
cannot live longer by eating less.)
In short, animals die after they metabolized 1 GJ/kg. In other words, once we ate all
the calories we were designed for, we die.
∗∗
A car at a certain speed uses 7 litres of gasoline per 100 km. What is the combined air
Challenge 223 s and rolling resistance? (Assume that the engine has an efficiency of 25 %.)
∗∗
A cork is attached to a thin string a metre long. The string is passed over a long rod
126 4 from objects and images to conservation

F I G U R E 84 Is it safe to let the cork go?

atmosphere
mountain
height h
ocean plain ocean
ocean

Motion Mountain – The Adventure of Physics
ocean crust solid
solidcontinental
continentalcrust
crust ocean crust

depth d
liquid
magma
of the mantle

F I G U R E 85 A simple model for continents and mountains.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
held horizontally, and a wine glass is attached at the other end. If you let go the cork in
Challenge 224 s Figure 84, nothing breaks. Why not? And what happens exactly?
∗∗
In 1907, Duncan MacDougalls, a medical doctor, measured the weight of dying people,
Ref. 101 in the hope to see whether death leads to a mass change. He found a sudden decrease
between 10 and 20 g at the moment of death. He attributed it to the soul exiting the body.
Challenge 225 s Can you find a more satisfying explanation?
∗∗
It is well known that the weight of a one-year-old child depends on whether it wants to
be carried or whether it wants to reach the floor. Does this contradict mass conservation?
Challenge 226 e

∗∗
The Earth’s crust is less dense (2.7 kg/l) than the Earth’s mantle (3.1 kg/l) and floats on
it. As a result, the lighter crust below a mountain ridge must be much deeper than below
a plain. If a mountain rises 1 km above the plain, how much deeper must the crust be
Challenge 227 s below it? The simple block model shown in Figure 85 works fairly well; first, it explains
why, near mountains, measurements of the deviation of free fall from the vertical line
4 from objects and images to conservation 127

lead to so much lower values than those expected without a deep crust. Later, sound
measurements confirmed directly that the continental crust is indeed thicker beneath
mountains.
∗∗
All homogeneous cylinders roll down an inclined plane in the same way. True or false?
Challenge 228 e And what about spheres? Can you show that spheres roll faster than cylinders?
∗∗
Challenge 229 s Which one rolls faster: a soda can filled with liquid or a soda can filled with ice? (And
how do you make a can filled with ice?)
∗∗
Take two cans of the same size and weight, one full of ravioli and one full of peas. Which
Challenge 230 e one rolls faster on an inclined plane?

Motion Mountain – The Adventure of Physics
∗∗
Another difference between matter and images: matter smells. In fact, the nose is a matter
sensor. The same can be said of the tongue and its sense of taste.
∗∗
Take a pile of coins. You can push out the coins, starting with the one at the bottom,
by shooting another coin over the table surface. The method also helps to visualize two-
Challenge 231 e dimensional momentum conservation.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In early 2004, two men and a woman earned £ 1.2 million in a single evening in a Lon-
don casino. They did so by applying the formulae of Galilean mechanics. They used the
method pioneered by various physicists in the 1950s who built various small computers
that could predict the outcome of a roulette ball from the initial velocity imparted by the
Ref. 102 croupier. In the case of Britain, the group added a laser scanner to a smartphone that
measured the path of a roulette ball and predicted the numbers where it would arrive.
In this way, they increased the odds from 1 in 37 to about 1 in 6. After six months of
investigations, Scotland Yard ruled that they could keep the money they won.
In fact around the same time, a few people earned around 400 000 euro over a few
weeks by using the same method in Germany, but with no computer at all. In certain
casinos, machines were throwing the roulette ball. By measuring the position of the zero
to the incoming ball with the naked eye, these gamblers were able to increase the odds
of the bets they placed during the last allowed seconds and thus win a considerable sum
purely through fast reactions.
∗∗
Challenge 232 s Does the universe rotate?
∗∗
The toy of Figure 86 shows interesting behaviour: when a number of spheres are lifted
128 4 from objects and images to conservation

before the hit observed after the hit

F I G U R E 86 A
well-known toy.

before the hit observed after the hit
V=0 v V‘ v’ 0
F I G U R E 87 An
elastic collision
2L,2M L, M that seems not to

Motion Mountain – The Adventure of Physics
obey energy
conservation.

and dropped to hit the resting ones, the same number of spheres detach on the other side,
whereas the previously dropped spheres remain motionless. At first sight, all this seems
to follow from energy and momentum conservation. However, energy and momentum
conservation provide only two equations, which are insufficient to explain or determine
the behaviour of five spheres. Why then do the spheres behave in this way? And why do

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 233 d they all swing in phase when a longer time has passed?
∗∗
A surprising effect is used in home tools such as hammer drills. We remember that when
a small ball elastically hits a large one at rest, both balls move after the hit, and the small
Ref. 103 one obviously moves faster than the large one. Despite this result, when a short cylinder
hits a long one of the same diameter and material, but with a length that is some integer
multiple of that of the short one, something strange happens. After the hit, the small
cylinder remains almost at rest, whereas the large one moves, as shown in Figure 87.
Even though the collision is elastic, conservation of energy seems not to hold in this
case. (In fact this is the reason that demonstrations of elastic collisions in schools are
Challenge 234 d always performed with spheres.) What happens to the energy?
∗∗
Is the structure shown in Figure 88 possible?
∗∗
Does a wall get a stronger jolt when it is hit by a ball rebounding from it or when it is hit
Challenge 235 s by a ball that remains stuck to it?
∗∗
4 from objects and images to conservation 129

wall

ladder

F I G U R E 88 Is this possible? F I G U R E 89 How does the ladder
fall?

Motion Mountain – The Adventure of Physics
Housewives know how to extract a cork of a wine bottle using a cloth or a shoe. Can you
Challenge 236 s imagine how? They also know how to extract the cork with the cloth if the cork has fallen
inside the bottle. How?
∗∗
The sliding ladder problem, shown schematically in Figure 89, asks for the detailed mo-
tion of the ladder over time. The problem is more difficult than it looks, even if friction
is not taken into account. Can you say whether the lower end always touches the floor,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 237 s or if is lifted into the air for a short time interval?
∗∗
A homogeneous ladder of length 5 m and mass 30 kg leans on a wall. The angle is 30°; the
static friction coefficient on the wall is negligible, and on the floor it is 0.3. A person of
mass 60 kg climbs the ladder. What is the maximum height the person can climb before
the ladder starts sliding? This and many puzzles about ladders can be found on www.
mathematische-basteleien.de/leiter.htm.
∗∗
Ref. 104 A common fly on the stern of a 30 000 ton ship of 100 m length tilts it by less than the
diameter of an atom. Today, distances that small are easily measured. Can you think of
Challenge 238 s at least two methods, one of which should not cost more than 2000 euro?
∗∗
Is the image of three stacked spinning tops shown in Figure 90 a true photograph, show-
ing a real observation, or is it the result of digital composition, showing an impossible
Challenge 239 ny situation?
∗∗
Challenge 240 s How does the kinetic energy of a rifle bullet compare to that of a running man?
130 4 from objects and images to conservation

F I G U R E 90 Is this a possible situation or is it a fake photograph?
(© Wikimedia)

Motion Mountain – The Adventure of Physics
∗∗
Challenge 241 s What happens to the size of an egg when one places it in a jar of vinegar for a few days?
∗∗
What is the amplitude of a pendulum oscillating in such a way that the absolute value of
Challenge 242 s its acceleration at the lowest point and at the return point are equal?
∗∗
Can you confirm that the value of the acceleration of a drop of water falling through mist

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 243 d is 𝑔/7?
∗∗
You have two hollow spheres: they have the same weight, the same size and are painted in
the same colour. One is made of copper, the other of aluminium. Obviously, they fall with
Challenge 244 s the same speed and acceleration. What happens if they both roll down a tilted plane?
∗∗
Challenge 245 s What is the shape of a rope when rope jumping?
∗∗
Challenge 246 s How can you determine the speed of a rifle bullet with only a scale and a metre stick?
∗∗
Why does a gun make a hole in a door but cannot push it open, in exact contrast to what
Challenge 247 e a finger can do?
∗∗
Challenge 248 s What is the curve described by the midpoint of a ladder sliding down a wall?
∗∗
4 from objects and images to conservation 131

Motion Mountain – The Adventure of Physics
F I G U R E 91 A commercial clock that needs no special energy source, because it takes its energy from
the environment (© Jaeger-LeCoultre).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
A high-tech company, see www.enocean.com, sells electric switches for room lights that
have no cables and no power cell (battery). You can glue such a switch to the centre of a
Challenge 249 s window pane. How is this possible?
∗∗
For over 50 years now, a famous Swiss clockmaker is selling table clocks with a rotating
pendulum that need no battery and no manual rewinding, as they take up energy from
the environment. A specimen is shown in Figure 91. Can you imagine how this clock
Challenge 250 s works?
∗∗
Ship lifts, such as the one shown in Figure 92, are impressive machines. How does the
Challenge 251 s weight of the lift change when the ship enters?
∗∗
Challenge 252 e How do you measure the mass of a ship?
∗∗
All masses are measured by comparing them, directly or indirectly, to the standard kilo-
gram in Sèvres near Paris. For a number of years, there was serious doubt that the stand-
132 4 from objects and images to conservation

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 92 The spectacular ship lift at Strépy-Thieux in Belgium. What engine power is needed to lift a
ship, if the right and left lifts were connected by ropes or by a hydraulic system? (© Jean-Marie
Hoornaert)
4 from objects and images to conservation 133

F I G U R E 93 The famous Celtic wobble stone – above and right – and a version made by bending a
spoon – bottom left (© Ed Keath).

Motion Mountain – The Adventure of Physics
ard kilogram was losing weight, possibly through outgassing, with an estimated rate of
around 0.5 μg/a. This was an awkward situation, and there has been a vast, worldwide
effort to find a better definition of the kilogram. Such an improved definition had to be
simple, precise, and make trips to Sèvres unnecessary. Finally, an alternative was found
and defined in 2019.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Which engine is more efficient: a moped or a human on a bicycle?
∗∗
Both mass and moment of inertia can be defined and measured both with and without
Challenge 253 e contact. Can you do so?
∗∗
Figure 93 shows the so-called Celtic wobble stone, also called anagyre or rattleback, a stone
Ref. 103 that starts rotating on a plane surface when it is put into up-and-down oscillation. The
size can vary between a few centimetres and a few metres. By simply bending a spoon
one can realize a primitive form of this strange device, if the bend is not completely
symmetrical. The rotation is always in the same direction. If the stone is put into rotation
in the wrong direction, after a while it stops and starts rotating in the other sense! Can
Challenge 254 d you explain the effect that seems to contradict the conservation of angular momentum?
∗∗
A beautiful effect, the chain fountain, was discovered in 2013 by Steve Mould. Certain
chains, when flowing out of a container, first shoot up in the air. See the video at www.
youtube.com/embed/_dQJBBklpQQ and the story of the discovery at stevemould.com.
Challenge 255 ny Can you explain the effect to your grandmother?
134 4 from objects and images to conservation

Summary on conservation in motion

“
The gods are not as rich as one might think:
what they give to one, they take away from the

”
other.
Antiquity

We have encountered four conservation principles that are valid for the motion of all
closed systems in everyday life:
— conservation of total linear momentum,
— conservation of total angular momentum,
— conservation of total energy,
— conservation of total mass.
None of these conservation principles applies to the motion of images. These principles
thus allow us to distinguish objects from images.
The conservation principles are among the great results in science. They limit the sur-
prises that nature can offer: conservation means that linear momentum, angular mo-

Motion Mountain – The Adventure of Physics
mentum, and mass–energy can neither be created from nothing, nor can they disappear
into nothing. Conservation limits creation. The quote below the section title expresses
this idea.
Page 280 Later on we will find out that these results could have been deduced from three simple
observations: closed systems behave the same independently of where they are, in what
direction they are oriented and of the time at which they are set up. In more abstract
terms, physicists like to say that all conservation principles are consequences of the in-
variances, or symmetries, of nature.
Later on, the theory of special relativity will show that energy and mass are conserved
only when taken together. Many adventures still await us.

F ROM T H E ROTAT ION OF T H E E A RT H
TO T H E R E L AT I V I T Y OF MOT ION

“ ”
Eppur si muove!
Anonymous**

I
s the Earth rotating? The search for definite answers to this question gives an
nteresting cross-section of the history of classical physics. In the fourth century,

Motion Mountain – The Adventure of Physics
n ancient Greece, Hicetas and Philolaus, already stated that the Earth rotates. Then,
in the year 265 b ce, Aristarchus of Samos was the first to explore the issue in detail. He
had measured the parallax of the Moon (today known to be up to 0.95°) and of the Sun
(today known to be 8.8 󸀠 ).*** The parallax is an interesting effect; it is the angle describing
the difference between the directions of a body in the sky when seen by an observer on
the surface of the Earth and when seen by a hypothetical observer at the Earth’s centre.
(See Figure 94.) Aristarchus noticed that the Moon and the Sun wobble across the sky,
and this wobble has a period of 24 hours. He concluded that the Earth rotates. It seems
that Aristarchus received death threats for his conclusion.
Aristarchus’ observation yields an even more powerful argument than the trails of the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 256 e stars shown in Figure 95. Can you explain why? (And how do the trails look at the most
Challenge 257 s populated places on Earth?)
Experiencing Figure 95 might be one reason that people dreamt and still dream about
reaching the poles. Because the rotation and the motion of the Earth make the poles ex-
tremely cold places, the adventure of reaching them is not easy. Many tried unsuccess-
Ref. 136 fully. A famous crook, Robert Peary, claimed to have reached the North Pole in 1909. (In
fact, Roald Amundsen reached both the South and the North Pole first.) Among oth-
ers, Peary claimed to have taken a picture there, but that picture, which went round the
Challenge 258 s world, turned out to be one of the proofs that he had not been there. Can you imagine
how?
If the Earth rotates instead of being at rest, said the unconvinced, the speed at the
equator has the substantial value of 0.46 km/s. How did Galileo explain why we do not
Challenge 259 e feel or notice this speed?
Vol. II, page 17 Measurements of the aberration of light also show the rotation of the Earth; it can
be detected with a telescope while looking at the stars. The aberration is a change of the

** ‘And yet she moves’ is the sentence about the Earth attributed, most probably incorrectly, to Galileo since
the 1640s. It is true, however, that at his trial he was forced to publicly retract the statement of a moving
Earth to save his life. For more details of this famous story, see the section on page 335.
*** For the definition of the concept of angle, see page 67, and for the definition of the measurement units
for angle see Appendix B.
136 5 from the rotation of the earth

rotating Moon sky
Earth or and
Sun stars

F I G U R E 94 The parallax – not drawn
to scale.

expected light direction, which we will discuss shortly. At the Equator, Earth rotation
adds an angular deviation of 0.32 󸀠 , changing sign every 12 hours, to the aberration due to

Motion Mountain – The Adventure of Physics
the motion of the Earth around the Sun, about 20.5 󸀠 . In modern times, astronomers have
found a number of additional proofs for the rotation of the Earth, but none is accessible
to the man on the street.
Also the measurements showing that the Earth is not a sphere, but is flattened at the
poles, confirmed the rotation of the Earth. Figure 96 illustrates the situation. Again, how-
ever, this eighteenth-century measurement by Maupertuis* is not accessible to everyday
observation.
Then, in the years 1790 to 1792 in Bologna, Giovanni Battista Guglielmini (1763–1817)
finally succeeded in measuring what Galileo and Newton had predicted to be the simplest

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
proof for the Earth’s rotation. On the rotating Earth, objects do not fall vertically, but are
slightly deviated to the east. This deviation appears because an object keeps the larger ho-
rizontal velocity it had at the height from which it started falling, as shown in Figure 97.
Guglielmini’s result was the first non-astronomical proof of the Earth’s rotation. The
experiments were repeated in 1802 by Johann Friedrich Benzenberg (1777–1846). Using
metal balls which he dropped from the Michaelis tower in Hamburg – a height of 76 m –
Benzenberg found that the deviation to the east was 9.6 mm. Can you confirm that the
value measured by Benzenberg almost agrees with the assumption that the Earth turns
Challenge 260 d once every 24 hours? There is also a much smaller deviation towards the Equator, not
measured by Guglielmini, Benzenberg or anybody after them up to this day; however, it
completes the list of effects on free fall by the rotation of the Earth.
Both deviations from vertical fall are easily understood if we use the result (described
Page 192 below) that falling objects describe an ellipse around the centre of the rotating Earth.
The elliptical shape shows that the path of a thrown stone does not lie on a plane for an

* Pierre Louis Moreau de Maupertuis (1698–1759), physicist and mathematician, was one of the key figures
in the quest for the principle of least action, which he named in this way. He was also the founding president
of the Berlin Academy of Sciences. Maupertuis thought that the principle reflected the maximization of
goodness in the universe. This idea was thoroughly ridiculed by Voltaire in his Histoire du Docteur Akakia
et du natif de Saint-Malo, 1753. (Read it at gallica.bnf.fr/ark:/12148/bpt6k6548988f.texteImage.) Maupertuis
performed his measurement of the Earth to distinguish between the theory of gravitation of Newton and
that of Descartes, who had predicted that the Earth is elongated at the poles, instead of flattened.
to the relativity of motion 137

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 95 The motion of the stars during the night, observed on 1 May 2012 from the South Pole,
together with the green light of an aurora australis (© Robert Schwartz).

sphere

Earth

5 km 5 km
Equator

F I G U R E 96 Earth’s deviation from spherical shape
due to its rotation (exaggerated).

observer standing on Earth; for such an observer, the exact path of a stone thus cannot
138 5 from the rotation of the earth

𝑣ℎ = 𝜔(𝑅 + ℎ)
ℎ North ℎ

𝑣 = 𝜔𝑅
𝜑

North
Equator
East F I G U R E 97 The
deviations of free fall
towards the east and
South towards the Equator due
to the rotation of the
Earth.

Motion Mountain – The Adventure of Physics
c. 0.2 Hz

F I G U R E 98 A typical carousel allows observing the Coriolis effect in its most striking appearance: if a
person lets a ball roll with the proper speed and direction, the ball is deﬂected so strongly that it comes
back to her.

be drawn on a flat piece of paper!
In 1798, Pierre Simon Laplace* explained how bodies move on the rotating Earth and
Ref. 106 showed that they feel an apparent force. In 1835, Gustave-Gaspard Coriolis then reformu-
lated and simplified the description. Imagine a ball that rolls over a table. For a person on
the floor, the ball rolls in a straight line. Now imagine that the table rotates. For the per-
son on the floor, the ball still rolls in a straight line. But for a person on the rotating table,
the ball traces a curved path. In short, any object that travels in a rotating background is
subject to a transversal acceleration. The acceleration, discovered by Laplace, is nowadays
called Coriolis acceleration or Coriolis effect. On a rotating background, travelling objects
deviate from the straight line. The best way to understand the Coriolis effect is to exper-

* Pierre Simon Laplace (b. 1749 Beaumont-en-Auge, d. 1827 Paris), important mathematician. His famous
treatise Traité de mécanique céleste appeared in five volumes between 1798 and 1825. He was the first to
propose that the Solar System was formed from a rotating gas cloud, and one of the first people to imagine
and explore black holes.
to the relativity of motion 139

Motion Mountain – The Adventure of Physics
F I G U R E 99 Cyclones, with their low pressure centre, differ in rotation sense between the southern
hemisphere, here cyclone Larry in 2006, and the northern hemisphere, here hurricane Katrina in 2005.
(Courtesy NOAA)

ience it yourself; this can be done on a carousel, as shown in Figure 98. Watching films

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 107 on the internet on the topic is also helpful. You will notice that on a rotating carousel it
is not easy to hit a target by throwing or rolling a ball.
Also the Earth is a rotating background. On the northern hemisphere, the rotation is
anticlockwise. As the result, any moving object is slightly deviated to the right (while the
magnitude of its velocity stays constant). On Earth, like on all rotating backgrounds, the
Coriolis acceleration 𝑎C results from the change of distance to the rotation axis. Can you
Challenge 261 s deduce the analytical expression for the Coriolis effect, namely 𝑎C = −2𝜔 × 𝑣?
On Earth, the Coriolis acceleration generally has a small value. Therefore it is best ob-
served either in large-scale or high-speed phenomena. Indeed, the Coriolis acceleration
determines the handedness of many large-scale phenomena with a spiral shape, such as
the directions of cyclones and anticyclones in meteorology – as shown in Figure 99 – the
general wind patterns on Earth and the deflection of ocean currents and tides. These phe-
nomena have opposite handedness on the northern and the southern hemisphere. Most
beautifully, the Coriolis acceleration explains why icebergs do not follow the direction
Ref. 108 of the wind as they drift away from the polar caps. The Coriolis acceleration also plays
a role in the flight of cannon balls (that was the original interest of Coriolis), in satellite
Ref. 109 launches, in the motion of sunspots and even in the motion of electrons in molecules.
All these Coriolis accelerations are of opposite sign on the northern and southern hemi-
spheres and thus prove the rotation of the Earth. For example, in the First World War,
many naval guns missed their targets in the southern hemisphere because the engineers
had compensated them for the Coriolis effect in the northern hemisphere.
140 5 from the rotation of the earth

𝜓1
N

Earth's
centre 𝜑
Eq
u ato
r
𝜓0
𝜓1 F I G U R E 100 The
turning motion of a
pendulum showing the
rotation of the Earth.

Only in 1962, after several earlier attempts by other researchers, Asher Shapiro was the

Motion Mountain – The Adventure of Physics
Ref. 110
first to verify that the Coriolis effect has a tiny influence on the direction of the vortex
formed by the water flowing out of a bath-tub. Instead of a normal bath-tub, he had to use
a carefully designed experimental set-up because, contrary to an often-heard assertion,
no such effect can be seen in a real bath-tub. He succeeded only by carefully eliminat-
ing all disturbances from the system; for example, he waited 24 hours after the filling of
the reservoir (and never actually stepped in or out of it!) in order to avoid any left-over
motion of water that would disturb the effect, and built a carefully designed, completely
rotationally-symmetric opening mechanism. Others have repeated the experiment in the
Ref. 110 southern hemisphere, finding opposite rotation direction and thus confirming the result.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In other words, the handedness of usual bath-tub vortices is not caused by the rotation
of the Earth, but results from the way the water starts to flow out. (A number of crooks
in Quito, a city located on the Equator, show gullible tourists that the vortex in a sink
changes when crossing the Equator line drawn on the road.) But let us go on with the
story about the Earth’s rotation.
In 1851, the physician-turned-physicist Jean Bernard Léon Foucault (b. 1819 Paris,
d. 1868 Paris) performed an experiment that removed all doubts and rendered him
world-famous practically overnight. He suspended a 67 m long pendulum* in the
Panthéon in Paris and showed the astonished public that the direction of its swing
changed over time, rotating slowly. To anybody with a few minutes of patience to watch
the change of direction, the experiment proved that the Earth rotates. If the Earth did
not rotate, the swing of the pendulum would always continue in the same direction. On
a rotating Earth, in Paris, the direction changes to the right, in clockwise sense, as shown
in Figure 100. The swing direction does not change if the pendulum is located at the
Equator, and it changes to the left in the southern hemisphere.** A modern version of the

Challenge 262 d * Why was such a long pendulum necessary? Understanding the reasons allows one to repeat the experiment
Ref. 111 at home, using a pendulum as short as 70 cm, with the help of a few tricks. To observe Foucault’s effect with
a simple set-up, attach a pendulum to your office chair and rotate the chair slowly. Several pendulum an-
imations, with exaggerated deviation, can be found at commons.wikimedia.org/wiki/Foucault_pendulum.
** The discovery also shows how precision and genius go together. In fact, the first person to observe the
to the relativity of motion 141

pendulum can be observed via the web cam at pendelcam.kip.uni-heidelberg.de; high
speed films of the pendulum’s motion during day and night can also be downloaded at
www.kip.uni-heidelberg.de/oeffwiss/pendel/zeitraffer/.
The time over which the orientation of the pendulum’s swing performs a full turn –
the precession time – can be calculated. Study a pendulum starting to swing in the North–
Challenge 263 d South direction and you will find that the precession time 𝑇Foucault is given by
23 h 56 min
𝑇Foucault = (34)
sin 𝜑

where 𝜑 is the latitude of the location of the pendulum, e.g. 0° at the Equator and 90° at
the North Pole. This formula is one of the most beautiful results of Galilean kinematics.*
Foucault was also the inventor and namer of the gyroscope. He built the device, shown
in Figure 101 and Figure 102, in 1852, one year after his pendulum. With it, he again
demonstrated the rotation of the Earth. Once a gyroscope rotates, the axis stays fixed in
space – but only when seen from distant stars or galaxies. (By the way, this is not the
same as talking about absolute space. Why?) For an observer on Earth, the axis direction

Motion Mountain – The Adventure of Physics
Challenge 264 s
changes regularly with a period of 24 hours. Gyroscopes are now routinely used in ships
and in aeroplanes to give the direction of north, because they are more precise and more
reliable than magnetic compasses. The most modern versions use laser light running in
circles instead of rotating masses.**
In 1909, Roland von Eötvös measured a small but surprising effect: due to the rotation
of the Earth, the weight of an object depends on the direction in which it moves. As a
result, a balance in rotation around the vertical axis does not stay perfectly horizontal:
Challenge 266 s the balance starts to oscillate slightly. Can you explain the origin of the effect?
Ref. 112 In 1910, John Hagen published the results of an even simpler experiment, proposed

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
by Louis Poinsot in 1851. Two masses are put on a horizontal bar that can turn around a
vertical axis, a so-called isotomeograph. Its total mass was 260 kg. If the two masses are
slowly moved towards the support, as shown in Figure 103, and if the friction is kept low
enough, the bar rotates. Obviously, this would not happen if the Earth were not rotating.
Challenge 267 s Can you explain the observation? This little-known effect is also useful for winning bets
between physicists.
In 1913, Arthur Compton showed that a closed tube filled with water and some small
Ref. 113 floating particles (or bubbles) can be used to show the rotation of the Earth. The device is
called a Compton tube or Compton wheel. Compton showed that when a horizontal tube
filled with water is rotated by 180°, something happens that allows one to prove that the
Earth rotates. The experiment, shown in Figure 104, even allows measuring the latitude
Challenge 268 d of the point where the experiment is made. Can you guess what happens?
Another method to detect the rotation of the Earth using light was first realized in
1913 Georges Sagnac:*** he used an interferometer to produce bright and dark fringes
effect was Vincenzo Viviani, a student of Galileo, as early as 1661! Indeed, Foucault had read about Viviani’s
work in the publications of the Academia dei Lincei. But it took Foucault’s genius to connect the effect to
the rotation of the Earth; nobody had done so before him.
* The calculation of the period of Foucault’s pendulum assumes that the precession rate is constant during
a rotation. This is only an approximation (though usually a good one).
Challenge 265 s ** Can you guess how rotation is detected in this case?
*** Georges Sagnac (b. 1869 Périgeux, d. 1928 Meudon-Bellevue) was a physicist in Lille and Paris, friend
142 5 from the rotation of the earth

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 101 The gyroscope: the original system by Foucault with its freely movable spinning top, the
mechanical device to bring it to speed, the optical device to detect its motion, the general construction
principle, and a modern (triangular) ring laser gyroscope, based on colour change of rotating laser light
instead of angular changes of a rotating mass (© CNAM, JAXA).

Vol. IV, page 56 of light with two light beams, one circulating in clockwise direction, and the second
circulating in anticlockwise direction. The interference fringes are shifted when the whole
Ref. 114 system rotates; the faster it rotates, the larger the shift. A modern, high-precision version
of the experiment, which uses lasers instead of lamps, is shown in Figure 105. (More
Vol. III, page 104 details on interference and fringes are found in volume III.) Sagnac also determined the
relationship between the fringe shift and the details of the experiment. The rotation of a
complete ring interferometer with angular frequency (vector) Ω produces a fringe shift
Challenge 269 s of angular phase Δ𝜑 given by
8π Ω 𝑎
Δ𝜑 = (35)
𝑐𝜆
where 𝑎 is the area (vector) enclosed by the two interfering light rays, 𝜆 their wavelength
and 𝑐 the speed of light. The effect is now called the Sagnac effect after its discoverer. It had
of the Curies, Langevin, Perrin, and Borel. Sagnac also deduced from his experiment that the speed of light
was independent of the speed of its source, and thus confirmed a prediction of special relativity.
to the relativity of motion 143

Motion Mountain – The Adventure of Physics
F I G U R E 102 A
three-dimensional
model of Foucault’s
original gyroscope: in
the pdf verion of this
text, the model can
be rotated and
zoomed by moving

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the cursor over it
(© Zach Joseph
Espiritu).
144 5 from the rotation of the earth

North

water- 𝑟
𝑚 𝑚 filled East
tube

West

South

Motion Mountain – The Adventure of Physics
F I G U R E 103 Showing the rotation of the Earth F I G U R E 104 Demonstrating the rotation of the
through the rotation of an axis. Earth with water.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 105 A modern precision ring laser interferometer (© Bundesamt für Kartographie und
Geodäsie, Carl Zeiss).

Ref. 115 already been predicted 20 years earlier by Oliver Lodge.* Today, Sagnac interferometers
are the central part of laser gyroscopes – shown in Figure 101 – and are found in every
passenger aeroplane, missile and submarine, in order to measure the changes of their
motion and thus to determine their actual position.
A part of the fringe shift is due to the rotation of the Earth. Modern high-precision
Sagnac interferometers use ring lasers with areas of a few square metres, as shown in

* Oliver Lodge (b. 1851, Stoke, d. on-Trent-1940, Wiltshire) was a physicist and spiritualist who studied
electromagnetic waves and tried to communicate with the dead. A strange but influential figure, his ideas
are often cited when fun needs to be made of physicists; for example, he was one of those (rare) physicists
who believed that at the end of the nineteenth-century physics was complete.
to the relativity of motion 145

mirror

massive metal rod

typically 1.5 m F I G U R E 106 Observing the
rotation of the Earth in two
seconds.

Motion Mountain – The Adventure of Physics
Figure 105. Such a ring interferometer is able to measure variations in the rotation rates
of the Earth of less than one part per million. Indeed, over the course of a year, the length
of a day varies irregularly by a few milliseconds, mostly due to influences from the Sun or
Ref. 116 the Moon, due to weather changes and due to hot magma flows deep inside the Earth.*
But also earthquakes, the El Niño effect in the climate and the filling of large water dams
have effects on the rotation of the Earth. All these effects can be studied with such high-
precision interferometers; they can also be used for research into the motion of the soil
due to lunar tides or earthquakes, and for checks on the theory of special relativity.
Finally, in 1948, Hans Bucka developed the simplest experiment so far to show the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 117 rotation of the Earth. A metal rod allows anybody to detect the rotation of the Earth
after only a few seconds of observation, using the set-up of Figure 106. The experiment
Challenge 270 s can be easily be performed in class. Can you guess how it works?
In summary: all observations show that the Earth surface rotates at 464 m/s at the
Equator, a larger value than that of the speed of sound in air, which is about 340 m/s at
usual conditions. The rotation of the Earth also implies an acceleration, at the Equator,
of 0.034 m/s2 . We are in fact whirling through the universe.

How d oes the E arth rotate?
Is the rotation of the Earth, the length of the day, constant over geological time scales?
That is a hard question. If you find a method leading to an answer, publish it! (The same
Ref. 118 is true for the question of whether the length of the year is constant.) Only a few methods
are known, as we will find out shortly.
The rotation of the Earth is not even constant during a human lifespan. It varies by a
few parts in 108 . In particular, on a ‘secular’ time scale, the length of the day increases by
about 1 to 2 ms per century, mainly because of the friction by the Moon and the melting
of the polar ice caps. This was deduced by studying historical astronomical observations

* The growth of leaves on trees and the consequent change in the Earth’s moment of inertia, already thought
of in 1916 by Harold Jeffreys, is way too small to be seen, as it is hidden by larger effects.
146 5 from the rotation of the earth

Ref. 119 of the ancient Babylonian and Arab astronomers. Additional ‘decadic’ changes have an
amplitude of 4 or 5 ms and are due to the motion of the liquid part of the Earth’s core.
(The centre of the Earth’s core is solid; this was discovered in 1936 by the Danish seis-
mologist Inge Lehmann (1888–1993); her discovery was confirmed most impressively
by two British seismologists in 2008, who detected shear waves of the inner core, thus
confirming Lehmann’s conclusion. There is a liquid core around the solid core.)
The seasonal and biannual changes of the length of the day – with an amplitude of
0.4 ms over six months, another 0.5 ms over the year, and 0.08 ms over 24 to 26 months
– are mainly due to the effects of the atmosphere. In the 1950s the availability of precision
measurements showed that there is even a 14 and a 28 day period with an amplitude
of 0.2 ms, due to the Moon. In the 1970s, when wind oscillations with a length scale of
about 50 days were discovered, they were also found to alter the length of the day, with
an amplitude of about 0.25 ms. However, these last variations are quite irregular.
Also the oceans influence the rotation of the Earth, due to the tides, ocean currents,
Ref. 120 wind forcing, and atmospheric pressure forcing. Further effects are due to the ice sheet
variations and due to water evaporation and rainfalls. Last but not least, flows in the

Motion Mountain – The Adventure of Physics
interior of the Earth, both in the mantle and in the core, change the rotation. For example,
earthquakes, plate motion, post-glacial rebound and volcanic eruptions all influence the
rotation.
But why does the Earth rotate at all? The rotation originated in the rotating gas cloud
at the origin of the Solar System. This connection explains that the Sun and all planets,
except two, turn around their axes in the same direction, and that they also all orbit the
Ref. 121 Sun in that same direction. But the complete story is outside the scope of this text.
The rotation around its axis is not the only motion of the Earth; it performs other
motions as well. This was already known long ago. In 128 b ce, the Greek astronomer

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Hipparchos discovered what is today called the (equinoctial) precession. He compared a
measurement he made himself with another made 169 years before. Hipparchos found
that the Earth’s axis points to different stars at different times. He concluded that the
sky was moving. Today we prefer to say that the axis of the Earth is moving. (Why?)
Challenge 271 e During a period of 25 800 years the axis draws a cone with an opening angle of 23.5°.
This motion, shown in Figure 107, is generated by the tidal forces of the Moon and the
Sun on the equatorial bulge of the Earth that result from its flattening. The Sun and the
Moon try to align the axis of the Earth at right angles to the Earth’s path; this torque
leads to the precession of the Earth’s axis.
Precession is a motion common to all rotating systems: it appears in planets, spinning
tops and atoms. (Precession is also at the basis of the surprise related to the suspended
wheel shown on page 244.) Precession is most easily seen in spinning tops, be they sus-
pended or not. An example is shown in Figure 108; for atomic nuclei or planets, just
imagine that the suspending wire is missing and the rotating body less flattened. On the
Earth, precession leads to the upwelling of deep water in the equatorial Atlantic Ocean
and regularly changes the ecology of algae.
In addition, the axis of the Earth is not even fixed relative to the Earth’s surface.
In 1884, by measuring the exact angle above the horizon of the celestial North Pole,
Friedrich Küstner (1856–1936) found that the axis of the Earth moves with respect to
the Earth’s crust, as Bessel had suggested 40 years earlier. As a consequence of Küstner’s
discovery, the International Latitude Service was created. The polar motion Küstner dis-
to the relativity of motion 147

nutation period
year 15000: is 18.6 years year 2000:
North pole is North pole is
Vega in Polaris in
Lyra Ursa minor

precession

N
Moon’s path
Moon

equatorial

Motion Mountain – The Adventure of Physics
bulge
Eq equatorial
uat bulge
or

S
Earth’s path

F I G U R E 107 The precession and the nutation of the Earth’s axis.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
covered turned out to consist of three components: a small linear drift – not yet under-
stood – a yearly elliptical motion due to seasonal changes of the air and water masses,
and a circular motion* with a period of about 1.2 years due to fluctuations in the pres-
sure at the bottom of the oceans. In practice, the North Pole moves with an amplitude
Ref. 122 of about 15 m around an average central position, as shown in Figure 109. Short-term
variations of the North Pole position, due to local variations in atmospheric pressure, to
Ref. 123 weather changes and to the tides, have also been measured. The high precision of the GPS
system is possible only with the help of the exact position of the Earth’s axis; and only
with this knowledge can artificial satellites be guided to Mars or other planets.
The details of the motion of the Earth have been studied in great detail. Table 25 gives
an overview of the knowledge and precision that is available today.
In 1912, the meteorologist and geophysicist Alfred Wegener (1880–1930) discovered
an even larger effect. After studying the shapes of the continental shelves and the geolo-
gical layers on both sides of the Atlantic, he conjectured that the continents move, and
that they are all fragments of a single continent that broke up 200 million years ago.**

* The circular motion, a wobble, was predicted by the great Swiss mathematician Leonhard Euler (1707–
1783). In a disgusting story, using Euler’s and Bessel’s predictions and Küstner’s data, in 1891 Seth Chandler
claimed to be the discoverer of the circular component.
** In this old continent, called Gondwanaland, there was a huge river that flowed westwards from Chad
148 5 from the rotation of the earth

F I G U R E 108
Precession of
a suspended
spinning top

Motion Mountain – The Adventure of Physics
(mpg ﬁlm
© Lucas
Barbosa)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 109 The motion of the North Pole – roughly speaking, the Earth’s effective polhode – from
2003 to 2007, including the prediction until 2008 (left) and the average position since 1900 (right) –
with 0.1 arcsecond being around 3.1 m on the surface of the Earth – not showing the diurnal and
semidiurnal variations of a fraction of a millisecond of arc due to the tides (from hpiers.obspm.fr/
eop-pc).

Even though at first derided across the world, Wegener’s discoveries were correct.
Modern satellite measurements, shown in Figure 110, confirm this model. For example,
the American continent moves away from the European continent by about 23 mm every

to Guayaquil in Ecuador. After the continent split up, this river still flowed to the west. When the Andes
appeared, the water was blocked, and many millions of years later, it reversed its flow. Today, the river still
flows eastwards: it is called the Amazon River.
to the relativity of motion 149

F I G U R E 110 The continental plates are
the objects of tectonic motion (HoloGlobe

Motion Mountain – The Adventure of Physics
project, NASA).

F I G U R E 111 The tectonic plates of the Earth, with the relative speeds at the boundaries. (© NASA)

year, as shown in Figure 111. There are also speculations that this velocity may have been
much higher at certain periods in the past. The way to check this is to look at the mag-
netization of sedimental rocks. At present, this is still a hot topic of research. Following
the modern version of the model, called plate tectonics, the continents (with a density of
150 5 from the rotation of the earth

F I G U R E 112 The angular size of the Sun
changes due to the elliptical motion of the
Earth (© Anthony Ayiomamitis).

Motion Mountain – The Adventure of Physics
F I G U R E 113 Friedrich Wilhelm Bessel (1784–1846).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Page 126 2.7 ⋅ 103 kg/m3 ) float on the fluid mantle of the Earth (with a density of 3.1 ⋅ 103 kg/m3 )
Vol. III, page 223 like pieces of cork on water, and the convection inside the mantle provides the driving
Ref. 124 mechanism for the motion.

Does the E arth move?
Also the centre of the Earth is not at rest in the universe. In the third century b ce
Aristarchus of Samos maintained that the Earth turns around the Sun. Experiments such
as that of Figure 112 confirm that the orbit is an ellipse. However, a fundamental difficulty
of the heliocentric system is that the stars look the same all year long. How can this be,
if the Earth travels around the Sun? The distance between the Earth and the Sun has
been known since the seventeenth century, but it was only in 1837 that Friedrich Wil-
helm Bessel* became the first person to observe the parallax of a star. This was a result
of extremely careful measurements and complex calculations: he discovered the Bessel
functions in order to realize it. He was able to find a star, 61 Cygni, whose apparent pos-
ition changed with the month of the year. Seen over the whole year, the star describes a
small ellipse in the sky, with an opening of 0.588 󸀠󸀠 (this is the modern value). After care-

* Friedrich Wilhelm Bessel (1784–1846), Westphalian astronomer who left a successful business career to
dedicate his life to the stars, and became the foremost astronomer of his time.
to the relativity of motion 151

TA B L E 25 Modern measurement data about the motion of the Earth (from hpiers.obspm.fr/eop-pc).

O b s e r va b l e Symbol Va l u e
Mean angular velocity of Earth Ω 72.921 150(1) μrad/s
Nominal angular velocity of Earth (epoch ΩN 72.921 151 467 064 μrad/s
1820)
Conventional mean solar day (epoch 1820) d 86 400 s
Conventional sidereal day dsi 86 164.090 530 832 88 s
Ratio conv. mean solar day to conv. sidereal day 𝑘 = d/dsi 1.002 737 909 350 795
Conventional duration of the stellar day dst 86 164.098 903 691 s
Ratio conv. mean solar day to conv. stellar day 𝑘󸀠 = d/dst 1.002 737 811 911 354 48
General precession in longitude 𝑝 5.028 792(2) 󸀠󸀠 /a
Obliquity of the ecliptic (epoch 2000) 𝜀0 23° 26 󸀠 21.4119 󸀠󸀠
Küstner-Chandler period in terrestrial frame 𝑇KC 433.1(1.7) d
Quality factor of the Küstner-Chandler peak 𝑄KC 170

Motion Mountain – The Adventure of Physics
Free core nutation period in celestial frame 𝑇F 430.2(3) d
Quality factor of the free core nutation 𝑄F 2 ⋅ 104
Astronomical unit AU 149 597 870.691(6) km
Sidereal year (epoch 2000) 𝑎si 365.256 363 004 d
= 365 d 6 h 9 min 9.76 s
Tropical year 𝑎tr 365.242 190 402 d
= 365 d 5 h 48 min 45.25 s
Mean Moon period 𝑇M 27.321 661 55(1) d
Earth’s equatorial radius 𝑎 6 378 136.6(1) m
8.0101(2) ⋅ 1037 kg m2

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
First equatorial moment of inertia 𝐴
Longitude of principal inertia axis 𝐴 𝜆𝐴 −14.9291(10)°
Second equatorial moment of inertia 𝐵 8.0103(2) ⋅ 1037 kg m2
Axial moment of inertia 𝐶 8.0365(2) ⋅ 1037 kg m2
Equatorial moment of inertia of mantle 𝐴m 7.0165 ⋅ 1037 kg m2
Axial moment of inertia of mantle 𝐶m 7.0400 ⋅ 1037 kg m2
Earth’s flattening 𝑓 1/298.25642(1)
Astronomical Earth’s dynamical flattening ℎ = (𝐶 − 𝐴)/𝐶 0.003 273 794 9(1)
Geophysical Earth’s dynamical flattening 𝑒 = (𝐶 − 𝐴)/𝐴 0.003 284 547 9(1)
Earth’s core dynamical flattening 𝑒f 0.002 646(2)
Second degree term in Earth’s gravity potential 𝐽2 = −(𝐴 + 𝐵 − 1.082 635 9(1) ⋅ 10−3
2𝐶)/(2𝑀𝑅2 )
Secular rate of 𝐽2 d𝐽2 /d𝑡 −2.6(3) ⋅ 10−11 /a
Love number (measures shape distortion by 𝑘2 0.3
tides)
Secular Love number 𝑘s 0.9383
Mean equatorial gravity 𝑔eq 9.780 3278(10) m/s2
Geocentric constant of gravitation 𝐺𝑀 3.986 004 418(8) ⋅ 1014 m3 /s2
Heliocentric constant of gravitation 𝐺𝑀⊙ 1.327 124 420 76(50) ⋅ 1020 m3 /s2
Moon-to-Earth mass ratio 𝜇 0.012 300 038 3(5)
152 5 from the rotation of the earth

precession ellipticity change
rotation axis

Earth
Sun

Sun

rotation tilt change
axis

Earth perihelion shift
Sun

Motion Mountain – The Adventure of Physics
Earth
orbital inclination change
P
Sun
P
Sun

F I G U R E 114 Changes in the Earth’s motion around the Sun, as seen from different observers outside
the orbital plane.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
fully eliminating all other possible explanations, he deduced that the change of position
was due to the motion of the Earth around the Sun, when seen from distant stars. From
the size of the ellipse he determined the distance to the star to be 105 Pm, or 11.1 light
Challenge 272 s years.
Bessel had thus managed, for the first time, to measure the distance of a star. By doing
so he also proved that the Earth is not fixed with respect to the stars in the sky. The motion
of the Earth was not a surprise. It confirmed the result of the mentioned aberration of
Vol. II, page 17 light, discovered in 1728 by James Bradley* and to be discussed below. When seen from
the sky, the Earth indeed revolves around the Sun.
With the improvement of telescopes, other motions of the Earth were discovered. In
1748, James Bradley announced that there is a small regular change of the precession,
which he called nutation, with a period of 18.6 years and an angular amplitude of 19.2 󸀠󸀠 .
Nutation occurs because the plane of the Moon’s orbit around the Earth is not exactly

* James Bradley (b. 1693 Sherborne , d. 1762 Chalford), was an important astronomer. He was one of the first
astronomers to understand the value of precise measurement, and thoroughly modernized the Greenwich
observatory. He discovered, independently of Eustachio Manfredi, the aberration of light, and showed with
it that the Earth moves. In particular, the discovery allowed him to measure the speed of light and confirm
the value of 0.3 Gm/s. He later discovered the nutation of the Earth’s axis.
to the relativity of motion 153

the same as the plane of the Earth’s orbit around the Sun. Are you able to confirm that
Challenge 273 e this situation produces nutation?
Astronomers also discovered that the 23.5° tilt – or obliquity – of the Earth’s axis,
the angle between its intrinsic and its orbital angular momentum, actually changes from
22.1° to 24.5° with a period of 41 000 years. This motion is due to the attraction of the Sun
and the deviations of the Earth from a spherical shape. In 1941, during the Second World
War, the Serbian astronomer Milutin Milankovitch (1879–1958) retreated into solitude
and explored the consequences. In his studies he realized that this 41 000 year period of
the obliquity, together with an average period of 22 000 years due to precession,* gives
rise to more than 20 ice ages in the last 2 million years. This happens through stronger
or weaker irradiation of the poles by the Sun. The changing amounts of melted ice then
lead to changes in average temperature. The last ice age had its peak about 20 000 years
ago and ended around 11 800 years ago; the next is still far away. A spectacular confirm-
ation of the relation between ice age cycles and astronomy came through measurements
of oxygen isotope ratios in ice cores and sea sediments, which allow the average temper-
Ref. 125 ature over the past million years to be tracked. Figure 115 shows how closely the temper-

Motion Mountain – The Adventure of Physics
ature follows the changes in irradiation due to changes in obliquity and precession.
The Earth’s orbit also changes its eccentricity with time, from completely circular to
slightly oval and back. However, this happens in very complex ways, not with periodic
regularity, and is due to the influence of the large planets of the solar system on the
Earth’s orbit. The typical time scale is 100 000 to 125 000 years.
In addition, the Earth’s orbit changes in inclination with respect to the orbits of the
other planets; this seems to happen regularly every 100 000 years. In this period the in-
clination changes from +2.5° to −2.5° and back.
Even the direction in which the ellipse points changes with time. This so-called peri-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
helion shift is due in large part to the influence of the other planets; a small remaining
part will be important in the chapter on general relativity. The perihelion shift of Mercury
was the first piece of data confirming Einstein’s theory.
Obviously, the length of the year also changes with time. The measured variations are
of the order of a few parts in 1011 or about 1 ms per year. However, knowledge of these
changes and of their origins is much less detailed than for the changes in the Earth’s
rotation.
The next step is to ask whether the Sun itself moves. Indeed it does. Locally, it moves
with a speed of 19.4 km/s towards the constellation of Hercules. This was shown by Wil-
liam Herschel in 1783. But globally, the motion is even more interesting. The diameter of
the galaxy is at least 100 000 light years, and we are located 26 000 light years from the
centre. (This has been known since 1918; the centre of the galaxy is located in the direc-
tion of Sagittarius.) At our position, the galaxy is 1 300 light years thick; presently, we are
Ref. 126 68 light years ‘above’ the centre plane. The Sun, and with it the Solar System, takes about
225 million years to turn once around the galactic centre, its orbital velocity being around
220 km/s. It seems that the Sun will continue moving away from the galaxy plane until it
is about 250 light years above the plane, and then move back, as shown in Figure 116. The
oscillation period is estimated to be around 62 million years, and has been suggested as

* In fact, the 25 800 year precession leads to three insolation periods, of 23 700, 22 400 and 19 000 years, due
to the interaction between precession and perihelion shift.
154 5 from the rotation of the earth

P re c e s s ion P a ra me te r ( 1 0 −3 )
A −40
−20
0
20
40
4
B
0
ΔT S ( ° C )

−4

−8

C
0

R F (W m )

Motion Mountain – The Adventure of Physics
−2
−1

−2

D 2
24. 0 0. 4

(W m )
−2
O bliquity ( ° )

(° C )
23. 5

obl
0 0. 0
23. 0

obl
ΔT S

RF
22.5 −0. 4
−2
0 100 200 300 400 500 600 700 800
Age (1000 years before present)
F I G U R E 115 Modern measurements showing how Earth’s precession parameter (black curve A) and
obliquity (black curve D) inﬂuence the average temperature (coloured curve B) and the irradiation of
the Earth (blue curve C) over the past 800 000 years: the obliquity deduced by Fourier analysis from the
irradiation data RF (blue curve D) and the obliquity deduced by Fourier analysis from the temperature
(red curve D) match the obliquity known from astronomical data (black curve D); sharp cooling events
took place whenever the obliquity rose while the precession parameter was falling (marked red below
the temperature curve) (© Jean Jouzel/Science from Ref. 125).

the mechanism for the mass extinctions of animal life on Earth, possibly because some
gas cloud or some cosmic radiation source may be periodically encountered on the way.
The issue is still a hot topic of research.
The motion of the Sun around the centre of the Milky Way implies that the planets
of the Solar System can be seen as forming helices around the Sun. Figure 117 illustrates
their helical path.
We turn around the galaxy centre because the formation of galaxies, like that of plan-
etary systems, always happens in a whirl. By the way, can you confirm from your own
to the relativity of motion 155

120 000 al = 1.2 Zm

our galaxy

orbit of our local star system

500 al = 5 Em Sun's path
F I G U R E 116 The
50 000 al = 500 Em
motion of the Sun
around the galaxy.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 117 The helical motion of the ﬁrst four planets around the path traced by the Sun during its
travel around the centre of the Milky Way. Brown: Mercury, white: Venus, blue: Earth, red: Mars.
(QuickTime ﬁlm © Rhys Taylor at www.rhysy.net).

Challenge 274 s observation that our galaxy itself rotates?
Finally, we can ask whether the galaxy itself moves. Its motion can indeed be observed
because it is possible to give a value for the motion of the Sun through the universe,
defining it as the motion against the background radiation. This value has been measured
Ref. 127 to be 370 km/s. (The velocity of the Earth through the background radiation of course
depends on the season.) This value is a combination of the motion of the Sun around
the galaxy centre and of the motion of the galaxy itself. This latter motion is due to the
156 5 from the rotation of the earth

Motion Mountain – The Adventure of Physics
F I G U R E 118 Driving through snowﬂakes shows the effects of relative motion in three dimensions.
Page 206 Similar effects are seen when the Earth speeds through the universe. (© Neil Provo at neilprovo.com).

gravitational attraction of the other, nearby galaxies in our local group of galaxies.*
In summary, the Earth really moves, and it does so in rather complex ways. As Henri

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Poincaré would say, if we are in a given spot today, say the Panthéon in Paris, and come
back to the same spot tomorrow at the same time, we are in fact 31 million kilometres
away. This state of affairs would make time travel extremely difficult even if it were pos-
sible (which it is not); whenever you went back to the past, you would have to get to the
old spot exactly!

Is velo cit y absolu te? – The theory of everyday relativity
Why don’t we feel all the motions of the Earth? The two parts of the answer were already
given in 1632. First of all, as Galileo explained, we do not feel the accelerations of the
Earth because the effects they produce are too small to be detected by our senses. Indeed,
many of the mentioned accelerations do induce measurable effects only in high-precision
Vol. II, page 155 experiments, e.g. in atomic clocks.
But the second point made by Galileo is equally important: it is impossible to feel the
high speed at which we are moving. We do not feel translational, unaccelerated motions
because this is impossible in principle. Galileo discussed the issue by comparing the ob-
servations of two observers: one on the ground and another on the most modern means
of unaccelerated transportation of the time, a ship. Galileo asked whether a man on the

* This is roughly the end of the ladder. Note that the expansion of the universe, to be studied later, produces
no motion.
to the relativity of motion 157

ground and a man in a ship moving at constant speed experience (or ‘feel’) anything
different. Einstein used observers in trains. Later it became fashionable to use travellers
Challenge 275 e in rockets. (What will come next?) Galileo explained that only relative velocities between
bodies produce effects, not the absolute values of the velocities. For the senses and for all
measurements we find:

⊳ There is no difference between constant, undisturbed motion, however rapid
it may be, and rest. This is called Galileo’s principle of relativity.

Indeed, in everyday life we feel motion only if the means of transportation trembles –
thus if it accelerates – or if we move against the air. Therefore Galileo concludes that
two observers in straight and undisturbed motion against each other cannot say who is
‘really’ moving. Whatever their relative speed, neither of them ‘feels’ in motion.*
Rest is relative. Or more clearly: rest is an observer-dependent concept. This result
of Galilean physics is so important that Poincaré introduced the expression ‘theory of
relativity’ and Einstein repeated the principle explicitly when he published his famous

Motion Mountain – The Adventure of Physics
theory of special relativity. However, these names are awkward. Galilean physics is also
a theory of relativity! The relativity of rest is common to all of physics; it is an essential
aspect of motion.
In summary, undisturbed or uniform motion has no observable effect; only change
of motion does. Velocity cannot be felt; acceleration can. As a result, every physicist can
deduce something simple about the following statement by Wittgenstein:

* In 1632, in his Dialogo, Galileo writes: ‘Shut yourself up with some friend in the main cabin below decks
on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have
a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all
sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and,
in throwing something to your friend, you need throw it no more strongly in one direction than another,
the distances being equal: jumping with your feet together, you pass equal spaces in every direction. When
you have observed all these things carefully (though there is no doubt that when the ship is standing still
everything must happen in this way), have the ship proceed with any speed you like, so long as the motion
is uniform and not fluctuating this way and that, you will discover not the least change in all the effects
named, nor could you tell from any of them whether the ship was moving or standing still. In jumping,
you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than
toward the prow even though the ship is moving quite rapidly, despite the fact that during the time you are
in the air the floor under you will be going in a direction opposite to your jump. In throwing something
to your companion, you will need no more force to get it to him whether he is in the direction of the bow
or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without
dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in
their water will swim toward the front of their bowl with no more effort than toward the back, and will
go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies
will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated
toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been
separated during long intervals by keeping themselves in the air. And if smoke is made by burning some
incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward
one side than the other. The cause of all these correspondences of effects is the fact that the ship’s motion is
common to all the things contained in it, and to the air also. That is why I said you should be below decks;
for if this took place above in the open air, which would not follow the course of the ship, more or less
noticeable differences would be seen in some of the effects noted.’ (Translation by Stillman Drake)
158 5 from the rotation of the earth

Daß die Sonne morgen aufgehen wird, ist eine Hypothese; und das heißt:
wir wissen nicht, ob sie aufgehen wird.*

The statement is wrong. Can you explain why Wittgenstein erred here, despite his strong
Challenge 276 s desire not to?

Is rotation relative?
When we turn rapidly, our arms lift. Why does this happen? How can our body detect
whether we are rotating or not? There are two possible answers. The first approach, pro-
moted by Newton, is to say that there is an absolute space; whenever we rotate against
this space, the system reacts. The other answer is to note that whenever the arms lift,
the stars also rotate, and in exactly the same manner. In other words, our body detects
rotation because we move against the average mass distribution in space.
The most cited discussion of this question is due to Newton. Instead of arms, he ex-
plored the water in a rotating bucket. In a rotating bucket, the water surface forms a

Motion Mountain – The Adventure of Physics
concave shape, whereas the surface is flat for a non-rotating bucket. Newton asked why
this is the case. As usual for philosophical issues, Newton’s answer was guided by the
mysticism triggered by his father’s early death. Newton saw absolute space as a mystical
and religious concept and was not even able to conceive an alternative. Newton thus
saw rotation as an absolute type of motion. Most modern scientists have fewer personal
problems and more common sense than Newton; as a result, today’s consensus is that
rotation effects are due to the mass distribution in the universe:

⊳ Rotation is relative.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
A number of high-precision experiments confirm this conclusion; thus it is also part of
Einstein’s theory of relativity.

Curiosities and fun challenges ab ou t rotation and relativit y
When travelling in the train, you can test Galileo’s statement about the everyday relativ-
ity of motion. Close your eyes and ask somebody to turn you around several times: are
Challenge 277 e you able to say in which direction the train is running?
∗∗
A good bathroom scales, used to determine the weight of objects, does not show a con-
Challenge 278 s stant weight when you step on it and stay motionless. Why not?
∗∗
Challenge 279 s If a gun located at the Equator shoots a bullet vertically, where does the bullet fall?
∗∗
Challenge 280 s Why are most rocket launch sites as near as possible to the Equator?

* ‘That the Sun will rise to-morrow, is an hypothesis; and that means that we do not know whether it will
rise.’ This well-known statement is found in Ludwig Wittgenstein, Tractatus, 6.36311.
to the relativity of motion 159

∗∗
At the Equator, the speed of rotation of the Earth is 464 m/s, or about Mach 1.4; the latter
number means that it is 1.4 times the speed of sound. This supersonic motion has two
intriguing consequences.
First of all, the rotation speed determines the size of typical weather phenomena. This
size, the so-called Rossby radius, is given by the speed of sound (or some other typical
speed) divided by twice the local rotation speed, multiplied by the radius of the Earth.
At moderate latitudes, the Rossby radius is about 2000 km. This is a sizeable fraction
of the Earth’s radius, so that only a few large weather systems are present on Earth at
any specific time. If the Earth rotated more slowly, the weather would be determined by
short-lived, local flows and have no general regularities. If the Earth rotated more rapidly,
the weather would be much more violent – as on Jupiter – but the small Rossby radius
implies that large weather structures have a huge lifetime, such as the red spot on Jupiter,
which lasted for several centuries. In a sense, the rotation of the Earth has the speed that
provides the most interesting weather.
The other consequence of the value of the Earth’s rotation speed concerns the thick-

Motion Mountain – The Adventure of Physics
ness of the atmosphere. Mach 1 is also, roughly speaking, the thermal speed of air mo-
lecules. This speed is sufficient for an air molecule to reach the characteristic height of
the atmosphere, about 6 km. On the other hand, the speed of rotation Ω of the Earth de-
Page 137 termines its departure ℎ from sphericity: the Earth is flattened, as we saw above. Roughly
speaking, we have 𝑔ℎ = Ω2 𝑅2 /2, or about 12 km. (This is correct to within 50 %, the ac-
tual value is 21 km.) We thus find that the speed of rotation of the Earth implies that its
flattening is comparable to the thickness of the atmosphere.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The Coriolis effect influences rivers and their shores. This surprising connection was
made in 1860 by Karl Ernst von Baer who found that in Russia, many rivers flowing north
in lowlands had right shores that are steep and high, and left shores that are low and flat.
Challenge 281 e (Can you explain the details?) He also found that rivers in the southern hemisphere show
the opposite effect.
∗∗
The Coriolis effect saves lives and helps people. Indeed, it has an important application
for navigation systems; the typical uses are shown in Figure 119. Insects use vibrating
masses to stabilize their orientation, to determine their direction of travel and to find
their way. Most two-winged insects, or diptera, use vibrating halteres for navigation: in
particular, bees, house-flies, hover-flies and crane flies use them. Other insects, such as
moths, use vibrating antennae for navigation. Cars, satellites, mobile phones, remote-
controlled helicopter models, and computer games also use tiny vibrating masses as ori-
entation and navigation sensors, in exactly the same way as insects do.
In all these navigation applications, one or a few tiny masses are made to vibrate; if
the system to which they are attached turns, the change of orientation leads to a Coriolis
effect. The effect is measured by detecting the ensuing change in geometry; the change,
and thus the signal strength, depends on the angular velocity and its direction. Such
orientation sensors are therefore called vibrating Coriolis gyroscopes. Their development
160 5 from the rotation of the earth

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 119 The use of the Coriolis effect in insects – here a crane ﬂy and a hovering ﬂy – and in
micro-electromechanic systems (size about a few mm); all provide navigation signals to the systems to
which they are attached (© Pinzo, Sean McCann, ST Microelectronics).
to the relativity of motion 161

and production is a sizeable part of high-tech business – and of biological evolution.
∗∗
A wealthy and quirky customer asked his architect to plan and build a house whose four
Challenge 282 e walls all faced south. How did the architect realize the request?
∗∗
Would travelling through interplanetary space be healthy? People often fantasize about
long trips through the cosmos. Experiments have shown that on trips of long duration,
cosmic radiation, bone weakening, muscle degeneration and psychological problems are
the biggest dangers. Many medical experts question the viability of space travel lasting
longer than a couple of years. Other dangers are rapid sunburn, at least near the Sun,
and exposure to the vacuum. So far only one man has experienced vacuum without
Ref. 128 protection. He lost consciousness after 14 seconds, but survived unharmed.
∗∗

Motion Mountain – The Adventure of Physics
Challenge 283 s In which direction does a flame lean if it burns inside a jar on a rotating turntable?
∗∗
Galileo’s principle of everyday relativity states that it is impossible to determine an abso-
lute velocity. It is equally impossible to determine an absolute position, an absolute time
Challenge 284 s and an absolute direction. Is this correct?
∗∗
Does centrifugal acceleration exist? Most university students go through the shock of

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
meeting a teacher who says that it doesn’t because it is a ‘fictitious’ quantity, in the face
of what one experiences every day in a car when driving around a bend. Simply ask the
teacher who denies it to define ‘existence’. (The definition physicists usually use is given
Vol. III, page 324 later on.) Then check whether the definition applies to the term and make up your own
Challenge 285 s mind.
Whether you like the term ‘centrifugal acceleration’ or avoid it by using its negative,
the so-called centripetal acceleration, you should know how it is calculated. We use a
simple trick. For an object in circular motion of radius 𝑟, the magnitude 𝑣 of the velocity
𝑣 = d𝑥/d𝑡 is 𝑣 = 2π𝑟/𝑇. The vector 𝑣 behaves over time exactly like the position of the
object: it rotates continuously. Therefore, the magnitude 𝑎 of the centrifugal/centripetal
acceleration 𝑎 = d𝑣/d𝑡 is given by the corresponding expression, namely 𝑎 = 2π𝑣/𝑇.
Eliminating 𝑇, we find that the centrifugal/centripetal acceleration 𝑎 of a body rotating
at speed 𝑣 at radius 𝑟 is given by
𝑣2
𝑎= = 𝜔2 𝑟 . (36)
𝑟
This is the acceleration we feel when sitting in a car that goes around a bend.
∗∗
Rotational motion holds a little surprise for anybody who studies it carefully. Angular
momentum is a quantity with a magnitude and a direction. However, it is not a vector,
162 5 from the rotation of the earth

water
ping-pong ball
string
stone
F I G U R E 120 How does the ball move when the jar is
accelerated in direction of the arrow?

as any mirror shows. The angular momentum of a body circling in a plane parallel to a
mirror behaves in a different way from a usual arrow: its mirror image is not reflected
Challenge 286 e if it points towards the mirror! You can easily check this for yourself. For this reason,
Challenge 287 e angular momentum is called a pseudo-vector. (Are rotations pseudo-vectors?) The fact
Page 284 has no important consequences in classical physics; but we have to keep it in mind for
Vol. V, page 245 later, when we explore nuclear physics.

Motion Mountain – The Adventure of Physics
∗∗
What is the best way to transport a number of full coffee or tea cups while at the same
Challenge 288 s time avoiding spilling any precious liquid?
∗∗
A ping-pong ball is attached by a string to a stone, and the whole is put under water
in a jar. The set-up is shown in Figure 120. Now the jar is accelerated horizontally, for
Challenge 289 s example in a car. In which direction does the ball move? What do you deduce for a jar

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
at rest?
∗∗
The Moon recedes from the Earth by 3.8 cm a year, due to friction. Can you find the
Challenge 290 s mechanism responsible for the effect?
∗∗
What are earthquakes? Earthquakes are large examples of the same process that make a
door squeak. The continental plates correspond to the metal surfaces in the joints of the
door.
Earthquakes can be described as energy sources. The Richter scale is a direct measure
of this energy. The Richter magnitude 𝑀s of an earthquake, a pure number, is defined
from its energy 𝐸 in joule via

log(𝐸/1 J) − 4.8
𝑀s = . (37)
1.5
The strange numbers in the expression have been chosen to put the earthquake values as
near as possible to the older, qualitative Mercalli scale (now called EMS98) that classifies
the intensity of earthquakes. However, this is not fully possible; the most sensitive instru-
ments today detect earthquakes with magnitudes of −3. The highest value ever measured
to the relativity of motion 163

F I G U R E 121 What happens when the ape climbs?

Motion Mountain – The Adventure of Physics
was a Richter magnitude of 10, in Chile in 1960. Magnitudes above 12 are probably im-
Challenge 291 s possible. Can you show why?
∗∗
What is the motion of the point on the surface of the Earth that has Sun in its zenith –
i.e., vertically above it – when seen on a map of the Earth during one day? And day after
Challenge 292 ny day?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Can it happen that a satellite dish for geostationary TV satellites focuses the sunshine
Challenge 293 s onto the receiver?
∗∗
Why is it difficult to fire a rocket from an aeroplane in the direction opposite to the
Challenge 294 s motion of the plane?
∗∗
An ape hangs on a rope. The rope hangs over a wheel and is attached to a mass of equal
weight hanging down on the other side, as shown in Figure 121. The rope and the wheel
Challenge 295 s are massless and frictionless. What happens when the ape climbs the rope?
∗∗
Challenge 296 s Can a water skier move with a higher speed than the boat pulling him?
∗∗
You might know the ‘Dynabee’, a hand-held gyroscopic device that can be accelerated
Challenge 297 d to high speed by proper movements of the hand. How does it work?
∗∗
164 5 from the rotation of the earth

Motion Mountain – The Adventure of Physics
F I G U R E 122 The motion of the angular velocity of a tumbling brick. The tip of the angular velocity
vector moves along the yellow curve, the polhode. It moves together with the tumbling object, as does
the elliptical mesh representing the energy ellipsoid; the herpolhode is the white curve and lies in the
plane bright blue cross pattern that represents the invariable plane (see text). The animation behind the
screenshot illustrates the well-known nerd statement: the polhode rolls without slipping on the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
herpolhode lying in the invariable plane. The full animation is available online at www.ialms.net/sim/
3d-rigid-body-simulation/. Any rotating, irregular, free rigid body follows such a motion. For the Earth,
which is rotating and irregular, but neither fully free nor fully rigid, this description is only an
approximation. (© Svetoslav Zabunov)

It is possible to make a spinning top with a metal paper clip. It is even possible to make
Challenge 298 s one of those tops that turn onto their head when spinning. Can you find out how?
∗∗
The moment of inertia of a body depends on the shape of the body; usually, the angular
momentum and the angular velocity do not point in the same direction. Can you confirm
Challenge 299 s this with an example?
∗∗
Challenge 300 s What is the moment of inertia of a homogeneous sphere?
∗∗
The complete moment of inertia of a rigid body is determined by the values along its
three principal axes. These values are all equal for a sphere and for a cube. Does it mean
Challenge 301 s that it is impossible to distinguish a sphere from a cube by their inertial behaviour?
to the relativity of motion 165

∗∗
Here is some mathematical fun about the rotation of a free rigid body. Even for a free ro-
tating rigid body, such as a brick rotating in free space, the angular velocity is, in general,
not constant: the brick tumbles while rotating. In this motion, both energy and angular
momentum are constant, but the angular velocity is not. In particular, not even the dir-
ection of angular velocity is constant; in other words, the north pole changes with time.
In fact, the north pole changes with time both for an observer on the body and for an
observer in free space. How does the north pole, the end point of the angular velocity
vector, move?
Page 276 The moment of inertia is a tensor and can thus represented by an ellipsoid. In addition,
the motion of a free rigid body is described, in the angular velocity space, by the kinetic
energy ellipsoid – because its rotation energy is constant. When a free rigid body moves,
the energy ellipsoid – not the moment of inertia ellipsoid – rolls on the invariable plane
Challenge 302 e that is perpendicular to the initial (and constant) angular momentum of the body. This
is the mathematical description of the tumbling motion. The curve traced by the angular
velocity, or the extended north pole of the body, on the invariable plane is called the

Motion Mountain – The Adventure of Physics
herpolhode. It is an involved curve that results from two superposed conical motions.
For an observer on the rotating body, another curve is more interesting. The pole of
the rotating body traces a curve on the energy ellipsoid – which is itself attached to the
body. This curve is called the polhode. The polhode is a closed curve i three dimensions,
the herpolhode is an open curve in two dimensions. The curves were named by Louis
Poinsot and are especially useful for the description of the motion of rotating irregularly
shaped bodies. For an complete visualization, see the excellent website www.ialms.net/
sim/3d-rigid-body-simulation/.
The polhode is a circle only if the rigid body has rotational symmetry; the pole motion

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Page 147 is then called precession; we encountered it above. As shown in Figure 109, the measured
polhode of the Earth is not of the expected shape; this irregularity is due to several effects,
among them the non-rigidity of our planet.
Even though the angular momentum of a freely tumbling brick is fixed in space, it is
Challenge 303 e not fixed in the body frame. Can you confirm this? In the body frame, the end of the
angular momentum of a tumbling brick moves along still another curve, given by the
intersection of a sphere and an ellipsoid, as Jacques Binet pointed out.
∗∗
Is it true that the Moon in the first quarter in the northern hemisphere looks like the
Challenge 304 s Moon in the last quarter in the southern hemisphere?
∗∗
An impressive confirmation that the Earth is a sphere can be seen at sunset, if we turn,
against our usual habit, our back to the Sun. On the eastern sky we can then see the
impressive rise of the Earth’s shadow. We can admire the vast shadow rising over the
whole horizon, clearly having the shape of a segment of a huge circle. Figure 124 shows
an example. In fact, more precise investigations show that it is not the shadow of the
Earth alone, but the shadow of its ionosphere.
∗∗
166 5 from the rotation of the earth

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 123 Long exposures of the stars at night – one when facing north, above the Gemini
telescope in Hawaii, and one above the Alps that includes the celestial equator, with the geostationary
satellites on it (© Gemini Observatory/AURA, Michael Kunze).
to the relativity of motion 167

F I G U R E 124 The shadow of the Earth – here a panoramic photograph taken at the South Pole – shows

Motion Mountain – The Adventure of Physics
that the Earth is round (© Ian R. Rees).

Challenge 305 s How would Figure 123 look if taken at the Equator?
∗∗
Precision measurements show that not all planets move in exactly the same plane. Mer-
cury shows the largest deviation. In fact, no planet moves exactly in an ellipse, nor even
in a plane around the Sun. Almost all of these effects are too small and too complex to
explain here.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Since the Earth is round, there are many ways to drive from one point on the Earth to
another along a circle segment. This freedom of choice has interesting consequences for
volleyballs and for men watching women. Take a volleyball and look at its air inlet. If
you want to move the inlet to a different position with a simple rotation, you can choose
Challenge 306 e the rotation axis in many different ways. Can you confirm this? In other words, when we
look in a given direction and then want to look in another, the eye can accomplish this
change in different ways. The option chosen by the human eye had already been studied
by medical scientists in the eighteenth century. It is called Listing’s ‘law’.* It states that all
Challenge 307 s axes that nature chooses lie in one plane. Can you imagine its position in space? Many
men have a real interest that this mechanism is strictly followed; if not, looking at women
on the beach could cause the muscles moving their eyes to get knotted up.
∗∗
Imagine to cut open a soft mattress, glue a steel ball into it, and glue the mattress together
again. Now imagine that we use a magnetic field to rotate the steel ball glued inside. Intu-

* If you are interested in learning in more detail how nature and the eye cope with the complexities
of three dimensions, see the schorlab.berkeley.edu/vilis/whatisLL.htm and www.physpharm.fmd.uwo.ca/
undergrad/llconsequencesweb/ListingsLaw/perceptual1.htm websites.
168 5 from the rotation of the earth

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 125 A steel ball glued in a mattress can rotate forever. (QuickTime ﬁlm © Jason Hise).

itively, we think that the ball can only be rotated by a finite angle, whose value would be
limited by the elasticity of the mattress. But in reality, the steel ball can be rotated infin-
itely often! This surprising possibility is a consequence of the tethered rotation shown in
Page 90 Figure 54 and Figure 55. Such a continuous rotation in a mattress is shown in Figure 125.
Challenge 308 r And despite its fascination, nobody has yet realized the feat. Can you?

Legs or wheels? – Again
The acceleration and deceleration of standard wheel-driven cars is never much greater
than about 1 𝑔 = 9.8 m/s2 , the acceleration due to gravity on our planet. Higher accel-
erations are achieved by motorbikes and racing cars through the use of suspensions that
divert weight to the axes and by the use of spoilers, so that the car is pushed downwards
to the relativity of motion 169

F I G U R E 126 A basilisk lizard (Basiliscus basiliscus)
running on water, with a total length of about
25 cm, showing how the propulsing leg pushes
into the water (© TERRA).

with more than its own weight. Modern spoilers are so efficient in pushing a car towards
the track that racing cars could race on the roof of a tunnel without falling down.

Motion Mountain – The Adventure of Physics
Through the use of special tyres the downward forces produced by aerodynamic ef-
fects are transformed into grip; modern racing tyres allow forward, backward and side-
ways accelerations (necessary for speed increase, for braking and for turning corners) of
about 1.1 to 1.3 times the load. Engineers once believed that a factor 1 was the theoretical
limit and this limit is still sometimes found in textbooks; but advances in tyre technology,
mostly by making clever use of interlocking between the tyre and the road surface as in a
gear mechanism, have allowed engineers to achieve these higher values. The highest ac-
celerations, around 4 𝑔, are achieved when part of the tyre melts and glues to the surface.
Special tyres designed to make this happen are used for dragsters, but high-performance
radio-controlled model cars also achieve such values.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
How do wheels compare to using legs? High jump athletes can achieve peak ac-
celerations of about 2 to 4 𝑔, cheetahs over 3 𝑔, bushbabies up to 13 𝑔, locusts about
Ref. 129 18 𝑔, and fleas have been measured to accelerate about 135 𝑔. The maximum accelera-
tion known for animals is that of click beetles, a small insect able to accelerate at over
2000 m/s2 = 200 𝑔, about the same as an airgun pellet when fired. Legs are thus definit-
ively more efficient accelerating devices than wheels – a cheetah can easily beat any car
or motorbike – and evolution developed legs, instead of wheels, to improve the chances
of an animal in danger getting to safety.
In short, legs outperform wheels. But there are other reasons for using legs instead of
Challenge 309 s wheels. (Can you name some?) For example, legs, unlike wheels, allow walking on water.
Most famous for this ability is the basilisk, * a lizard living in Central America and shown
in Figure 126. This reptile is up to 70 cm long and has a mass of up to 500 g. It looks like a
miniature Tyrannosaurus rex and is able to run over water surfaces on its hind legs. The
motion has been studied in detail with high-speed cameras and by measurements using
aluminium models of the animal’s feet. The experiments show that the feet slapping on
Ref. 130 the water provides only 25 % of the force necessary to run above water; the other 75 % is
provided by a pocket of compressed air that the basilisks create between their feet and
the water once the feet are inside the water. In fact, basilisks mainly walk on air. (Both
* In the Middle Ages, the term ‘basilisk’ referred to a mythical monster supposed to appear shortly before
the end of the world. Today, it is a small reptile in the Americas.
170 5 from the rotation of the earth

F I G U R E 127 A water strider, total size F I G U R E 128 A water walking robot, total
about 10 mm (© Charles Lewallen). size about 20 mm (© AIP).

Ref. 131 effects used by basilisks are also found in fast canoeing.) It was calculated that humans
are also able to walk on water, provided their feet hit the water with a speed of 100 km/h

Motion Mountain – The Adventure of Physics
using the simultaneous physical power of 15 sprinters. Quite a feat for all those who ever
did so.
There is a second method of walking and running on water; this second method
even allows its users to remain immobile on top of the water surface. This is what water
striders, insects of the family Gerridae with an overall length of up to 15 mm, are able to
do (together with several species of spiders), as shown in Figure 127. Like all insects, the
water strider has six legs (spiders have eight). The water strider uses the back and front
legs to hover over the surface, helped by thousands of tiny hairs attached to its body. The
hairs, together with the surface tension of water, prevent the strider from getting wet. If

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
you put shampoo into the water, the water strider sinks and can no longer move.
The water strider uses its large middle legs as oars to advance over the surface, reach-
ing speeds of up to 1 m/s doing so. In short, water striders actually row over water. The
same mechanism is used by the small robots that can move over water and were de-
Ref. 132 veloped by Metin Sitti and his group, as shown in Figure 128.
Robot design is still in its infancy. No robot can walk or even run as fast as the animal
system it tries to copy. For two-legged robots, the most difficult kind, the speed record
is around 3.5 leg lengths per second. In fact, there is a race going on in robotics depart-
ments: each department tries to build the first robot that is faster, either in metres per
second or in leg lengths per second, than the original four-legged animal or two-legged
human. The difficulties of realizing this development goal show how complicated walk-
ing motion is and how well nature has optimized living systems.
Legs pose many interesting problems. Engineers know that a staircase is comfortable
to walk only if for each step the depth 𝑙 plus twice the height ℎ is a constant: 𝑙 + 2ℎ =
Challenge 310 s 0.63 ± 0.02 m. This is the so-called staircase formula. Why does it hold?
Challenge 311 s Most animals have an even number of legs. Do you know an exception? Why not? In
fact, one can argue that no animal has less than four legs. Why is this the case?
On the other hand, all animals with two legs have legs side by side, whereas most
systems with two wheels have them one behind the other. Why is this not the other way
Challenge 312 e round?
Legs are very efficient actuators. As Figure 129 shows, most small animals can run
to the relativity of motion 171

100

Relative running speed (body length/s)

Motion Mountain – The Adventure of Physics
1

0.01 0.1 1 10 100 1000 10000
Body mass (kg)
F I G U R E 129 The graph shows how the relative running speed changes with the mass of terrestrial
mammal species, for 142 different species. The graph also illustrates how the running performance
changes above 30 kg. Filled squares show Rodentia; open squares show Primata; ﬁlled diamonds
Proboscidae; open diamonds Marsupialia; ﬁlled triangles Carnivora; open triangles Artiodactyla; ﬁlled
circles Perissodactyla; open circles Lagomorpha (© José Iriarte-Díaz/JEB).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 133 with about 25 body lengths per second. For comparison, almost no car achieves such a
speed. Only animals that weigh more than about 30 kg, including humans, are slower.
Legs also provide simple distance rulers: just count your steps. In 2006, it was dis-
covered that this method is used by certain ant species, such as Cataglyphis fortis. They
Ref. 134 can count to at least 25 000, as shown by Matthias Wittlinger and his team. These ants
use the ability to find the shortest way back to their home even in structureless desert
terrain.
Ref. 135 Why do 100 m sprinters run faster than ordinary people? A thorough investigation
shows that the speed 𝑣 of a sprinter is given by

𝐹c
𝑣 = 𝑓 𝐿 stride = 𝑓 𝐿 c , (38)
𝑊
where 𝑓 is the frequency of the legs, 𝐿 stride is the stride length, 𝐿 c is the contact length –
the length that the sprinter advances during the time the foot is in contact with the floor
– 𝑊 the weight of the sprinter, and 𝐹c the average force the sprinter exerts on the floor
during contact. It turns out that the frequency 𝑓 is almost the same for all sprinters; the
only way to be faster than the competition is to increase the stride length 𝐿 stride . Also the
contact length 𝐿 c varies little between athletes. Increasing the stride length thus requires
that the athlete hits the ground with strong strokes. This is what athletic training for
172 5 from the rotation of the earth

sprinters has to achieve.

Summary on Galilean relativit y
The Earth rotates. The acceleration is so small that we do not feel it. The speed of rotation
is large, but we do not feel it, because there is no way to do so.
Undisturbed or inertial motion cannot be felt or measured. It is thus impossible to
distinguish motion from rest. The distinction between rest and motion depends on the
observer: Motion of bodies is relative. That is why the soil below our feet seems so stable
to us, even though it moves with high speed across the universe.
Since motion is relative, speed values depend on the observer. Later on, we will dis-
cover that one example of motion in nature has a speed value that is not relative: the mo-
tion of light. But we continue first with the study of motion transmitted over distance,
without the use of any contact at all.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Chapter 6

MOT ION DU E TO G R AV I TAT ION

“ ”
Caddi come corpo morto cade.
Dante, Inferno, c. V, v. 142.**

T
he first and main method to generate motion without any contact
hat we discover in our environment is height. Waterfalls, snow, rain,

Motion Mountain – The Adventure of Physics
he ball of your favourite game and falling apples all rely on it. It was one of
the fundamental discoveries of physics that height has this property because there is
an interaction between every body and the Earth. Gravitation produces an acceleration
along the line connecting the centres of gravity of the body and the Earth. Note that in
order to make this statement, it is necessary to realize that the Earth is a body in the
same way as a stone or the Moon, that this body is finite and that therefore it has a centre
and a mass. Today, these statements are common knowledge, but they are by no means
evident from everyday personal experience.
In several myths about the creation or the organization of the world, such as the bib-
lical one or the Indian one, the Earth is not an object, but an imprecisely defined entity,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
such as an island floating or surrounded by water with unclear boundaries and unclear
Challenge 313 s method of suspension. Are you able to convince a friend that the Earth is round and not
flat? Can you find another argument apart from the roundness of the Earth’s shadow
when it is visible on the Moon, shown in Figure 134?

Gravitation as a limit to uniform motion
A productive way to define gravitation, or gravity for short, appears when we note that
no object around us moves along a straight line. In nature, there is a limit to steady, or
constant motion:

⊳ Gravity prevents uniform motion, i.e., it prevents constant and straight mo-
tion.

In nature, we never observe bodies moving at constant speed along a straight line. Speak-
ing with the vocabulary of kinematics: Gravity introduces an acceleration for every phys-
ical body. The gravitation of the objects in the environment curves the path of a body,
changes its speed, or both. This limit has two aspects. First, gravity prevents unlimited
uniform motion:

** ‘I fell like dead bodies fall.’ Dante Alighieri (1265, Firenze–1321, Ravenna), the powerful Italian poet.
174 6 motion due to gravitation

culmination
zenith of a star or
n
idia planet:
celestial mer me 90° – ϕ + δ
North rid
ian
Pole
star or planet
nia
rid

celestial
t
me

e Equator
plan
r or
sta
of

merid
90° – ϕ ϕ–δ δ
ation
lin

ian
c
de
latitude ϕ 90° – ϕ

Motion Mountain – The Adventure of Physics
North observer South

F I G U R E 130 Some important concepts when observing the stars and at night.

⊳ Motion cannot be straight for ever. Motion is not boundless.

We will learn later what this means for the universe as a whole. Secondly,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
⊳ Motion cannot be constant and straight even during short time intervals.

In other words, if we measure with sufficient precision, we will always find deviations
from uniform motion. (Physicist also say that motion is never inertial.) These limits apply
no matter how powerful the motion is and how free the moving body is from external
influence. In nature, gravitation prevents the steady, uniform motion of atoms, billiard
balls, planets, stars and even galaxies.
Gravitation is the first limitation to motion that we discover in nature. We will dis-
cover two additional limits to motion later on in our walk. These three fundamental
Page 8 limits are illustrated in Figure 1. To achieve a precise description of motion, we need to
take each limit of motion into account. This is our main aim in the rest of our adventure.
Gravity affects all bodies, even if they are distant from each other. How exactly does
gravitation affect two bodies that are far apart? We ask the experts for measuring distant
objects: the astronomers.

Gravitation in the sky
The gravitation of the Earth forces the Moon in an orbit around it. The gravitation of the
Sun forces the Earth in an orbit around it and sets the length of the year. Similarly, the
gravitation of the Sun determines the motion of all the other planets across the sky. We
6 motion due to gravitation 175

Motion Mountain – The Adventure of Physics
F I G U R E 131 ‘Planet’ means ‘wanderer’. This composed image shows the retrograde motion of planet
Mars across the sky – the Pleiades star cluster is at the top left – when the planet is on the other side of
the Sun. The pictures were taken about a week apart and superimposed. The motion is one of the
many examples that are fully explained by universal gravitation (© Tunc Tezel).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
usually imagine to be located at the centre of the Sun and then say that the planets ‘orbit
the Sun’. The Sun thus prevents the planets from moving in straight lines and forces them
into orbits. How can we check this?
First of all, looking at the sky at night, we can check that the planets always stay within
Page 222 the zodiac, a narrow stripe across the sky. The centre line of the zodiac gives the path of
the Sun and is called the ecliptic, since the Moon must be located on it to produce an
Page 223 eclipse. This shows that planets move (approximately) in a single, common plane.*
To learn more about the motion in the sky, astronomers have performed numerous
measurements of the movements of the Moon and the planets. The most industrious
of all was Tycho Brahe,** who organized an industrial-scale search for astronomical
facts sponsored by his king. His measurements were the basis for the research of his
young assistant, the Swabian astronomer Johannes Kepler*** who found the first precise

* The apparent height of the ecliptic changes with the time of the year and is the reason for the changing
seasons. Therefore seasons are a gravitational effect as well.
** Tycho Brahe (b. 1546 Scania, d. 1601 Prague), famous astronomer, builder of Uraniaborg, the astronom-
ical castle. He consumed almost 10 % of the Danish gross national product for his research, which produced
the first star catalogue and the first precise position measurements of planets.
*** Johannes Kepler (1571 Weil der Stadt–1630 Regensburg) studied Protestant theology and became a
teacher of mathematics, astronomy and rhetoric. He helped his mother to defend herself successfully in
a trial where she was accused of witchcraft. His first book on astronomy made him famous, and he became
assistant to Tycho Brahe and then, at his teacher’s death, the Imperial Mathematician. He was the first to
176 6 motion due to gravitation

d
d
Sun

planet
F I G U R E 132 The motion of a planet around
the Sun, showing its semimajor axis 𝑑, which is
also the spatial average of its distance from the
Sun.

Vol. III, page 324 description of planetary motion. This is not an easy task, as the observation of Figure 131
shows. In his painstaking research on the movements of the planets in the zodiac, Kepler

Motion Mountain – The Adventure of Physics
discovered several ‘laws’, i.e., patterns or rules. The motion of all the planets follow the
same rules, confirming that the Sun determines their orbits. The three main ones are as
follows:

1. Planets move on ellipses with the Sun located at one focus (1609).
2. Planets sweep out equal areas in equal times (1609).
3. All planets have the same ratio 𝑇2 /𝑑3 between the orbit duration 𝑇 and the semimajor
axis 𝑑 (1619).

Kepler’s results are illustrated in Figure 132. The sheer work required to deduce the three

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
‘laws’ was enormous. Kepler had no calculating machine available. The calculation tech-
nology he used was the recently discovered logarithms. Anyone who has used tables of
logarithms to perform calculations can get a feeling for the amount of work behind these
three discoveries.
Finally, in 1684, all observations by Kepler about planets and stones were condensed
into an astonishingly simple result by the English physicist Robert Hooke and a few oth-
ers:*

⊳ Every body of mass 𝑀 attracts any other body towards its centre with an
acceleration whose magnitude 𝑎 is given by

𝑀
𝑎=𝐺 (39)
𝑟2
where 𝑟 is the centre-to-centre distance of the two bodies.

use mathematics in the description of astronomical observations, and introduced the concept and field of
‘celestial physics’.
* Robert Hooke (1635–1703), important English physicist and secretary of the Royal Society. Apart from
discovering the inverse square relation and many others, such as Hooke’s ‘law’, he also wrote the Micro-
graphia, a beautifully illustrated exploration of the world of the very small.
6 motion due to gravitation 177

Orbit of planet

Sun (origin)
S F Reflection of
Circle of largest point C
possible planet along tangent:
distance from Sun fixed in space
at energy E < 0 Tangent
P
to planet
planet

Motion Mountain – The Adventure of Physics
R = – k/E motion
position
= constant

C
Projection of planet
position P onto
circle of largest
planet distance

F I G U R E 133 The proof that a planet moves in an ellipse (magenta) around the Sun, given an inverse
square distance relation for gravitation. The proof – detailed in the text – is based on the relation

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
SP+PF=R. Since R is constant, the orbit is an ellipse

This is called universal gravitation, or the universal ‘law’ of gravitation, because it is valid
both in the sky and on Earth, as we will see shortly. The proportionality constant 𝐺 is
called the gravitational constant; it is one of the fundamental constants of nature, like the
Page 182 speed of light 𝑐 or the quantum of action ℏ. More about 𝐺 will be said shortly.
The effect of gravity thus decreases with increasing distance; the effect depends on the
inverse distance squared of the bodies under consideration. If bodies are small compared
with the distance 𝑟, or if they are spherical, expression (39) is correct as it stands; for
non-spherical shapes the acceleration has to be calculated separately for each part of the
bodies and then added together.
Ref. 147 Why is the usual planetary orbit an ellipse? The simplest argument is given in Fig-
ure 133. We know that the acceleration due to gravity varies as 𝑎 = 𝐺𝑀/𝑟2 . We also know
that an orbiting body of mass 𝑚 has a constant energy 𝐸 < 0. We then can draw, around
the Sun, the circle with radius 𝑅 = −𝐺𝑀𝑚/𝐸, which gives the largest distance that a
body with energy 𝐸 can be from the Sun. We now project the planet position 𝑃 onto
this circle, thus constructing a position 𝐶. We then reflect 𝐶 along the tangent to get a
position 𝐹. This last position 𝐹 is fixed in space and time, as a simple argument shows.
Challenge 314 s (Can you find it?) As a result of the construction, the distance sum SP+PF is constant
in time, and given by the radius 𝑅 = −𝐺𝑀𝑚/𝐸. Since the distance sum is constant, the
178 6 motion due to gravitation

orbit is an ellipse, because an ellipse is precisely the curve that appears when this sum is
constant. (Remember that an ellipse can be drawn with a piece of rope in this way.) Point
𝐹, like the Sun, is a focus of the ellipse. The construction thus shows that the motion of a
planet defines two foci and follows an elliptical orbit defined by these two foci. In short,
we have deduced the first of Kepler’s ‘laws’ from the expression of universal gravitation.
The second of Kepler’s ‘laws’, about equal swept areas, implies that planets move
faster when they are near the Sun. It is a simple way to state the conservation of angular
Challenge 315 e momentum. What does the third ‘law’ state?
Can you confirm that also the second and third of Kepler’s ‘laws’ follow from Hooke’s
Challenge 316 s expression of universal gravity? Publishing this result – which was obvious to Hooke –
was one of the achievements of Newton. Try to repeat this achievement; it will show you
not only the difficulties, but also the possibilities of physics, and the joy that puzzles give.
Newton solved these puzzles with geometric drawings – though in quite a complex
manner. It is well known that Newton was not able to write down, let alone handle, dif-
Ref. 28 ferential equations at the time he published his results on gravitation. In fact, Newton’s
notation and calculation methods were poor. (Much poorer than yours!) The English

Motion Mountain – The Adventure of Physics
mathematician Godfrey Hardy* used to say that the insistence on using Newton’s in-
tegral and differential notation, rather than the earlier and better method, still common
today, due to his rival Leibniz – threw back English mathematics by 100 years.
To sum up, Kepler, Hooke and Newton became famous because they brought order to
the description of planetary motion. They showed that all motion due to gravity follows
from the same description, the inverse square distance. For this reason, the inverse square
distance relation 𝑎 = 𝐺𝑀/𝑟2 is called the universal law of gravity. Achieving this unifica-
tion of motion description, though of small practical significance, was widely publicized.
The main reasons were the age-old prejudices and fantasies linked with astrology.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In fact, the inverse square distance relation explains many additional phenomena. It
explains the motion and shape of the Milky Way and of the other galaxies, the motion of
many weather phenomena, and explains why the Earth has an atmosphere but the Moon
Challenge 317 s does not. (Can you explain this?)

Gravitation on E arth
This inverse square dependence of gravitational acceleration is often, but incorrectly,
called Newton’s ‘law’ of gravitation. Indeed, the occultist and physicist Isaac Newton
proved more elegantly than Hooke that the expression agreed with all astronomical and
terrestrial observations. Above all, however, he organized a better public relations cam-
Ref. 137 paign, in which he falsely claimed to be the originator of the idea.
Newton published a simple proof showing that the description of astronomical grav-
itation also gives the correct description for stones thrown through the air, down here
on ‘father Earth’. To achieve this, he compared the acceleration 𝑎m of the Moon with
that of stones 𝑔. For the ratio between these two accelerations, the inverse square rela-
tion predicts a value 𝑔/𝑎m = 𝑑2m/𝑅2 , where 𝑑m the distance of the Moon and 𝑅 is the
radius of the Earth. The Moon’s distance can be measured by triangulation, comparing

* Godfrey Harold Hardy (1877–1947) was an important number theorist, and the author of the well-known
A Mathematician’s Apology. He also ‘discovered’ the famous Indian mathematician Srinivasa Ramanujan,
and brought him to Britain.
6 motion due to gravitation 179

F I G U R E 134 How to compare
the radius of the Earth with
that of the Moon during a
partial lunar eclipse
(© Anthony Ayiomamitis).

Motion Mountain – The Adventure of Physics
the position of the Moon against the starry background from two different points on
Earth.* The result is 𝑑m /𝑅 = 60 ± 3, depending on the orbital position of the Moon,
so that an average ratio 𝑔/𝑎m = 3.6 ⋅ 103 is predicted from universal gravity. But both
accelerations can also be measured directly. At the surface of the Earth, stones are sub-
ject to an acceleration due to gravitation with magnitude 𝑔 = 9.8 m/s2 , as determined by
measuring the time that stones need to fall a given distance. For the Moon, the definition
of acceleration, 𝑎 = d𝑣/d𝑡, in the case of circular motion – roughly correct here – gives
𝑎m = 𝑑m(2π/𝑇)2 , where 𝑇 = 2.4 Ms is the time the Moon takes for one orbit around

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the Earth.** The measurement of the radius of the Earth*** yields 𝑅 = 6.4 Mm, so that

* The first precise – but not the first – measurement was achieved in 1752 by the French astronomers Lalande
and La Caille, who simultaneously measured the position of the Moon seen from Berlin and from Le Cap.
** This expression for the centripetal acceleration is deduced easily by noting that for an object in circular
motion, the magnitude 𝑣 of the velocity 𝑣 = d𝑥/d𝑡 is given as 𝑣 = 2π𝑟/𝑇. The drawing of the vector 𝑣 over
Challenge 318 s time, the so-called hodograph, shows that it behaves exactly like the position of the object. Therefore the
magnitude 𝑎 of the acceleration 𝑎 = d𝑣/d𝑡 is given by the corresponding expression, namely 𝑎 = 2π𝑣/𝑇.
*** This is the hardest quantity to measure oneself. The most surprising way to determine the Earth’s size
is the following: watch a sunset in the garden of a house, with a stopwatch in hand, such as the one in your
Ref. 138 mobile phone. When the last ray of the Sun disappears, start the stopwatch and run upstairs. There, the Sun
is still visible; stop the stopwatch when the Sun disappears again and note the time 𝑡. Measure the height
difference ℎ between the two eye positions where the Sun was observed. The Earth’s radius 𝑅 is then given
Challenge 319 s by 𝑅 = 𝑘 ℎ/𝑡2 , with 𝑘 = 378 ⋅ 106 s2 .
Ref. 139 There is also a simple way to measure the distance to the Moon, once the size of the Earth is known.
Take a photograph of the Moon when it is high in the sky, and call 𝜃 its zenith angle, i.e., its angle from
the vertical above you. Make another photograph of the Moon a few hours later, when it is just above the
Page 69 horizon. On this picture, unlike the common optical illusion, the Moon is smaller, because it is further
away. With a sketch the reason becomes immediately clear. If 𝑞 is the ratio of the two angular diameters,
the Earth–Moon distance 𝑑m is given by the relation 𝑑2m = 𝑅2 + (2𝑅𝑞 cos 𝜃/(1 − 𝑞2 ))2 . Enjoy finding its
Challenge 320 s derivation from the sketch.
Another possibility is to determine the size of the Moon by comparing it with the size of the Earth in
a lunar exclipe, as shown in Figure 134. The distance to the Moon is then computed from its angular size,
about 0.5°.
180 6 motion due to gravitation

Moon

Earth

figure to be inserted

F I G U R E 135 A physicist’s and an artist’s view of the fall of the Moon: a diagram by Christiaan Huygens
(not to scale) and a marble statue by Auguste Rodin.

Motion Mountain – The Adventure of Physics
the average Earth–Moon distance is 𝑑m = 0.38 Gm. One thus has 𝑔/𝑎m = 3.6 ⋅ 103 , in
agreement with the above prediction. With this famous ‘Moon calculation’ we have thus
shown that the inverse square property of gravitation indeed describes both the motion
of the Moon and that of stones. You might want to deduce the value of the product 𝐺𝑀
Challenge 321 s for Earth.
Universal gravitation thus describes all motion due to gravity – both on Earth and in
the sky. This was an important step towards the unification of physics. Before this discov-
ery, from the observation that on the Earth all motion eventually comes to rest, whereas

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in the sky all motion is eternal, Aristotle and many others had concluded that motion
in the sublunar world has different properties from motion in the translunar world. Sev-
eral thinkers had criticized this distinction, notably the philosopher and rector of the
Ref. 140 University of Paris, Jean Buridan.* The Moon calculation was the most important result
showing this distinction to be wrong. This is the reason for calling Hooke’s expression
(39) the universal gravitation.
Universal gravitation allows us to answer another old question. Why does the Moon
not fall from the sky? Well, the preceding discussion showed that fall is motion due to
gravitation. Therefore the Moon actually is falling, with the peculiarity that instead of
falling towards the Earth, it is continuously falling around it. Figure 135 illustrates the
idea. The Moon is continuously missing the Earth.**
The Moon is not the only object that falls around the Earth. Figure 137 shows another.

* Jean Buridan (c. 1295 to c. 1366) was also one of the first modern thinkers to discuss the rotation of the
Earth about an axis.
** Another way to put it is to use the answer of the Dutch physicist Christiaan Huygens (1629–1695): the
Moon does not fall from the sky because of the centrifugal acceleration. As explained on page 161, this
explanation is often out of favour at universities.
Ref. 141 There is a beautiful problem connected to the left side of the figure: Which points on the surface of the
Challenge 322 d Earth can be hit by shooting from a mountain? And which points can be hit by shooting horizontally?
6 motion due to gravitation 181

F I G U R E 136 A precision second pendulum, thus about 1 m in length; almost

Motion Mountain – The Adventure of Physics
at the upper end, the vacuum chamber that compensates for changes in
atmospheric pressure; towards the lower end, the wide construction that
compensates for temperature variations of pendulum length; at the very
bottom, the screw that compensates for local variations of the gravitational
acceleration, giving a ﬁnal precision of about 1 s per month (© Erwin Sattler
OHG).

F I G U R E 137 The man in
orbit feels no weight, the
blue atmosphere, which is
not, does (NASA).
182 6 motion due to gravitation

TA B L E 26 Some measured values of the acceleration due to gravity.

Place Va l u e

Poles 9.83 m/s2
Trondheim 9.8215243 m/s2
Hamburg 9.8139443 m/s2
Munich 9.8072914 m/s2
Rome 9.8034755 m/s2
Equator 9.78 m/s2
Moon 1.6 m/s2
Sun 273 m/s2

Motion Mountain – The Adventure of Physics
Properties of gravitation: 𝐺 and 𝑔
Gravitation implies that the path of a stone is not a parabola, as stated earlier, but actually

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Page 77
an ellipse around the centre of the Earth. This happens for exactly the same reason that
Page 177 the planets move in ellipses around the Sun. Are you able to confirm this statement?
Universal gravitation allows us to understand the puzzling acceleration value
𝑔 = 9.8 m/s2 we encountered in equation (6). The value is due to the relation

𝑔 = 𝐺𝑀Earth /𝑅2Earth . (40)

The expression can be deduced from equation (39), universal gravity, by taking the Earth
to be spherical. The everyday acceleration of gravity 𝑔 thus results from the size of the
Earth, its mass, and the universal constant of gravitation 𝐺. Obviously, the value for 𝑔
is almost constant on the surface of the Earth, as shown in Table 26, because the Earth
is almost a sphere. Expression (40) also explains why 𝑔 gets smaller as one rises above
the Earth, and the deviations of the shape of the Earth from sphericity explain why 𝑔 is
different at the poles and higher on a plateau. (What would 𝑔 be on the Moon? On Mars?
Challenge 323 s On Jupiter?)
By the way, it is possible to devise a simple machine, other than a yo-yo, that slows
down the effective acceleration of gravity by a known amount, so that one can measure
Challenge 324 s its value more easily. Can you imagine this machine?
Note that 9.8 is roughly π2 . This is not a coincidence: the metre has been chosen in
such a way to make this (roughly) correct. The period 𝑇 of a swinging pendulum, i.e., a
6 motion due to gravitation 183

Challenge 325 s back and forward swing, is given by*

𝑙
𝑇 = 2π√ , (41)
𝑔

where 𝑙 is the length of the pendulum, and 𝑔 = 9.8 m/s2 is the gravitational acceleration.
(The pendulum is assumed to consist of a compact mass attached to a string of negligible
mass.) The oscillation time of a pendulum depends only on the length of the string and
on 𝑔, thus on the planet it is located on.
If the metre had been defined such that 𝑇/2 = 1 s, the value of the normal acceleration
Challenge 327 e 𝑔 would have been exactly π2 m/s2 = 9.869 604 401 09 m/s2 . Indeed, this was the first
proposal for the definition of the metre; it was made in 1673 by Huygens and repeated
in 1790 by Talleyrand, but was rejected by the conference that defined the metre because
variations in the value of 𝑔 with geographical position, temperature-induced variations
of the length of a pendulum and even air pressure variations induce errors that are too

Motion Mountain – The Adventure of Physics
large to yield a definition of useful precision. (Indeed, all these effects must be corrected
in pendulum clocks, as shown in Figure 136.)
Finally, the proposal was made to define the metre as 1/40 000 000 of the circum-
ference of the Earth through the poles, a so-called meridian. This proposal was almost
identical to – but much more precise than – the pendulum proposal. The meridian defin-
ition of the metre was then adopted by the French national assembly on 26 March 1791,
with the statement that ‘a meridian passes under the feet of every human being, and all
meridians are equal’. (Nevertheless, the distance from the Equator to the poles is not
Ref. 142 exactly 10 Mm; that is a strange story. One of the two geographers who determined the
size of the first metre stick was dishonest. The data he gave for his measurements – the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
general method of which is shown in Figure 138 – was fabricated. Thus the first official
metre stick in Paris was shorter than it should be.)
Continuing our exploration of the gravitational acceleration 𝑔, we can still ask: Why
does the Earth have the mass and size it has? And why does 𝐺 have the value it has? The
first question asks for a history of the Solar System; it is still unanswered and is topic of
research. The second question is addressed in Appendix B.
If gravitation is indeed universal, and if all objects really attract each other, attraction
should also occur between any two objects of everyday life. Gravity must also work side-
ways. This is indeed the case, even though the effects are extremely small. Indeed, the
effects are so small that they were measured only long after universal gravity had pre-
dicted them. On the other hand, measuring this effect is the only way to determine the
gravitational constant 𝐺. Let us see how to do it.

* Formula (41) is noteworthy mainly for all that is missing. The period of a pendulum does not depend on
the mass of the swinging body. In addition, the period of a pendulum does not depend on the amplitude.
(This is true as long as the oscillation angle is smaller than about 15°.) Galileo discovered this as a student,
when observing a chandelier hanging on a long rope in the dome of Pisa. Using his heartbeat as a clock he
found that even though the amplitude of the swing got smaller and smaller, the time for the swing stayed
the same.
A leg also moves like a pendulum, when one walks normally. Why then do taller people tend to walk
Challenge 326 s faster? Is the relation also true for animals of different size?
184 6 motion due to gravitation

Motion Mountain – The Adventure of Physics
F I G U R E 138 The measurements that lead to the deﬁnition of the
metre (© Ken Alder).

We note that measuring the gravitational constant 𝐺 is also the only way to determ-
ine the mass of the Earth. The first to do so, in 1798, was the English physicist Henry
Cavendish; he used the machine, ideas and method of John Michell who died when
attempting the experiment. Michell and Cavendish* called the aim and result of their

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
experiments ‘weighing the Earth’.
The idea of Michell was to suspended a horizontal handle, with two masses at the
end, at the end of a long metal wire. He then approached two additional large masses
at the two ends of the handle, avoiding any air currents, and measured how much the
handle rotated. Figure 139 shows how to repeat this experiment in your basement, and
Figure 140 how to perform it when you have a larger budget.
The value the gravitational constant 𝐺 found in more elaborate versions of the
Michell–Cavendish experiments is

𝐺 = 6.7 ⋅ 10−11 Nm2 /kg2 = 6.7 ⋅ 10−11 m3 /kg s2 . (42)

Cavendish’s experiment was thus the first to confirm that gravity also works sideways.
The experiment also allows deducing the mass 𝑀 of the Earth from its radius 𝑅 and the
Challenge 328 e relation 𝑔 = 𝐺𝑀/𝑅2 . Therefore, the experiment also allows to deduce the average density
of the Earth. Finally, as we will see later on, this experiment proves, if we keep in mind
Vol. II, page 141 that the speed of light is finite and invariant, that space is curved. All this is achieved
with this simple set-up!
* Henry Cavendish (b. 1731 Nice, d. 1810 London) was one of the great geniuses of physics; rich, autistic,
misogynist, unmarried and solitary, he found many rules of nature, but never published them. Had he done
so, his name would be much more well-known. John Michell (1724–1793) was church minister, geologist
and amateur astronomer.
6 motion due to gravitation 185

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 139 Home experiments that allow determining the gravitational constant 𝐺, weighing the
Earth, proving that gravity also works sideways and thus showing that gravity curves space. Top left and
right: a torsion balance made of foam and lead, with pétanque (boules) masses as ﬁxed masses; centre
right: a torsion balance made of wood and lead, with stones as ﬁxed masses; bottom: a time sequence
showing how the stones do attract the lead (© John Walker).

Ref. 143 Cavendish found a mass density of the Earth of 5.5 times that of water. This was a
surprising result because rock only has 2.8 times the density of water. What is the origin
Challenge 329 e of the large density value?
We note that 𝐺 has a small value. Above, we mentioned that gravity limits motion. In
fact, we can write the expression for universal gravitation in the following way:

𝑎𝑟2
=𝐺>0 (43)
𝑀
Gravity prevents uniform motion. In fact, we can say more: Gravitation is the smallest
186 6 motion due to gravitation

Motion Mountain – The Adventure of Physics
F I G U R E 140 A modern precision torsion balance experiment to measure the gravitational constant,
performed at the University of Washington (© Eöt-Wash Group).

𝜑(𝑥, 𝑦)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
𝑦
𝑥
grad 𝜑
F I G U R E 141 The potential and the gradient, visualized
for two spatial dimensions.

possible effect of the environment on a moving body. All other effects come on top of grav-
Challenge 330 d ity. However, it is not easy to put this statement in a simple formula.
Gravitation between everyday objects is weak. For example, two average people 1 m
apart feel an acceleration towards each other that is less than that exerted by a common
Challenge 331 s fly when landing on the skin. Therefore we usually do not notice the attraction to other
people. When we notice it, it is much stronger than that. The measurement of 𝐺 thus
proves that gravitation cannot be the true cause of people falling in love, and also that
erotic attraction is not of gravitational origin, but of a different source. The physical basis
Vol. III, page 15 for love will be studied later in our walk: it is called electromagnetism.

The gravitational potential
Gravity has an important property: all effects of gravitation can also be described by an-
other observable, namely the (gravitational) potential 𝜑. We then have the simple relation
6 motion due to gravitation 187

that the acceleration is given by the gradient of the potential

𝑎 = −∇𝜑 or 𝑎 = −grad 𝜑 . (44)

The gradient is just a learned term for ‘slope along the steepest direction’. The gradient is
defined for any point on a slope, is large for a steep one and small for a shallow one. The
gradient points in the direction of steepest ascent, as shown in Figure 141. The gradient is
abbreviated ∇, pronounced ‘nabla’, and is mathematically defined through the relation
∇𝜑 = (∂𝜑/∂𝑥, ∂𝜑/∂𝑦, ∂𝜑/∂𝑧) = grad 𝜑.* The minus sign in (44) is introduced by con-
vention, in order to have higher potential values at larger heights. In everyday life, when
the spherical shape of the Earth can be neglected, the gravitational potential is given by

𝜑 = 𝑔ℎ . (45)

The potential 𝜑 is an interesting quantity; with a single number at every position in space
we can describe the vector aspects of gravitational acceleration. It automatically gives that

Motion Mountain – The Adventure of Physics
gravity in New Zealand acts in the opposite direction to gravity in Paris. In addition, the
potential suggests the introduction of the so-called potential energy 𝑈 by setting

𝑈 = 𝑚𝜑 (46)

and thus allowing us to determine the change of kinetic energy 𝑇 of a body falling from
a point 1 to a point 2 via

𝑇1 − 𝑇2 = 𝑈2 − 𝑈1 or 1
𝑚𝑣2
2 1 1
− 12 𝑚2 𝑣2 2 = 𝑚𝜑2 − 𝑚𝜑1 . (47)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In other words, the total energy, defined as the sum of kinetic and potential energy, is con-
served in motion due to gravity. This is a characteristic property of gravitation. Gravity
conserves energy and momentum.
Page 110 Not all accelerations can be derived from a potential; systems with this property are
called conservative. Observation shows that accelerations due to friction are not conser-
vative, but accelerations due to electromagnetism are. In short, we can either say that
gravity can be described by a potential, or say that it conserves energy and momentum.
Both mean the same. When the non-spherical shape of the Earth can be neglected, the
potential energy of an object at height ℎ is given by

𝑈 = 𝑚𝑔ℎ . (48)

To get a feeling of how much energy this is, answer the following question. A car with
mass 1 Mg falls down a cliff of 100 m. How much water can be heated from freezing point
Challenge 332 s to boiling point with the energy of the car?

* In two or more dimensions slopes are written ∂𝜑/∂𝑧 – where ∂ is still pronounced ‘d’ – because in those
cases the expression 𝑑𝜑/𝑑𝑧 has a slightly different meaning. The details lie outside the scope of this walk.
188 6 motion due to gravitation

The shape of the E arth
Universal gravity also explains why the Earth and most planets are (almost) spherical.
Since gravity increases with decreasing distance, a liquid body in space will always try to
form a spherical shape. Seen on a large scale, the Earth is indeed liquid. We also know
that the Earth is cooling down – that is how the crust and the continents formed. The
sphericity of smaller solid objects encountered in space, such as the Moon, thus means
that they used to be liquid in older times.
The Earth is thus not flat, but roughly spherical. Therefore, the top of two tall buildings
Challenge 333 s is further apart than their base. Can this effect be measured?
Sphericity considerably simplifies the description of motion. For a spherical or a
Challenge 334 e point-like body of mass 𝑀, the potential 𝜑 is

𝑀
𝜑 = −𝐺 . (49)
𝑟
A potential considerably simplifies the description of motion, since a potential is addit-

Motion Mountain – The Adventure of Physics
ive: given the potential of a point particle, we can calculate the potential and then the
motion around any other irregularly shaped object.* Interestingly, the number 𝑑 of di-
mensions of space is coded into the potential 𝜑 of a spherical mass: the dependence of 𝜑
Challenge 336 s on the radius 𝑟 is in fact 1/𝑟𝑑−2 . The exponent 𝑑 − 2 has been checked experimentally to
Ref. 144 extremely high precision; no deviation of 𝑑 from 3 has ever been found.
The concept of potential helps in understanding the shape of the Earth in more detail.
Ref. 145 Since most of the Earth is still liquid when seen on a large scale, its surface is always hori-
zontal with respect to the direction determined by the combination of the accelerations
of gravity and rotation. In short, the Earth is not a sphere. It is not an ellipsoid either.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 146 The mathematical shape defined by the equilibrium requirement is called a geoid. The
geoid shape, illustrated in Figure 142, differs from a suitably chosen ellipsoid by at most
Challenge 337 ny 50 m. Can you describe the geoid mathematically? The geoid is an excellent approxima-
tion to the actual shape of the Earth; sea level differs from it by less than 20 metres. The
differences can be measured with satellite radar and are of great interest to geologists
and geographers. For example, it turns out that the South Pole is nearer to the equatorial
* Alternatively, for a general, extended body, the potential is found by requiring that the divergence of its
gradient is given by the mass (or charge) density times some proportionality constant. More precisely, we
have
Δ𝜑 = 4π𝐺𝜌 (50)
where 𝜌 = 𝜌(𝑥, 𝑡) is the mass volume density of the body and the so-called Laplace operator Δ, pronounced
‘delta’, is defined as Δ𝑓 = ∇∇𝑓 = ∂2 𝑓/∂𝑥2 +∂2 𝑓/∂𝑦2 +∂2 𝑓/∂𝑧2 . Equation (50) is called the Poisson equation
for the potential 𝜑. It is named after Siméon-Denis Poisson (1781–1840), eminent French mathematician
and physicist. The positions at which 𝜌 is not zero are called the sources of the potential. The so-called
source term Δ𝜑 of a function is a measure for how much the function 𝜑(𝑥) at a point 𝑥 differs from the
Challenge 335 e average value in a region around that point. (Can you show this, by showing that Δ𝜑 ∼ 𝜑̄ − 𝜑(𝑥)?) In other
words, the Poisson equation (50) implies that the actual value of the potential at a point is the same as the
average value around that point minus the mass density multiplied by 4π𝐺. In particular, in the case of
empty space the potential at a point is equal to the average of the potential around that point.
Often the concept of gravitational field is introduced, defined as 𝑔 = −∇𝜑. We avoid this in our walk
because we will discover that, following the theory of relativity, gravity is not due to a field at all; in fact,
even the concept of gravitational potential turns out to be only an approximation.
6 motion due to gravitation 189

F I G U R E 142 The shape of the Earth, with
exaggerated height scale
(© GeoForschungsZentrum Potsdam).

Motion Mountain – The Adventure of Physics
plane than the North Pole by about 30 m. This is probably due to the large land masses
in the northern hemisphere.
Page 137 Above we saw how the inertia of matter, through the so-called ‘centrifugal force’,
increases the radius of the Earth at the Equator. In other words, the Earth is flattened
at the poles. The Equator has a radius 𝑎 of 6.38 Mm, whereas the distance 𝑏 from the
poles to the centre of the Earth is 6.36 Mm. The precise flattening (𝑎 − 𝑏)/𝑎 has the value
Appendix B 1/298.3 = 0.0034. As a result, the top of Mount Chimborazo in Ecuador, even though
its height is only 6267 m above sea level, is about 20 km farther away from the centre of

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the Earth than the top of Mount Sagarmatha* in Nepal, whose height above sea level is
8850 m. The top of Mount Chimborazo is in fact the point on the surface most distant
from the centre of the Earth.
The shape of the Earth has another important consequence. If the Earth stopped ro-
tating (but kept its shape), the water of the oceans would flow from the Equator to the
poles; all of Europe would be under water, except for the few mountains of the Alps that
are higher than about 4 km. The northern parts of Europe would be covered by between
6 km and 10 km of water. Mount Sagarmatha would be over 11 km above sea level. We
would also walk inclined. If we take into account the resulting change of shape of the
Earth, the numbers come out somewhat smaller. In addition, the change in shape would
produce extremely strong earthquakes and storms. As long as there are none of these
Page 157 effects, we can be sure that the Sun will indeed rise tomorrow, despite what some philo-
sophers pretended.

Dynamics – how d o things move in various dimensions?
The concept of potential is a powerful tool. If a body can move only along a – straight
or curved – line, the concepts of kinetic and potential energy are sufficient to determine
completely the way the body moves.

* Mount Sagarmatha is sometimes also called Mount Everest.
190 6 motion due to gravitation

F I G U R E 143 The change of the
moon during the month,
showing its libration (QuickTime

Motion Mountain – The Adventure of Physics
ﬁlm © Martin Elsässer)

In fact, motion in one dimension follows directly from energy conservation. For a body
moving along a given curve, the speed at every instant is given by energy conservation.
If a body can move in two dimensions – i.e., on a flat or curved surface – and if the
forces involved are internal (which is always the case in theory, but not in practice), the
conservation of angular momentum can be used. The full motion in two dimensions thus

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
follows from energy and angular momentum conservation. For example, all properties
of free fall follow from energy and angular momentum conservation. (Are you able to
Challenge 338 s show this?) Again, the potential is essential.
In the case of motion in three dimensions, a more general rule for determining motion
is necessary. If more than two spatial dimensions are involved conservation is insufficient
to determine how a body moves. It turns out that general motion follows from a simple
principle: the time average of the difference between kinetic and potential energy must
be as small as possible. This is called the least action principle. We will explain the details
Page 248 of this calculation method later. But again, the potential is the main ingredient in the
calculation of change, and thus in the description of any example of motion.
For simple gravitational motions, motion is two-dimensional, in a plane. Most three-
dimensional problems are outside the scope of this text; in fact, some of these problems
are so hard that they still are subjects of research. In this adventure, we will explore three-
dimensional motion only for selected cases that provide important insights.

The Mo on
How long is a day on the Moon? The answer is roughly 29 Earth-days. That is the time
that it takes for an observer on the Moon to see the Sun again in the same position in the
sky.
One often hears that the Moon always shows the same side to the Earth. But this is
6 motion due to gravitation 191

F I G U R E 144 High resolution maps (not photographs) of the near side (left) and far side (right) of the

Motion Mountain – The Adventure of Physics
moon, showing how often the latter saved the Earth from meteorite impacts (courtesy USGS).

wrong. As one can check with the naked eye, a given feature in the centre of the face of the
Moon at full Moon is not at the centre one week later. The various motions leading to this
change are called librations; they are shown in the film in Figure 143. The motions appear
mainly because the Moon does not describe a circular, but an elliptical orbit around the
Earth and because the axis of the Moon is slightly inclined, when compared with that
of its rotation around the Earth. As a result, only around 45 % of the Moon’s surface is
permanently hidden from Earth.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The first photographs of the hidden area of the Moon were taken in the 1960s by a
Soviet artificial satellite; modern satellites provided exact maps, as shown in Figure 144.
Challenge 339 e (Just zoom into the figure for fun.) The hidden surface is much more irregular than the
visible one, as the hidden side is the one that intercepts most asteroids attracted by the
Earth. Thus the gravitation of the Moon helps to deflect asteroids from the Earth. The
number of animal life extinctions is thus reduced to a small, but not negligible number.
In other words, the gravitational attraction of the Moon has saved the human race from
extinction many times over.*
The trips to the Moon in the 1970s also showed that the Moon originated from the
Earth itself: long ago, an object hit the Earth almost tangentially and threw a sizeable
fraction of material up into the sky. This is the only mechanism able to explain the large
Ref. 148 size of the Moon, its low iron content, as well as its general material composition.
Ref. 149 The Moon is receding from the Earth at 3.8 cm a year. This result confirms the old de-
duction that the tides slow down the Earth’s rotation. Can you imagine how this meas-
Challenge 340 s urement was performed? Since the Moon slows down the Earth, the Earth also changes
shape due to this effect. (Remember that the shape of the Earth depends on its speed of

* The web pages www.minorplanetcenter.net/iau/lists/Closest.html and InnerPlot.html give an impression
of the number of objects that almost hit the Earth every year. Without the Moon, we would have many
additional catastrophes.
192 6 motion due to gravitation

rotation.) These changes in shape influence the tectonic activity of the Earth, and maybe
also the drift of the continents.
The Moon has many effects on animal life. A famous example is the midge Clunio,
Ref. 150 which lives on coasts with pronounced tides. Clunio spends between six and twelve
weeks as a larva, sure then hatches and lives for only one or two hours as an adult flying
insect, during which time it reproduces. The midges will only reproduce if they hatch
during the low tide phase of a spring tide. Spring tides are the especially strong tides
during the full and new moons, when the solar and lunar effects combine, and occur
only every 14.8 days. In 1995, Dietrich Neumann showed that the larvae have two built-
in clocks, a circadian and a circalunar one, which together control the hatching to pre-
cisely those few hours when the insect can reproduce. He also showed that the circalunar
clock is synchronized by the brightness of the Moon at night. In other words, the larvae
monitor the Moon at night and then decide when to hatch: they are the smallest known
astronomers.
If insects can have circalunar cycles, it should come as no surprise that women also
have such a cycle; however, in this case the precise origin of the cycle length is still un-

Motion Mountain – The Adventure of Physics
Ref. 151 known and a topic of research.
The Moon also helps to stabilize the tilt of the Earth’s axis, keeping it more or less
fixed relative to the plane of motion around the Sun. Without the Moon, the axis would
change its direction irregularly, we would not have a regular day and night rhythm, we
would have extremely large climate changes, and the evolution of life would have been
Ref. 152 impossible. Without the Moon, the Earth would also rotate much faster and we would
Ref. 153 have much less clement weather. The Moon’s main remaining effect on the Earth, the
Page 146 precession of its axis, is responsible for the ice ages.
The orbit of the Moon is still a topic of research. It is still not clear why the Moon orbit

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
is at a 5° to the ecliptic and how the orbit changed since the Moon formed. Possibly, the
collision that led to the formation of the Moon tilted the rotation axis of the Earth and the
Ref. 154 original Moon; then over thousands of millions of years, both axes moved in complicated
ways towards the ecliptic, one more than the other. During this evolution, the distance
to the Moon is estimated to have increased by a factor of 15.

Orbits – conic sections and more
The path of a body continuously orbiting another under the influence of gravity is an
ellipse with the central body at one focus. A circular orbit is also possible, a circle being a
special case of an ellipse. Single encounters of two objects can also be parabolas or hyper-
bolas, as shown in Figure 145. Circles, ellipses, parabolas and hyperbolas are collectively
known as conic sections. Indeed each of these curves can be produced by cutting a cone
Challenge 341 e with a knife. Are you able to confirm this?
If orbits are mostly ellipses, it follows that comets return. The English astronomer Ed-
mund Halley (1656–1742) was the first to draw this conclusion and to predict the return
of a comet. It arrived at the predicted date in 1756, after his death, and is now named after
him. The period of Halley’s comet is between 74 and 80 years; the first recorded sighting
was 22 centuries ago, and it has been seen at every one of its 30 passages since, the last
time in 1986.
Depending on the initial energy and the initial angular momentum of the body with
6 motion due to gravitation 193

hyperbola

parabola

mass

circle ellipse

Motion Mountain – The Adventure of Physics
F I G U R E 145 The possible orbits, due to universal gravity, of a small mass around a single large mass
(left) and a few recent examples of measured orbits (right), namely those of some extrasolar planets
and of the Earth, all drawn around their respective central star, with distances given in astronomical
units (© Geoffrey Marcy).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
respect to the central planet, paths are either elliptic, parabolic or hyperbolic. Can you
determine the conditions for the energy and the angular momentum needed for these
Challenge 342 s paths to appear?
In practice, parabolic orbits do not exist in nature. (Some comets seem to approach
this case when moving around the Sun; but almost all comets follow elliptical paths –
as long as they are far from other planets.). Hyperbolic paths do exist; artificial satellites
follow them when they are shot towards a planet, usually with the aim of changing the
direction of the satellite’s journey across the Solar System.
Why does the inverse square ‘law’ lead to conic sections? First, for two bodies, the
total angular momentum 𝐿 is a constant:

d𝜑
𝐿 = 𝑚𝑟2 𝜑̇ = 𝑚𝑟2 ( ) (51)
d𝑡

and therefore the motion lies in a plane. Also the energy 𝐸 is a constant

d𝑟 2 1 d𝜑 2 𝑚𝑀
𝐸 = 12 𝑚 ( ) + 2 𝑚 (𝑟 ) − 𝐺 . (52)
d𝑡 d𝑡 𝑟
194 6 motion due to gravitation

Challenge 343 e Together, the two equations imply that

𝐿2 1
𝑟= . (53)
𝐺𝑚2 𝑀 2𝐸𝐿2
1 + √1 + 2 3 2 cos 𝜑
𝐺𝑚𝑀

Now, any curve defined by the general expression

𝐶 𝐶
𝑟= or 𝑟 = (54)
1 + 𝑒 cos 𝜑 1 − 𝑒 cos 𝜑

is an ellipse for 0 < 𝑒 < 1, a parabola for 𝑒 = 1 and a hyperbola for 𝑒 > 1, one focus being
at the origin. The quantity 𝑒, called the eccentricity, describes how squeezed the curve is.
In other words, a body in orbit around a central mass follows a conic section.
In all orbits, also the heavy mass moves. In fact, both bodies orbit around the common
centre of mass. Both bodies follow the same type of curve – ellipse, parabola or hyperbola

Motion Mountain – The Adventure of Physics
Challenge 344 e – but the sizes of the two curves differ.
If more than two objects move under mutual gravitation, many additional possibilities
for motions appear. The classification and the motions are quite complex. In fact, this
so-called many-body problem is still a topic of research, both for astronomers and for
mathematicians. Let us look at a few observations.
When several planets circle a star, they also attract each other. Planets thus do not
move in perfect ellipses. The largest deviation is a perihelion shift, as shown in Figure 114.
Page 152 It is observed for Mercury and a few other planets, including the Earth. Other deviations
from elliptical paths appear during a single orbit. In 1846, the observed deviations of

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the motion of the planet Uranus from the path predicted by universal gravity were used
to predict the existence of another planet, Neptune, which was discovered shortly after-
wards.
Page 106 We have seen that mass is always positive and that gravitation is thus always attractive;
there is no antigravity. Can gravity be used for levitation nevertheless, using more than
two bodies? Yes; there are two examples.* The first are the geostationary satellites, which
are used for easy transmission of television and other signals from and towards Earth.
The Lagrangian libration points are the second example. Named after their discoverer,
these are points in space near a two-body system, such as Moon–Earth or Earth–Sun,
in which small objects have a stable equilibrium position. An overview is given in Fig-
ure 147. Can you find their precise position, remembering to take rotation into account?
Challenge 345 s There are three additional Lagrangian points on the Earth–Moon line (or Sun–planet
Challenge 346 d line). How many of them are stable?
There are thousands of asteroids, called Trojan asteroids, at and around the Lagrangian
points of the Sun–Jupiter system. In 1990, a Trojan asteroid for the Mars–Sun system was
discovered. Finally, in 1997, an ‘almost Trojan’ asteroid was found that follows the Earth
on its way around the Sun (it is only transitionary and follows a somewhat more complex
Ref. 156 orbit). This ‘second companion’ of the Earth has a diameter of 5 km. Similarly, on the

Vol. III, page 226 * Levitation is discussed in detail in the section on electrodynamics.
6 motion due to gravitation 195

Motion Mountain – The Adventure of Physics
F I G U R E 146 Geostationary satellites, seen here in the upper left quadrant, move against the other stars
and show the location of the celestial Equator. (MP4 ﬁlm © Michael Kunze)

geostationary

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
satellite
planet (or Sun)

fixed N L5
parabolic π/3
antenna
Earth π/3
π/3 π/3
L4
moon (or planet)

F I G U R E 147 Geostationary satellites (left) and the main stable Lagrangian points (right).

main Lagrangian points of the Earth–Moon system a high concentration of dust has been
observed.
Astronomers know that many other objects follow irregular orbits, especially aster-
Ref. 155 oids. For example, asteroid 2003 YN107 followed an irregular orbit, shown in Figure 148,
that accompanied the Earth for a number of years.
To sum up, the single equation 𝑎 = −𝐺𝑀𝑟/𝑟3 correctly describes a large number
of phenomena in the sky. The first person to make clear that this expression describes
196 6 motion due to gravitation

0.2

0.15

0.1

0.05

y
0
Size of Moon's orbit

Motion Mountain – The Adventure of Physics
Exit 12 years later
Start
-0.05

-0.1

-0.15
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15

F I G U R E 148 An example of irregular orbit, partly measured and partly calculated, due to the
gravitational attraction of several masses: the orbit of the temporary Earth quasi-satellite 2003 YN107 in
geocentric coordinates. This asteroid, with a diameter of 20(10) m, became orbitally trapped near the
Earth around 1995 and remained so until 2006. The black circle represents the Moon’s orbit around the
Earth. (© Seppo Mikkola).

everything happening in the sky was Pierre Simon Laplace in his famous treatise Traité
de mécanique céleste. When Napoleon told him that he found no mention about the cre-
ator in the book, Laplace gave a famous, one sentence summary of his book: Je n’ai pas
eu besoin de cette hypothèse. ‘I had no need for this hypothesis.’ In particular, Laplace
studied the stability of the Solar System, the eccentricity of the lunar orbit, and the ec-
centricities of the planetary orbits, always getting full agreement between calculation and
measurement.
These results are quite a feat for the simple expression of universal gravitation; they
also explain why it is called ‘universal’. But how accurate is the formula? Since astronomy
allows the most precise measurements of gravitational motion, it also provides the most
stringent tests. In 1849, Urbain Le Verrier concluded after intensive study that there was
only one known example of a discrepancy between observation and universal gravity,
namely one observation for the planet Mercury. (Nowadays a few more are known.) The
6 motion due to gravitation 197

Motion Mountain – The Adventure of Physics
F I G U R E 149 Tides at Saint-Valéry en Caux on 20 September 2005 (© Gilles Régnier).

point of least distance to the Sun of the orbit of planet Mercury, its perihelion, rotates
around the Sun at a rate that is slightly less than that predicted: he found a tiny differ-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 157 ence, around 38 󸀠󸀠 per century. (This was corrected to 43 󸀠󸀠 per century in 1882 by Simon
Newcomb.) Le Verrier thought that the difference was due to a planet between Mercury
and the Sun, Vulcan, which he chased for many years without success. Indeed, Vulcan
Vol. II, page 164 does not exist. The correct explanation of the difference had to wait for Albert Einstein.

Tides
Ref. 158 Why do physics texts always talk about tides? Because, as general relativity will show,
tides prove that space is curved! It is thus useful to study them in a bit more detail. Fig-
ure 149 how striking tides can be. Gravitation explains the sea tides as results of the at-
traction of the ocean water by the Moon and the Sun. Tides are interesting; even though
the amplitude of the tides is only about 0.5 m on the open sea, it can be up to 20 m at
Challenge 347 s special places near the coast. Can you imagine why? The soil is also lifted and lowered by
Ref. 57 the Sun and the Moon, by about 0.3 m, as satellite measurements show. Even the atmo-
sphere is subject to tides, and the corresponding pressure variations can be filtered out
Ref. 159 from the weather pressure measurements.
Tides appear for any extended body moving in the gravitational field of another. To
understand the origin of tides, picture a body in orbit, like the Earth, and imagine its
components, such as the segments of Figure 150, as being held together by springs. Uni-
versal gravity implies that orbits are slower the more distant they are from a central body.
As a result, the segment on the outside of the orbit would like to be slower than the cent-
198 6 motion due to gravitation

Sun before
𝑡1
deformed

𝑡=0 after
spherical

F I G U R E 150 Tidal deformations due to F I G U R E 151 The origin of tides.
gravity.

Motion Mountain – The Adventure of Physics
ral one; but it is pulled by the rest of the body through the springs. In contrast, the inside
segment would like to orbit more rapidly but is retained by the others. Being slowed
down, the inside segments want to fall towards the Sun. In sum, both segments feel a
pull away from the centre of the body, limited by the springs that stop the deformation.
Therefore, extended bodies are deformed in the direction of the field inhomogeneity.
For example, as a result of tidal forces, the Moon always has (roughly) the same face
to the Earth. In addition, its radius in direction of the Earth is larger by about 5 m than
the radius perpendicular to it. If the inner springs are too weak, the body is torn into

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
pieces; in this way a ring of fragments can form, such as the asteroid ring between Mars
and Jupiter or the rings around Saturn.
Let us return to the Earth. If a body is surrounded by water, it will form bulges in the
direction of the applied gravitational field. In order to measure and compare the strength
of the tides from the Sun and the Moon, we reduce tidal effects to their bare minimum.
As shown in Figure 151, we can study the deformation of a body due to gravity by studying
the arrangement of four bodies. We can study the free fall case because orbital motion
and free fall are equivalent. Now, gravity makes some of the pieces approach and others
diverge, depending on their relative positions. The figure makes clear that the strength of
the deformation – water has no built-in springs – depends on the change of gravitational
acceleration with distance; in other words, the relative acceleration that leads to the tides
is proportional to the derivative of the gravitational acceleration.
Page 453 Using the numbers from Appendix B, the gravitational accelerations from the Sun
and the Moon measured on Earth are
𝐺𝑀Sun
𝑎Sun = = 5.9 mm/s2
𝑑2Sun
𝐺𝑀
𝑎Moon = 2 Moon = 0.033 mm/s2 (55)
𝑑Moon

and thus the attraction from the Moon is about 178 times weaker than that from the Sun.
6 motion due to gravitation 199

Moon

Earth’s rotation drives
The Moon attracts the
the bulge forward.
tide bulge and thus slows
down the rotation of
the Earth.

Earth

Motion Mountain – The Adventure of Physics
The bulge of the tide attracts
the Moon and thus increases
the Moon’s orbit radius.

F I G U R E 152 The Earth, the Moon and the friction effects of the tides (not to scale).

When two nearby bodies fall near a large mass, the relative acceleration is pro-
portional to their distance, and follows 𝑑𝑎 = (𝑑𝑎/𝑑𝑟) 𝑑𝑟. The proportionality factor
𝑑𝑎/𝑑𝑟 = ∇𝑎, called the tidal acceleration (gradient), is the true measure of tidal effects.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 348 e Near a large spherical mass 𝑀, it is given by

𝑑𝑎 2𝐺𝑀
=− 3 (56)
𝑑𝑟 𝑟
which yields the values

𝑑𝑎Sun 2𝐺𝑀
= − 3 Sun = −0.8 ⋅ 10−13 /s2
𝑑𝑟 𝑑Sun
𝑑𝑎Moon 2𝐺𝑀
= − 3 Moon = −1.7 ⋅ 10−13 /s2 . (57)
𝑑𝑟 𝑑Moon

In other words, despite the much weaker pull of the Moon, its tides are predicted to be
over twice as strong as the tides from the Sun; this is indeed observed. When Sun, Moon
and Earth are aligned, the two tides add up; these so-called spring tides are especially
strong and happen every 14.8 days, at full and new moon.
Tides lead to a pretty puzzle. Moon tides are much stronger than Sun tides. This im-
Challenge 349 s plies that the Moon is much denser than the Sun. Why?
Tides also produce friction, as shown in Figure 152. The friction leads to a slowing of
the Earth’s rotation. Nowadays, the slowdown can be measured by precise clocks (even
200 6 motion due to gravitation

F I G U R E 153 A spectacular result of tides: volcanism
on Io (NASA).

Motion Mountain – The Adventure of Physics
Ref. 116 though short time variations due to other effects, such as the weather, are often larger).
The results fit well with fossil results showing that 400 million years ago, in the Devonian
Vol. II, page 231 period, a year had 400 days, and a day about 22 hours. It is also estimated that 900 million

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
years ago, each of the 481 days of a year were 18.2 hours long. The friction at the basis of
this slowdown also results in an increase in the distance of the Moon from the Earth by
Challenge 350 s about 3.8 cm per year. Are you able to explain why?
As mentioned above, the tidal motion of the soil is also responsible for the triggering
of earthquakes. Thus the Moon can have also dangerous effects on Earth. (Unfortunately,
knowing the mechanism does not allow predicting earthquakes.) The most fascinating
example of tidal effects is seen on Jupiter’s satellite Io. Its tides are so strong that they
induce intense volcanic activity, as shown in Figure 153, with eruption plumes as high
as 500 km. If tides are even stronger, they can destroy the body altogether, as happened
to the body between Mars and Jupiter that formed the planetoids, or (possibly) to the
moons that led to Saturn’s rings.
In summary, tides are due to relative accelerations of nearby mass points. This has an
Vol. II, page 136 important consequence. In the chapter on general relativity we will find that time multi-
plied by the speed of light plays the same role as length. Time then becomes an additional
dimension, as shown in Figure 154. Using this similarity, two free particles moving in the
same direction correspond to parallel lines in space-time. Two particles falling side-by-
side also correspond to parallel lines. Tides show that such particles approach each other.
Vol. II, page 190 In other words, tides imply that parallel lines approach each other. But parallel lines can
approach each other only if space-time is curved. In short, tides imply curved space-time
and space. This simple reasoning could have been performed in the eighteenth century;
however, it took another 200 years and Albert Einstein’s genius to uncover it.
6 motion due to gravitation 201

space
𝑡1

𝛼
𝑏

𝑡2 𝑀
time

F I G U R E 154 Particles falling F I G U R E 155 Masses bend light.
side-by-side approach over time.

C an light fall?

Motion Mountain – The Adventure of Physics
“
Die Maxime, jederzeit selbst zu denken, ist die

”
Aufklärung.
Immanuel Kant*

Towards the end of the seventeenth century people discovered that light has a finite velo-
Vol. II, page 15 city – a story which we will tell in detail later. An entity that moves with infinite velocity
cannot be affected by gravity, as there is no time to produce an effect. An entity with a
finite speed, however, should feel gravity and thus fall.
Does its speed increase when light reaches the surface of the Earth? For almost three
centuries people had no means of detecting any such effect; so the question was not in-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
vestigated. Then, in 1801, the Prussian astronomer Johann Soldner (1776–1833) was the
Ref. 160 first to put the question in a different way. Being an astronomer, he was used to measur-
ing stars and their observation angles. He realized that light passing near a massive body
would be deflected due to gravity.
Soldner studied a body on a hyperbolic path, moving with velocity 𝑐 past a spher-
ical mass 𝑀 at distance 𝑏 (measured from the centre), as shown in Figure 155. Soldner
Challenge 351 ny deduced the deflection angle
2 𝐺𝑀
𝛼univ. grav. = . (58)
𝑏 𝑐2
The value of the angle is largest when the motion is just grazing the mass 𝑀. For light
deflected by the mass of the Sun, the angle turns out to be at most a tiny 0.88 󸀠󸀠 = 4.3 μrad.
In Soldner’s time, this angle was too small to be measured. Thus the issue was forgotten.
Had it been pursued, general relativity would have begun as an experimental science, and
Vol. II, page 161 not as the theoretical effort of Albert Einstein! Why? The value just calculated is different
from the measured value. The first measurement took place in 1919;** it found the correct
dependence on the distance, but found a deflection of up to 1.75 󸀠󸀠 , exactly double that of
expression (58). The reason is not easy to find; in fact, it is due to the curvature of space,

* The maxim to think at all times for oneself is the enlightenment.
Challenge 352 s ** By the way, how would you measure the deflection of light near the bright Sun?
202 6 motion due to gravitation

as we will see. In summary, light can fall, but the issue hides some surprises.

Mass: inertial and gravitational
Mass describes how an object interacts with others. In our walk, we have encountered
two of its aspects. Inertial mass is the property that keeps objects moving and that offers
resistance to a change in their motion. Gravitational mass is the property responsible for
the acceleration of bodies nearby (the active aspect) or of being accelerated by objects
nearby (the passive aspect). For example, the active aspect of the mass of the Earth de-
termines the surface acceleration of bodies; the passive aspect of the bodies allows us
to weigh them in order to measure their mass using distances only, e.g. on a scale or a
balance. The gravitational mass is the basis of weight, the difficulty of lifting things.*
Is the gravitational mass of a body equal to its inertial mass? A rough answer is given
by the experience that an object that is difficult to move is also difficult to lift. The simplest
experiment is to take two bodies of different masses and let them fall. If the acceleration
is the same for all bodies, inertial mass is equal to (passive) gravitational mass, because

Motion Mountain – The Adventure of Physics
in the relation 𝑚𝑎 = ∇(𝐺𝑀𝑚/𝑟) the left-hand 𝑚 is actually the inertial mass, and the
right-hand 𝑚 is actually the gravitational mass.
Already in the seventeenth century Galileo had made widely known an even older
argument showing without a single experiment that the gravitational acceleration is in-
deed the same for all bodies. If larger masses fell more rapidly than smaller ones, then
the following paradox would appear. Any body can be seen as being composed of a large
fragment attached to a small fragment. If small bodies really fell less rapidly, the small
fragment would slow the large fragment down, so that the complete body would have
to fall less rapidly than the larger fragment (or break into pieces). At the same time, the
body being larger than its fragment, it should fall more rapidly than that fragment. This

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
is obviously impossible: all masses must fall with the same acceleration.
Many accurate experiments have been performed since Galileo’s original discussion.
In all of them, the independence of the acceleration of free fall from mass and material
Ref. 161 composition has been confirmed with the precision they allowed. In other words, exper-
iments confirm:

⊳ Gravitational mass and inertial mass are equal.

What is the origin of this mysterious equality?
The equality of gravitational and inertial mass is not a mystery at all. Let us go back
Page 100 to the definition of mass as a negative inverse acceleration ratio. We mentioned that the
physical origin of the accelerations does not play a role in the definition because the
origin does not appear in the expression. In other words, the value of the mass is by
definition independent of the interaction. That means in particular that inertial mass,
based on and measured with the electromagnetic interaction, and gravitational mass are
identical by definition.
The best proof of the equality of inertial and gravitational mass is illustrated in Fig-
ure 156: it shows that the two concepts only differ by the viewpoint of the observer. In-
ertial mass and gravitational mass describe the same observation.
Challenge 353 ny * What are the weight values shown by a balance for a person of 85 kg juggling three balls of 0.3 kg each?
6 motion due to gravitation 203

Motion Mountain – The Adventure of Physics
F I G U R E 156 The falling ball is in inertial motion for a falling observer and in gravitational motion for an
observer on the ground. Therefore, inertial mass is the same as gravitational mass.

We also note that we have not defined a separate concept of ‘passive gravitational
mass’. (This concept is sometimes found in research papers.) The mass being accelerated
by gravitation is the inertial mass. Worse, there is no way to define a ‘passive gravitational
Challenge 354 s mass’ that differs from inertial mass. Try it! All methods that measure a passive gravita-
tional mass, such as weighing an object, cannot be distinguished from the methods that

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
determine inertial mass from its reaction to acceleration. Indeed, all these methods use
the same non-gravitational mechanisms. Bathroom scales are a typical example.
Indeed, if the ‘passive gravitational mass’ were different from the inertial mass, we
would have strange consequences. Not only is it hard to distinguish the two in an exper-
iment; for those bodies for which it were different we would get into trouble with energy
Challenge 355 e conservation.
In fact, also assuming that (‘active’) ‘gravitational mass’ differs from inertial mass
gets us into trouble. How could ‘gravitational mass’ differ from inertial mass? Would the
difference depend on relative velocity, time, position, composition or on mass itself? No.
Challenge 356 s Each of these possibilities contradicts either energy or momentum conservation.
In summary, it is no wonder that all measurements confirm the equality of all mass
types: there is no other option – as Galileo pointed out. The lack of other options is due
to the fundamental equivalence of all mass definitions:

⊳ Mass ratios are acceleration ratios.

Vol. II, page 157 The topic is usually rehashed in general relativity, with no new results, because the defin-
ition of mass remains the same. Gravitational and inertial masses remain equal. In short:

⊳ Mass is a unique property of each body.
204 6 motion due to gravitation

Another, deeper issue remains, though. What is the origin of mass? Why does it exist?
This simple but deep question cannot be answered by classical physics. We will need
some patience to find out.

Curiosities and fun challenges ab ou t gravitation

“
Fallen ist weder gefährlich noch eine Schande;

”
Liegen bleiben ist beides.*
Konrad Adenauer

Cosmonauts on the International Space Station face two challenges: cancer-inducing
cosmic radiation and the lack of gravity. The lack of gravity often leads to orientation
problems and nausea in the first days, the so-called space sickness and motion sickness.
When these disappear, the muscles start to reduce in volume by a few % per month,
bones get weaker every week, the immune system is on permanent alarm state, blood
gets pumped into the head more than usual and produces round ‘baby’ faces – easily
seen on television – and strong headaches, legs loose blood and get thinner, body tem-

Motion Mountain – The Adventure of Physics
perature permanently increases by over one degree Celsius, the brain gets compressed by
the blood and spinal fluid, and the eyesight deteriorates, because also the eyes get com-
pressed. When cosmonauts return to Earth after six months in space, they have weak
bones and muscles, and they are unable to walk and stand. They need a day or two to
learn to do so again. Later, they often get hernias, and because of the bone reduction,
kidney stones. Other health issues are likely to exist; but they have been kept confiden-
tial by cosmonauts in order to maintain their image and their chances for subsequent
missions.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Gravity on the Moon is only one sixth of that on the Earth. Why does this imply that
it is difficult to walk quickly and to run on the Moon (as can be seen in the TV images
recorded there)?
∗∗
Understand and explain the following statement: a beam balance measures mass, a spring
Challenge 357 e scale measures weight.
∗∗
Does the Earth have other satellites apart from the Moon and the artificial satellites shot
into orbit up by rockets? Yes. The Earth has a number of mini-satellites and a large num-
ber of quasi-satellites. An especially long-lived quasi-satellite, an asteroid called 2016
HO3, has a size of about 60 m and was discovered in 2016. As shown in Figure 157, it
orbits the Earth and will continue to do so for another few hundred years, at a distance
from 40 to 100 times that of the Moon.
∗∗

* ‘Falling is neither dangerous nor a shame; to keep lying is both.’ Konrad Adenauer (b. 1876 Köln,
d. 1967 Rhöndorf ), West German Chancellor.
6 motion due to gravitation 205

Motion Mountain – The Adventure of Physics
F I G U R E 157 The calculated orbit of the quasi-satellite 2016 HO3, a temporary companion of the Earth
(courtesy NASA).

Show that a sphere bouncing – without energy loss – down an inclined plane hits the
plane in spots whose distances increase by a constant amount at every bounce.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Is the acceleration due to gravity constant over time? Not really. Every day, it is estimated
that 108 kg of material fall onto the Earth in the form of meteorites and asteroids. (Ex-
amples can be seen in Figure 158 and Figure 159.) Nevertheless, it is unknown whether
the mass of the Earth increases with time (due to the collection of meteorites and cosmic
dust) or decreases (due to gas loss). If you find a way to settle the issue, publish it.
∗∗
Incidentally, discovering objects hitting the Earth is not at all easy. Astronomers like to
point out that an asteroid as large as the one that led to the extinction of the dinosaurs
could hit the Earth without any astronomer noticing in advance, if the direction is slightly
unusual, such as from the south, where few telescopes are located.
∗∗
Several humans have survived free falls from aeroplanes for a thousand metres or more,
even though they had no parachute. A minority of them even did so without any harm
Challenge 358 s at all. How was this possible?
∗∗
Imagine that you have twelve coins of identical appearance, of which one is a forgery.
The forged one has a different mass from the eleven genuine ones. How can you decide
which is the forged one and whether it is lighter or heavier, using a simple balance only
206 6 motion due to gravitation

Motion Mountain – The Adventure of Physics
F I G U R E 158 A composite photograph of the Perseid meteor shower that is visible every year in mid
August. In that month, the Earth crosses the cloud of debris stemming from comet Swift–Tuttle, and
the source of the meteors appears to lie in the constellation of Perseus, because that is the direction in
which the Earth is moving in mid August. The effect and the picture are thus similar to what is seen on
Page 156 the windscreen when driving by car while it is snowing. (© Brad Goldpaint at goldpaintphotography.
com).

F I G U R E 159 Two photographs, taken about a second apart, showing a meteor break-up (© Robert
Mikaelyan).

Challenge 359 e three times?
You have nine identically-looking spheres, all of the same mass, except one, which is
heavier. Can you determine which one, using the balance only two times?
∗∗
For a physicist, antigravity is repulsive gravity – it does not exist in nature. Nevertheless,
6 motion due to gravitation 207

F I G U R E 160 Brooms fall more rapidly than stones (© Luca
Gastaldi).

Motion Mountain – The Adventure of Physics
the term ‘antigravity’ is used incorrectly by many people, as a short search on the in-
ternet shows. Some people call any effect that overcomes gravity, ‘antigravity’. However,
this definition implies that tables and chairs are antigravity devices. Following the defin-
ition, most of the wood, steel and concrete producers are in the antigravity business. The
internet definition makes absolutely no sense.
∗∗
Challenge 360 s What is the cheapest way to switch gravity off for 25 seconds?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Do all objects on Earth fall with the same acceleration of 9.8 m/s2 , assuming that air
resistance can be neglected? No; every housekeeper knows that. You can check this by
yourself. As shown in Figure 160, a broom angled at around 35° hits the floor before a
Challenge 361 s stone, as the sounds of impact confirm. Are you able to explain why?
∗∗
Also bungee jumpers are accelerated more strongly than 𝑔. For a bungee cord of mass 𝑚
and a jumper of mass 𝑀, the maximum acceleration 𝑎 is

𝑚 𝑚
𝑎 = 𝑔 (1 + (4 + )) . (59)
8𝑀 𝑀

Challenge 362 s Can you deduce the relation from Figure 161?
∗∗
Challenge 363 s Guess: What is the mass of a ball of cork with a radius of 1 m?
∗∗
Challenge 364 s Guess: One thousand 1 mm diameter steel balls are collected. What is the mass?
208 6 motion due to gravitation

1000 km

Motion Mountain – The Adventure of Physics
F I G U R E 161 The starting situation F I G U R E 162 An honest balance?
for a bungee jumper.

∗∗
How can you use your observations made during your travels with a bathroom scale to
Challenge 365 s show that the Earth is not flat?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Both the Earth and the Moon attract bodies. The centre of mass of the Earth–Moon
system is 4800 km away from the centre of the Earth, quite near its surface. Why do
Challenge 366 s bodies on Earth still fall towards the centre of the Earth?
∗∗
Does every spherical body fall with the same acceleration? No. If the mass of the object
is comparable to that of the Earth, the distance decreases in a different way. Can you
Challenge 367 e confirm this statement? Figure 162 shows a related puzzle. What then is wrong about
Page 202 Galileo’s argument about the constancy of acceleration of free fall?
∗∗
What is the fastest speed that a human can achieve making use of gravitational accele-
ration? There are various methods that try this; a few are shown in Figure 163. Terminal
speed of free falling skydivers can be even higher, but no reliable record speed value ex-
ists. The last word is not spoken yet, as all these records will be surpassed in the coming
years. It is important to require normal altitude; at stratospheric altitudes, speed values
Vol. II, page 136 can be four times the speed values at low altitude.
∗∗
It is easy to put a mass of a kilogram onto a table. Twenty kilograms is harder. A thousand
6 motion due to gravitation 209

Motion Mountain – The Adventure of Physics
F I G U R E 163 Reducing air resistance increases the terminal speed: left, the 2007 speed skiing world
record holder Simone Origone with 69.83 m/s and right, the 2007 speed world record holder for
bicycles on snow Éric Barone with 61.73 m/s (© Simone Origone, Éric Barone).

Challenge 368 s is impossible. However, 6 ⋅ 1024 kg is easy. Why?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Page 199 The friction between the Earth and the Moon slows down the rotation of both. The Moon
stopped rotating millions of years ago, and the Earth is on its way to doing so as well.
When the Earth stops rotating, the Moon will stop moving away from Earth. How far will
Challenge 369 ny the Moon be from the Earth at that time? Afterwards however, even further in the future,
the Moon will move back towards the Earth, due to the friction between the Earth–Moon
system and the Sun. Even though this effect would only take place if the Sun burned for
Challenge 370 s ever, which is known to be false, can you explain it?
∗∗
When you run towards the east, you lose weight. There are two different reasons for this:
the ‘centrifugal’ acceleration increases so that the force with which you are pulled down
diminishes, and the Coriolis force appears, with a similar result. Can you estimate the
Challenge 371 ny size of the two effects?
∗∗
Laboratories use two types of ultracentrifuges: preparative ultracentrifuges isolate vir-
uses, organelles and biomolecules, whereas analytical ultracentrifuges measure the
shape and mass of macromolecules. The fastest commercially available models achieve
200 000 rpm, or 3.3 kHz, and a centrifugal acceleration of 106 ⋅ 𝑔.
∗∗
210 6 motion due to gravitation

F I G U R E 164 The four satellites of Jupiter discovered by Galileo

Motion Mountain – The Adventure of Physics
and their motion (© Robin Scagell).

What is the relation between the time a stone takes falling through a distance 𝑙 and the
Challenge 372 s time a pendulum takes swinging though half a circle of radius 𝑙? (This problem is due to
Galileo.) How many digits of the number π can one expect to determine in this way?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Why can a spacecraft accelerate through the slingshot effect when going round a planet,
Challenge 373 s despite momentum conservation? It is speculated that the same effect is also the reason
for the few exceptionally fast stars that are observed in the galaxy. For example, the star
Ref. 162 HE0457-5439 moves with 720 km/s, which is much higher than the 100 to 200 km/s of
most stars in the Milky Way. It seems that the role of the accelerating centre was taken
by a black hole.
∗∗
Ref. 163 The orbit of a planet around the Sun has many interesting properties. What is the hodo-
Challenge 374 s graph of the orbit? What is the hodograph for parabolic and hyperbolic orbits?
∗∗
The Galilean satellites of Jupiter, shown in Figure 164, can be seen with small ama-
teur telescopes. Galileo discovered them in 1610 and called them the Medicean satel-
lites. (Today, they are named, in order of increasing distance from Jupiter, as Io, Europa,
Ganymede and Callisto.) They are almost mythical objects. They were the first bodies
found that obviously did not orbit the Earth; thus Galileo used them to deduce that the
Earth is not at the centre of the universe. The satellites have also been candidates to be
the first standard clock, as their motion can be predicted to high accuracy, so that the
‘standard time’ could be read off from their position. Finally, due to this high accuracy,
in 1676, the speed of light was first measured with their help, as told in the section on
6 motion due to gravitation 211

Earth Moon Earth

Moon

Sun

Motion Mountain – The Adventure of Physics
F I G U R E 165 Which of the two Moon paths is correct?

Vol. II, page 16 special relativity.
∗∗
A simple, but difficult question: if all bodies attract each other, why don’t or didn’t all
Challenge 375 s stars fall towards each other? Indeed, the inverse square expression of universal gravity
has a limitation: it does not allow one to make sensible statements about the matter in the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
universe. Universal gravity predicts that a homogeneous mass distribution is unstable;
indeed, an inhomogeneous distribution is observed. However, universal gravity does not
predict the average mass density, the darkness at night, the observed speeds of the distant
galaxies, etc. In fact, ‘universal’ gravity does not explain or predict a single property of
Vol. II, page 211 the universe. To do this, we need general relativity.
∗∗
The acceleration 𝑔 due to gravity at a depth of 3000 km is 10.05 m/s2 , over 2 % more
Ref. 164 than at the surface of the Earth. How is this possible? Also, on the Tibetan plateau, 𝑔 is
influenced by the material below it.
∗∗
When the Moon circles the Sun, does its path have sections concave towards the Sun, as
Challenge 376 s shown at the right of Figure 165, or not, as shown on the left? (Independent of this issue,
both paths in the diagram disguise that the Moon path does not lie in the same plane as
the path of the Earth around the Sun.)
∗∗
You can prove that objects attract each other (and that they are not only attracted by the
Earth) with a simple experiment that anybody can perform at home, as described on the
www.fourmilab.ch/gravitation/foobar website.
212 6 motion due to gravitation

∗∗
It is instructive to calculate the escape velocity from the Earth, i.e., that velocity with
which a body must be thrown so that it never falls back. It turns out to be around 11 km/s.
(This was called the second cosmic velocity in the past; the first cosmic velocity was the
name given to the lowest speed for an orbit, 7.9 km/s.) The exact value of the escape
Challenge 377 e velocity depends on the latitude of the thrower, and on the direction of the throw. (Why?)
What is the escape velocity from the Solar System? (It was once called the third cosmic
velocity.) By the way, the escape velocity from our galaxy is over 500 km/s. What would
happen if a planet or a system were so heavy that the escape velocity from it would be
Challenge 378 s larger than the speed of light?
∗∗
Challenge 379 s What is the largest asteroid one can escape from by jumping?
∗∗
For bodies of irregular shape, the centre of gravity of a body is not the same as the centre

Motion Mountain – The Adventure of Physics
Challenge 380 s of mass. Are you able to confirm this? (Hint: Find and use the simplest example possible.)
∗∗
Can gravity produce repulsion? What happens to a small test body on the inside of a
Challenge 381 ny large C-shaped mass? Is it pushed towards the centre of mass?
∗∗
A heavily disputed argument for the equality of inertial and gravitational mass was given
Ref. 165 by Chubykalo, and Vlaev. The total kinetic energy 𝑇 of two bodies circling around their

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
common centre of mass, like the Earth and the Moon, is given by 𝑇 = 𝐺𝑚𝑀/2𝑅, where
the two quantities 𝑚 and 𝑀 are the gravitational masses of the two bodies and 𝑅 their
distance. From this expression, in which the inertial masses do not appear on the right
side, they deduce that the inertial and gravitational mass must be proportional to each
Challenge 382 s other. Can you see how? Is the reasoning correct?
∗∗
Ref. 166 The shape of the Earth is not a sphere. As a consequence, a plumb line usually does not
Challenge 383 ny point to the centre of the Earth. What is the largest deviation in degrees?
∗∗
Owing to the slightly flattened shape of the Earth, the source of the Mississippi is about
20 km nearer to the centre of the Earth than its mouth; the water effectively runs uphill.
Challenge 384 s How can this be?
∗∗
If you look at the sky every day at 6 a.m., the Sun’s position varies during the year. The
result of photographing the Sun on the same film is shown in Figure 166. The curve,
called the analemma, is due to two combined effects: the inclination of the Earth’s axis
and the elliptical shape of the Earth’s orbit around the Sun. The top and the (hidden) bot-
tom points of the analemma correspond to the solstices. How does the analemma look
6 motion due to gravitation 213

Motion Mountain – The Adventure of Physics
F I G U R E 166 The analemma over Delphi, taken
between January and December 2002
(© Anthony Ayiomamitis).

Challenge 385 s if photographed every day at local noon? Why is it not a straight line pointing exactly
south?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
The constellation in which the Sun stands at noon (at the centre of the time zone) is sup-
posedly called the ‘zodiacal sign’ of that day. Astrologers say there are twelve of them,
namely Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius, Capri-
cornus, Aquarius and Pisces and that each takes (quite precisely) a twelfth of a year or
a twelfth of the ecliptic. Any check with a calendar shows that at present, the midday
Sun is never in the zodiacal sign during the days usually connected to it. The relation has
Page 152 shifted by about a month since it was defined, due to the precession of the Earth’s axis. A
check with a map of the star sky shows that the twelve constellations do not have the same
length and that on the ecliptic there are fourteen of them, not twelve. There is Ophiuchus
or Serpentarius, the serpent bearer constellation, between Scorpius and Sagittarius, and
Cetus, the whale, between Aquarius and Pisces. In fact, not a single astronomical state-
Ref. 167 ment about zodiacal signs is correct. To put it clearly, astrology, in contrast to its name,
is not about stars. (In German, the word ‘Strolch’, meaning ‘rogue’ or ‘scoundrel’, is
derived from the word ‘astrologer’.)
∗∗
For a long time, it was thought that there is no additional planet in our Solar System out-
Ref. 168 side Neptune and Pluto, because their orbits show no disturbances from another body.
Today, the view has changed. It is known that there are only eight planets: Pluto is not
214 6 motion due to gravitation

Sedna
Jupiter Kuiper Belt

Mars
Earth
Venus Uranus
Mercury
Saturn

Jupiter

Asteroids Pluto
Inner Outer
Solar System Solar System

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Inner extent
of Oort Cloud Orbit of
Sedna

F I G U R E 167 The orbit of Sedna in comparison with the orbits of the planets in the Solar System (NASA).

a planet, but the first of a set of smaller objects in the so-called Kuiper belt. Kuiper belt
objects are regularly discovered; over 1000 are known today.
In 2003, two major Kuiper objects were discovered; one, called Sedna, is almost as
Ref. 169 large as Pluto, the other, called Eris, is even larger than Pluto and has a moon. Both
have strongly elliptical orbits (see Figure 167). Since Pluto and Eris, like the asteroid
Ceres, have cleaned their orbit from debris, these three objects are now classified as dwarf
planets.
Outside the Kuiper belt, the Solar System is surrounded by the so-called Oort cloud. In
contrast to the flattened Kuiper belt, the Oort cloud is spherical in shape and has a radius
of up to 50 000 AU, as shown in Figure 167 and Figure 168. The Oort cloud consists of
a huge number of icy objects consisting of mainly of water, and to a lesser degree, of
methane and ammonia. Objects from the Oort cloud that enter the inner Solar System
become comets; in the distant past, such objects have brought water onto the Earth.
6 motion due to gravitation 215

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 168 The Kuiper belt, containing mainly planetoids, and the Oort cloud orbit, containing
comets, around the Solar System (NASA, JPL, Donald Yeoman).

∗∗
In astronomy new examples of motion are regularly discovered even in the present cen-
tury. Sometimes there are also false alarms. One example was the alleged fall of mini
comets on the Earth. They were supposedly made of a few dozen kilograms of ice, hitting
Ref. 170 the Earth every few seconds. It is now known not to happen.
∗∗
Universal gravity allows only elliptical, parabolic or hyperbolic orbits. It is impossible
for a small object approaching a large one to be captured. At least, that is what we have
learned so far. Nevertheless, all astronomy books tell stories of capture in our Solar Sys-
tem; for example, several outer satellites of Saturn have been captured. How is this pos-
Challenge 386 s sible?
∗∗
How would a tunnel have to be shaped in order that a stone would fall through it without
touching the walls? (Assume constant density.) If the Earth did not rotate, the tunnel
216 6 motion due to gravitation

would be a straight line through its centre, and the stone would fall down and up again,
in an oscillating motion. For a rotating Earth, the problem is much more difficult. What
Challenge 387 s is the shape when the tunnel starts at the Equator?
∗∗
The International Space Station circles the Earth every 90 minutes at an altitude of about
380 km. You can see where it is from the website www.heavens-above.com. By the way,
whenever it is just above the horizon, the station is the third brightest object in the night
Challenge 388 e sky, superseded only by the Moon and Venus. Have a look at it.
∗∗
Is it true that the centre of mass of the Solar System, its barycentre, is always inside the
Challenge 389 s Sun? Even though the Sun or a star move very little when planets move around them,
this motion can be detected with precision measurements making use of the Doppler
Vol. II, page 31 effect for light or radio waves. Jupiter, for example, produces a speed change of 13 m/s
in the Sun, the Earth 1 m/s. The first planets outside the Solar System, around the pulsar

Motion Mountain – The Adventure of Physics
PSR1257+12 and around the normal G-type star Pegasi 51, were discovered in this way,
in 1992 and 1995. In the meantime, several thousand so-called exoplanets have been
discovered with this and other methods. Some have even masses comparable to that of
the Earth. This research also showed that exoplanets are more numerous than stars, and
that earth-like planets are rare.
∗∗
Not all points on the Earth receive the same number of daylight hours during a year. The
Challenge 390 d effects are difficult to spot, though. Can you find one?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Can the phase of the Moon have a measurable effect on the human body, for example
Challenge 391 s through tidal effects?
∗∗
There is an important difference between the heliocentric system and the old idea that
all planets turn around the Earth. The heliocentric system states that certain planets,
such as Mercury and Venus, can be between the Earth and the Sun at certain times, and
behind the Sun at other times. In contrast, the geocentric system states that they are al-
ways in between. Why did such an important difference not immediately invalidate the
geocentric system? And how did the observation of phases, shown in Figure 169 and
Challenge 392 s Figure 170, invalidate the geocentric system?
∗∗
The strangest reformulation of the description of motion given by 𝑚𝑎 = ∇𝑈 is the almost
Ref. 171 absurd looking equation
∇𝑣 = d𝑣/d𝑠 (60)

where 𝑠 is the motion path length. It is called the ray form of the equation of motion. Can
Challenge 393 s you find an example of its application?
6 motion due to gravitation 217

Motion Mountain – The Adventure of Physics
F I G U R E 169 The
phases of the Moon
and of Venus, as
observed from Athens
in summer 2007
(© Anthony
Ayiomamitis).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 170 Universal gravitation also explains the observations of Venus, the evening and morning
star. In particular, universal gravitation, and the elliptical orbits it implies, explains its phases and its
change of angular size. The pictures shown here were taken in 2004 and 2005. The observations can
easily be made with a binocular or a small telescope (© Wah!; ﬁlm available at apod.nasa.gov/apod/
ap060110.html).

∗∗
Seen from Neptune, the size of the Sun is the same as that of Jupiter seen from the Earth
Challenge 394 s at the time of its closest approach. True?
∗∗
Ref. 172 The gravitational acceleration for a particle inside a spherical shell is zero. The vanishing
of gravity in this case is independent of the particle shape and its position, and independ-
Challenge 395 s ent of the thickness of the shell. Can you find the argument using Figure 171? This works
only because of the 1/𝑟2 dependence of gravity. Can you show that the result does not
218 6 motion due to gravitation

d𝑚
𝑟

𝑚
𝑅

d𝑀

F I G U R E 171 The vanishing of gravitational force inside a
spherical shell of matter.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 172 Le Sage’s own illustration of his model, showing the smaller density of ‘ultramondane
corpuscules’ between the attracting bodies and the higher density outside them (© Wikimedia)

hold for non-spherical shells? Note that the vanishing of gravity inside a spherical shell
usually does not hold if other matter is found outside the shell. How could one eliminate
Challenge 396 s the effects of outside matter?
∗∗
What is gravity? This simple question has a long history. In 1690, Nicolas Fatio de Duillier
Ref. 173 and in 1747, Georges-Louis Le Sage proposed an explanation for the 1/𝑟2 dependence. Le
Sage argued that the world is full of small particles – he called them ‘corpuscules ultra-
mondains’ – flying around randomly and hitting all objects. Single objects do not feel
the hits, since they are hit continuously and randomly from all directions. But when two
objects are near to each other, they produce shadows for part of the flux to the other body,
resulting in an attraction, as shown in Figure 172. Can you show that such an attraction
Challenge 397 e has a 1/𝑟2 dependence?
However, Le Sage’s proposal has a number of problems. First, the argument only
Challenge 398 e works if the collisions are inelastic. (Why?) However, that would mean that all bodies
6 motion due to gravitation 219

Ref. 2 would heat up with time, as Jean-Marc Lévy-Leblond explains. Secondly, a moving body
in free space would be hit by more or faster particles in the front than in the back; as
a result, the body should be decelerated. Finally, gravity would depend on size, but in a
strange way. In particular, three bodies lying on a line should not produce shadows, as
no such shadows are observed; but the naive model predicts such shadows.
Despite all criticisms, the idea that gravity is due to particles has regularly resurfaced
in physics research ever since. In the most recent version, the hypothetical particles are
called gravitons. On the other hand, no such particles have ever been observed. We will
understand the origin of gravitation in the final part of our mountain ascent.
∗∗
Challenge 399 ny For which bodies does gravity decrease as you approach them?
∗∗
Could one put a satellite into orbit using a cannon? Does the answer depend on the
Challenge 400 s direction in which one shoots?

Motion Mountain – The Adventure of Physics
∗∗
Two old computer users share experiences. ‘I threw my Pentium III and Pentium IV out
of the window.’ ‘And?’ ‘The Pentium III was faster.’
∗∗
Challenge 401 s How often does the Earth rise and fall when seen from the Moon? Does the Earth show
phases?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 402 ny What is the weight of the Moon? How does it compare with the weight of the Alps?
∗∗
If a star is made of high-density material, the speed of a planet orbiting near to it could
Challenge 403 s be greater than the speed of light. How does nature avoid this strange possibility?
∗∗
What will happen to the Solar System in the future? This question is surprisingly hard to
answer. The main expert of this topic, French planetary scientist Jacques Laskar, simu-
Ref. 174 lated a few hundred million years of evolution using computer-aided calculus. He found
Page 424 that the planetary orbits are stable, but that there is clear evidence of chaos in the evolu-
tion of the Solar System, at a small level. The various planets influence each other in
subtle and still poorly understood ways. Effects in the past are also being studied, such
as the energy change of Jupiter due to its ejection of smaller asteroids from the Solar
System, or the energy gains of Neptune. There is still a lot of research to be done in this
field.
∗∗
One of the open problems of the Solar System is the description of planet distances dis-
covered in 1766 by Johann Daniel Titius (1729–1796) and publicized by Johann Elert
220 6 motion due to gravitation

TA B L E 27 An unexplained property of nature: planet
distances from the Sun and the values resulting from the
Titius–Bode rule.

Planet 𝑛 predicted measured
d i s ta nc e i n AU

Mercury −∞ 0.4 0.4
Venus 0 0.7 0.7
Earth 1 1.0 1.0
Mars 2 1.6 1.5
Planetoids 3 2.8 2.2 to 3.2
Jupiter 4 5.2 5.2
Saturn 5 10.0 9.5
Uranus 6 19.6 19.2
Neptune 7 38.8 30.1
Pluto 8 77.2 39.5

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 173 The motion of the
planetoids compared to that of the
planets (Shockwave animation
© Hans-Christian Greier)

Bode (1747–1826). Titius discovered that planetary distances 𝑑 from the Sun can be ap-
proximated by
𝑑 = 𝑎 + 2𝑛 𝑏 with 𝑎 = 0.4 AU , 𝑏 = 0.3 AU (61)

where distances are measured in astronomical units and 𝑛 is the number of the planet.
The resulting approximation is compared with observations in Table 27.
Interestingly, the last three planets, as well as the planetoids, were discovered after
Bode’s and Titius’ deaths; the rule had successfully predicted Uranus’ distance, as well
as that of the planetoids. Despite these successes – and the failure for the last two planets
– nobody has yet found a model for the formation of the planets that explains Titius’
rule. The large satellites of Jupiter and of Uranus have regular spacing, but not according
6 motion due to gravitation 221

TA B L E 28 The orbital
periods known to the
Babylonians.

B ody Period

Saturn 29 a
Jupiter 12 a
Mars 687 d
Sun 365 d
Venus 224 d
Mercury 88 d
Moon 29 d

to the Titius–Bode rule.
Explaining or disproving the rule is one of the challenges that remain in classical

Motion Mountain – The Adventure of Physics
Ref. 175 mechanics. Some researchers maintain that the rule is a consequence of scale invari-
Ref. 176 ance, others maintain that it is an accident or even a red herring. The last interpretation
is also suggested by the non-Titius–Bode behaviour of practically all extrasolar planets.
The issue is not closed.
∗∗
Around 3000 years ago, the Babylonians had measured the orbital times of the seven
celestial bodies that move across the sky. Ordered from longest to shortest, they wrote
them down in Table 28. Six of the celestial bodies are visible in the beautiful Figure 174.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The Babylonians also introduced the week and the division of the day into 24 hours.
They dedicated every one of the 168 hours of the week to a celestial body, following the
order of Table 28. They also dedicated the whole day to that celestial body that corres-
ponds to the first hour of that day. The first day of the week was dedicated to Saturn;
Challenge 404 e the present ordering of the other days of the week then follows from Table 28. This story
Ref. 177 was told by Cassius Dio (c. 160 to c. 230). Towards the end of Antiquity, the ordering was
taken up by the Roman empire. In Germanic languages, including English, the Latin
names of the celestial bodies were replaced by the corresponding Germanic gods. The
order Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday and Friday is thus a
consequence of both the astronomical measurements and the astrological superstitions
of the ancients.
∗∗
In 1722, the great mathematician Leonhard Euler made a mistake in his calculation that
led him to conclude that if a tunnel, or better, a deep hole were built from one pole of
the Earth to the other, a stone falling into it would arrive at the Earth’s centre and then
immediately turn and go back up. Voltaire made fun of this conclusion for many years.
Can you correct Euler and show that the real motion is an oscillation from one pole
to the other, and can you calculate the time a pole-to-pole fall would take (assuming
Challenge 405 s homogeneous density)?
What would be the oscillation time for an arbitrary straight surface-to-surface tunnel
222 6 motion due to gravitation

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 174 These are the six
celestial bodies that are visible
at night with the naked eye and
whose positions vary over the
course of the year. The nearly
vertical line connecting them is
the ecliptic, the narrow stripe
around it the zodiac. Together
with the Sun, the seven celestial
bodies were used to name the
days of the week. (© Alex
Cherney)
6 motion due to gravitation 223

F I G U R E 175 The solar eclipse of 11 August 1999, photographed by Jean-Pierre Haigneré, member of
the Mir 27 crew, and the (enhanced) solar eclipse of 29 March 2006 (© CNES and Laurent
Laveder/PixHeaven.net).

Motion Mountain – The Adventure of Physics
Earth

wire

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 176 A wire
attached to the Earth’s
Equator.

Challenge 406 s of length 𝑙, thus not going from pole to pole?
The previous challenges circumvented the effects of the Earth’s rotation. The topic
Ref. 178 becomes much more interesting if rotation is included. What would be the shape of a
Challenge 407 s tunnel so that a stone falling through it never touches the wall?
∗∗
Figure 175 shows a photograph of a solar eclipse taken from the Russian space station
Mir and a photograph taken at the centre of the shadow from the Earth. Indeed, a global
view of a phenomenon can be quite different from a local one. What is the speed of the
Challenge 408 s shadow?
∗∗
In 2005, satellite measurements have shown that the water in the Amazon river presses
down the land up to 75 mm more in the season when it is full of water than in the season
Ref. 179 when it is almost empty.
224 6 motion due to gravitation

∗∗
Imagine that a wire existed that does not break. How long would such a wire have to
be so that, when attached to the Equator, it would stand upright in the air, as shown in
Challenge 409 s Figure 176? Could one build an elevator into space in this way?
∗∗
Usually there are roughly two tides per day. But there are places, such as on the coast
of Vietnam, where there is only one tide per day. See www.jason.oceanobs.com/html/
Challenge 410 ny applications/marees/marees_m2k1_fr.html. Why?
∗∗
It is sufficient to use the concept of centrifugal force to show that the rings of Saturn
cannot be made of massive material, but must be made of separate pieces. Can you find
Challenge 411 s out how?
∗∗

Motion Mountain – The Adventure of Physics
Why did Mars lose its atmosphere? Nobody knows. It has recently been shown that the
solar wind is too weak for this to happen. This is one of the many open riddles of the
solar system.
∗∗
All bodies in the Solar System orbit the Sun in the same direction. All? No; there are ex-
ceptions. One intriguing asteroid that orbits the Sun near Jupiter in the wrong direction
was discovered in 2015: it has a size of 3 km. For an animation of its astonishing orbit,
opposite to all Trojan asteroids, see www.astro.uwo.ca/~wiegert.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
The observed motion due to gravity can be shown to be the simplest possible, in the fol-
lowing sense. If we measure change of a falling object with the expression ∫ 𝑚𝑣2 /2 −
Page 253 𝑚𝑔ℎ d𝑡, then a constant acceleration due to gravity minimizes the change in every ex-
Challenge 412 e ample of fall. Can you confirm this?
∗∗
Motion due to gravity is fun: think about roller coasters. If you want to know more about
how they are built, visit www.vekoma.com.
∗∗
Gravity and air friction lead to interesting effects. What is the shape of a rope hanging
from a helicopter when the helicopter is flying? What is the shape when there is a weight
at the end of the rope? A small parachute? (The internet has the answer.)

“
The scientific theory I like best is that the rings

”
of Saturn are made of lost airline luggage.
Mark Russel
6 motion due to gravitation 225

Summary on gravitation
Spherical bodies of mass 𝑀 attract other bodies at a distance 𝑟 by inducing an accele-
ration towards them given by 𝑎 = 𝐺𝑀/𝑟2 . This expression, universal gravitation, de-
scribes snowboarders, skiers, paragliders, athletes, couch potatoes, pendula, stones, can-
ons, rockets, tides, eclipses, planet shapes, planet motion and much more. Universal grav-
itation is the first example of a unified description: it describes how everything falls. By the
acceleration it produces, gravitation limits the appearance of uniform motion in nature.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Chapter 7

C L A S SIC A L M E C HA N IC S , F ORC E A N D
T H E PR E DIC TA BI L I T Y OF MOT ION

A
ll those types of motion in which the only permanent property of
body is mass define the field of mechanics. The same name is given
lso to the experts studying the field. We can think of mechanics as the ath-
letic part of physics.** Both in athletics and in mechanics only lengths, times and masses
are measured – and of interest at all.

Motion Mountain – The Adventure of Physics
More specifically, our topic of investigation so far is called classical mechanics, to dis-
tinguish it from quantum mechanics. The main difference is that in classical physics ar-
bitrary small values are assumed to exist, whereas this is not the case in quantum physics.
Classical mechanics is often also called Galilean physics or Newtonian physics.***
Classical mechanics states that motion is predictable: it thus states that there are no
surprises in motion. Is this correct in all cases? Is predictability valid in the presence
of friction? Of free will? Are there really no surprises in nature? These issues merit a
discussion; they will accompany us for a stretch of our adventure.
We know that there is more to the world than gravity. Simple observations make this
point: floors and friction. Neither can be due to gravity. Floors do not fall, and thus are

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
not described by gravity; and friction is not observed in the skies, where motion is purely
due to gravity.**** Also on Earth, friction is unrelated to gravity, as you might want to
Challenge 413 e check yourself. There must be another interaction responsible for friction. We shall study
it in the third volume. But a few issues merit a discussion right away.

** This is in contrast to the actual origin of the term ‘mechanics’, which means ‘machine science’. It derives
from the Greek μηκανή, which means ‘machine’ and even lies at the origin of the English word ‘machine’
itself. Sometimes the term ‘mechanics’ is used for the study of the motion of solid bodies only, excluding,
e.g., hydrodynamics. This use fell out of favour in physics in the twentieth century.
*** The basis of classical mechanics, the description of motion using only space and time, is called kinemat-
ics. An example is the description of free fall by 𝑧(𝑡) = 𝑧0 + 𝑣0 (𝑡 − 𝑡0 ) − 12 𝑔(𝑡 − 𝑡0 )2 . The other, main part of
classical mechanics is the description of motion as a consequence of interactions between bodies; it is called
dynamics. An example of dynamics is the formula of universal gravity. The distinction between kinematics
and dynamics can also be made in relativity, thermodynamics and electrodynamics.
**** This is not completely correct: in the 1980s, the first case of gravitational friction was discovered: the
Vol. II, page 174 emission of gravity waves. We discuss it in detail in the chapter on general relativity. The discovery does not
change the main point, however.
7 classical mechanics, force and the predictability of motion 227

Motion Mountain – The Adventure of Physics
F I G U R E 177 The parabola shapes formed by accelerated water beams show that motion in everyday
life is predictable (© Oase GmbH).

TA B L E 29 Some force values in nature.

O b s e r va t i o n Force

Value measured in a magnetic resonance force microscope 820 zN
Force needed to rip a DNA molecule apart by pulling at its two ends 600 pN

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Maximum force exerted by human bite 2.1 kN
Force exerted by quadriceps up to 3 kN
Typical peak force exerted by sledgehammer 5 kN
Force sustained by 1 cm2 of a good adhesive up to 10 kN
Force needed to tear a good rope used in rock climbing 30 kN
Maximum force measurable in nature 3.0 ⋅ 1043 N

Should one use force? Power?

“
The direct use of physical force is so poor a
solution [...] that it is commonly employed only

”
by small children and great nations.
David Friedman

Everybody has to take a stand on this question, even students of physics. Indeed, many
types of forces are used and observed in daily life. One speaks of muscular, gravitational,
psychic, sexual, satanic, supernatural, social, political, economic and many others. Physi-
cists see things in a simpler way. They call the different types of forces observed between
objects interactions. The study of the details of all these interactions will show that, in
everyday life, they are of electrical or gravitational origin.
For physicists, all change is due to motion. The term force then also takes on a more
228 7 classical mechanics, force and the predictability of motion

restrictive definition. (Physical) force is defined as the change of momentum with time,
i.e., as
d𝑝
𝐹= . (62)
d𝑡
A few measured values are listed in Figure 29.
Ref. 87 A horse is running so fast that the hooves touch the ground only 20 % of the time.
Challenge 414 s What is the load carried by its legs during contact?
Force is the change of momentum. Since momentum is conserved, we can also say
that force measures the flow of momentum. As we will see in detail shortly, whenever a
force accelerates a body, momentum flows into it. Indeed, momentum can be imagined to
Ref. 87 be some invisible and intangible substance. Force measures how much of this substance
flows into or out of a body per unit time.

⊳ Force is momentum flow.

Motion Mountain – The Adventure of Physics
The conservation of momentum is due to the conservation of this liquid. Like any liquid,
momentum flows through a surface.
Using the Galilean definition of linear momentum 𝑝 = 𝑚𝑣, we can rewrite the defin-
ition of force (for constant mass) as

𝐹 = 𝑚𝑎 , (63)

where 𝐹 = 𝐹(𝑡, 𝑥) is the force acting on an object of mass 𝑚 and where 𝑎 = 𝑎(𝑡, 𝑥) =
d𝑣/d𝑡 = d2 𝑥/d𝑡2 is the acceleration of the same object, that is to say, its change of ve-
locity.* The expression states in precise terms that force is what changes the velocity of

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
masses. The quantity is called ‘force’ because it corresponds in many, but not all aspects
to everyday muscular force. For example, the more force is used, the further a stone can
be thrown. Equivalently, the more momentum is pumped into a stone, the further it can
be thrown. As another example, the concept of weight describes the flow of momentum
due to gravity.

⊳ Gravitation constantly pumps momentum into massive bodies.

Sand in an hourglass is running, and the hourglass is on a scale. Is the weight shown
Challenge 415 s on the scale larger, smaller or equal to the weight when the sand has stopped falling?
Forces are measured with the help of deformations of bodies. Everyday force values
can be measured by measuring the extension of a spring. Small force values, of the order
of 1 nN, can be detected by measuring the deflection of small levers with the help of a
reflected laser beam.

* This equation was first written down by the mathematician and physicist Leonhard Euler (b. 1707 Basel,
d. 1783 St. Petersburg) in 1747, 20 years after the death of Newton, to whom it is usually and falsely ascribed.
It was Euler, one of the greatest mathematicians of all time (and not Newton), who first understood that
this definition of force is useful in every case of motion, whatever the appearance, be it for point particles or
Ref. 28 extended objects, and be they rigid, deformable or fluid bodies. Surprisingly and in contrast to frequently-
Vol. II, page 83 made statements, equation (63) is even correct in relativity.
7 classical mechanics, force and the predictability of motion 229

Flow
of +p

Motion Mountain – The Adventure of Physics
F I G U R E 178 The pulling child pumps momentum into the chariot. In fact, some momentum ﬂows
back to the ground due to dynamic friction (not drawn).

However, whenever the concept of force is used, it should be remembered that physical
force is different from everyday force or everyday effort. Effort is probably best approxim-
ated by the concept of (physical) power, usually abbreviated 𝑃, and defined (for constant
force) as
d𝑊
𝑃= = 𝐹𝑣 (64)
d𝑡

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in which (physical) work 𝑊 is defined as 𝑊 = 𝐹𝑠, where 𝑠 is the distance along which
the force acts. Physical work is a form of energy, as you might want to check. Work, as a
form of energy, has to be taken into account when the conservation of energy is checked.
With the definition of work just given you can solve the following puzzles. What hap-
Challenge 416 s pens to the electricity consumption of an escalator if you walk on it instead of standing
Challenge 417 d still? What is the effect of the definition of power for the salary of scientists? A man who
Challenge 418 s walks carrying a heavy rucksack is hardly doing any work; why then does he get tired?
When students in exams say that the force acting on a thrown stone is least at the
Ref. 180 highest point of the trajectory, it is customary to say that they are using an incorrect
view, namely the so-called Aristotelian view, in which force is proportional to velocity.
Sometimes it is even said that they are using a different concept of state of motion. Critics
then add, with a tone of superiority, how wrong all this is. This is an example of intellec-
tual disinformation. Every student knows from riding a bicycle, from throwing a stone
or from pulling an object that increased effort results in increased speed. The student is
right; those theoreticians who deduce that the student has a mistaken concept of force
are wrong. In fact, instead of the physical concept of force, the student is just using the
everyday version, namely effort. Indeed, the effort exerted by gravity on a flying stone is
least at the highest point of the trajectory. Understanding the difference between physical
force and everyday effort is the main hurdle in learning mechanics.*
* This stepping stone is so high that many professional physicists do not really take it themselves; this is
230 7 classical mechanics, force and the predictability of motion

Description with Description with
forces at one single momentum flow
point

Alternatively:

+p -p

Motion Mountain – The Adventure of Physics
Flow
of -p
Flow
of +p

Flow
of +p

F I G U R E 179 The two equivalent descriptions of situations with zero net force, i.e., with a closed
momentum ﬂow. Compression occurs when momentum ﬂow and momentum point in the same
direction; extension occurs when momentum ﬂow and momentum point in opposite directions.

confirmed by the innumerable comments in papers that state that physical force is defined using mass,
and, at the same time, that mass is defined using force (the latter part of the sentence being a fundamental
mistake).
7 classical mechanics, force and the predictability of motion 231

Often the flow of momentum, equation (62), is not recognized as the definition of
force. This is mainly due to an everyday observation: there seem to be forces without
any associated acceleration or change in momentum, such as in a string under tension
or in water at high pressure. When one pushes against a tree, as shown in Figure 179,
there is no motion, yet a force is applied. If force is momentum flow, where does the
momentum go? It flows into the slight deformations of the arms and the tree. In fact,
when one starts pushing and thus deforming, the associated momentum change of the
molecules, the atoms, or the electrons of the two bodies can be observed. After the de-
formation is established a continuous and equal flow of momentum is going on in both
directions.
Ref. 87 Because force is net momentum flow, the concept of force is not really needed in the
description of motion. But sometimes the concept is practical. This is the case in everyday
life, where it is useful in situations where net momentum values are small or negligible.
For example, it is useful to define pressure as force per area, even though it is actually
a momentum flow per area. At the microscopic level, momentum alone suffices for the
description of motion.

Motion Mountain – The Adventure of Physics
In the section title we asked about the usefulness of force and power. Before we can
answer conclusively, we need more arguments. Through its definition, the concepts of
force and power are distinguished clearly from ‘mass’, ‘momentum’, ‘energy’ and from
each other. But where do forces originate? In other words, which effects in nature have
the capacity to accelerate bodies by pumping momentum into objects? Table 30 gives an
overview.

Forces, surfaces and conservation
We saw that force is the change of momentum. We also saw that momentum is conserved.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
How do these statements come together? The answer is the same for all conserved quant-
ities. We imagine a closed surface that is the boundary of a volume in space. Conserva-
tion implies that the conserved quantity enclosed inside the surface can only change by
flowing through that surface.*
All conserved quantities in nature – such as energy, linear momentum, electric charge,
and angular momentum – can only change by flowing through surfaces. In particular,
whenever the momentum of a body changes, this happens through a surface. Momentum
change is due to momentum flow. In other words, the concept of force always implies a
surface through which momentum flows.

* Mathematically, the conservation of a quantity 𝑞 is expressed with the help of the volume density 𝜌 = 𝑞/𝑉,
the current 𝐼 = 𝑞/𝑡, and the flow or flux 𝑗 = 𝜌𝑣, so that 𝑗 = 𝑞/𝐴𝑡. Conservation then implies

d𝑞 ∂𝜌
=∫ d𝑉 = − ∫ 𝑗d𝐴 = −𝐼 (65)
d𝑡 𝑉 ∂𝑡 𝐴=∂𝑉

or, equivalently,
∂𝜌
+ ∇𝑗 = 0 . (66)
∂𝑡
This is the continuity equation for the quantity 𝑞. All this only states that a conserved quantity in a closed
volume 𝑉 can only change by flowing through the surface 𝐴. This is a typical example of how complex
mathematical expressions can obfuscate the simple physical content.
232 7 classical mechanics, force and the predictability of motion

Motion Mountain – The Adventure of Physics
F I G U R E 180 Friction-based processes (courtesy Wikimedia).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
⊳ Force is the flow of momentum through a surface.

Ref. 299 This point is essential in understanding physical force. Every force requires a surface for
its definition.
To refine your own concept of force, you can search for the relevant surface when a
Challenge 419 e rope pulls a chariot, or when an arm pushes a tree, or when a car accelerates. It is also
helpful to compare the definition of force with the definition of power: both quantities
are flows through surfaces. As a result, we can say:

⊳ A motor is a momentum pump.

Friction and motion
Every example of motion, from the motion that lets us choose the direction of our gaze
to the motion that carries a butterfly through the landscape, can be put into one of the
two left-most columns of Table 30. Physically, those two columns are separated by the
following criterion: in the first class, the acceleration of a body can be in a different direc-
tion from its velocity. The second class of examples produces only accelerations that are
exactly opposed to the velocity of the moving body, as seen from the frame of reference
7 classical mechanics, force and the predictability of motion 233

TA B L E 30 Selected processes and devices changing the motion of bodies.

S i t uat i o n s t h at c a n S i t uat i o n s t h at Motors and
l e a d t o ac ce l e r at i on o nly l e a d t o a c t uat o r s
d e c e l e r at i o n
piezoelectricity
quartz under applied voltage thermoluminescence walking piezo tripod
collisions
satellite in planet encounter car crash rocket motor
growth of mountains meteorite crash swimming of larvae
magnetic effects
compass needle near magnet electromagnetic braking electromagnetic gun
magnetostriction transformer losses linear motor
current in wire near magnet electric heating galvanometer
electric effects

Motion Mountain – The Adventure of Physics
rubbed comb near hair friction between solids electrostatic motor
bombs fire muscles, sperm flagella
cathode ray tube electron microscope Brownian motor
light
levitating objects by light light bath stopping atoms (true) light mill
solar sail for satellites light pressure inside stars solar cell
elasticity
bow and arrow trouser suspenders ultrasound motor

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
bent trees standing up again pillow, air bag bimorphs
osmosis
water rising in trees salt conservation of food osmotic pendulum
electro-osmosis tunable X-ray screening
heat & pressure
freezing champagne bottle surfboard water resistance hydraulic engines
tea kettle quicksand steam engine
barometer parachute air gun, sail
earthquakes sliding resistance seismometer
attraction of passing trains shock absorbers water turbine
nuclei
radioactivity plunging into the Sun supernova explosion
biology
bamboo growth decreasing blood vessel molecular motors
diameter
gravitation
falling emission of gravity waves pulley
234 7 classical mechanics, force and the predictability of motion

of the braking medium. Such a resisting force is called friction, drag or a damping. All
Challenge 420 e examples in the second class are types of friction. Just check. Some examples of processes
based on friction are given in Figure 180.
Challenge 421 s Here is a puzzle on cycling: does side wind brake – and why?
Friction can be so strong that all motion of a body against its environment is made
impossible. This type of friction, called static friction or sticking friction, is common and
important: without it, turning the wheels of bicycles, trains or cars would have no effect.
Without static friction, wheels driven by a motor would have no grip. Similarly, not a
single screw would stay tightened and no hair clip would work. We could neither run
nor walk in a forest, as the soil would be more slippery than polished ice. In fact not only
our own motion, but all voluntary motion of living beings is based on friction. The same
is the case for all self-moving machines. Without static friction, the propellers in ships,
aeroplanes and helicopters would not have any effect and the wings of aeroplanes would
Challenge 422 s produce no lift to keep them in the air. (Why?)
In short, static friction is necessary whenever we or an engine want to move against
the environment.

Motion Mountain – The Adventure of Physics
Friction, sport, machines and predictabilit y
Once an object moves through its environment, it is hindered by another type of friction;
it is called dynamic friction and acts between all bodies in relative motion.* Without
Ref. 181 dynamic friction, falling bodies would always rebound to the same height, without ever
coming to a stop; neither parachutes nor brakes would work; and even worse, we would
have no memory, as we will see later.
All motion examples in the second column of Table 30 include friction. In these ex-
amples, macroscopic energy is not conserved: the systems are dissipative. In the first

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
column, macroscopic energy is constant: the systems are conservative.
The first two columns can also be distinguished using a more abstract, mathematical
criterion: on the left are accelerations that can be derived from a potential, on the right,
decelerations that can not. As in the case of gravitation, the description of any kind of
motion is much simplified by the use of a potential: at every position in space, one needs
only the single value of the potential to calculate the trajectory of an object, instead of
the three values of the acceleration or the force. Moreover, the magnitude of the velocity
of an object at any point can be calculated directly from energy conservation.
The processes from the second column cannot be described by a potential. These are
the cases where it is best to use force if we want to describe the motion of the system. For
example, the friction or drag force 𝐹 due to wind resistance of a body is roughly given by

1
𝐹 = 𝑐w 𝜌𝐴𝑣2 (67)
2
where 𝐴 is the area of its cross-section and 𝑣 its velocity relative to the air, 𝜌 is the density
of air. The drag coefficient 𝑐w is a pure number that depends on the shape of the moving

* There might be one exception. Recent research suggest that maybe in certain crystalline systems, such as
tungsten bodies on silicon, under ideal conditions gliding friction can be extremely small and possibly even
Ref. 182 vanish in certain directions of motion. This so-called superlubrication is presently a topic of research.
7 classical mechanics, force and the predictability of motion 235

typical passenger aeroplane cw = 0.03

typical sports car or van cw = 0.44
modern sedan cw = 0.28

dolphin and penguin cw = 0.035

soccer ball
turbulent (above c.10 m/s) cw = 0.2
laminar (below c.10 m/s) cw = 0.45

Motion Mountain – The Adventure of Physics
F I G U R E 181 Shapes and air/water resistance.

object. A few examples are given in Figure 181. The formula is valid for all fluids, not
only for air, below the speed of sound, as long as the drag is due to turbulence. This is
usually the case in air and in water. (At very low velocities, when the fluid motion is not
Page 361 turbulent but laminar, drag is called viscous and follows an (almost) linear relation with
speed.) You may check that drag, or aerodynamic resistance cannot be derived from a
Challenge 423 e potential.*
The drag coefficient 𝑐w is a measured quantity. Calculating drag coefficients with com-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
puters, given the shape of the body and the properties of the fluid, is one of the most
difficult tasks of science; the problem is still not solved. An aerodynamic car has a value
between 0.25 and 0.3; many sports cars share with vans values of 0.44 and higher, and ra-
cing car values can be as high as 1, depending on the amount of force that is used to keep
the car fastened to the ground. The lowest known values are for dolphins and penguins.**
Ref. 184 Wind resistance is also of importance to humans, in particular in athletics. It is estim-
ated that 100 m sprinters spend between 3 % and 6 % of their power overcoming drag.
This leads to varying sprint times 𝑡w when wind of speed 𝑤 is involved, related by the
expression
𝑡0 𝑤𝑡w 2
= 1.03 − 0.03 (1 − ) , (68)
𝑡w 100 m

* Such a statement about friction is correct only in three dimensions, as is the case in nature; in the case of
Challenge 424 s a single dimension, a potential can always be found.
** It is unclear whether there is, in nature, a smallest possible value for the drag coefficient.
The topic of aerodynamic shapes is also interesting for fluid bodies. They are kept together by surface
tension. For example, surface tension keeps the wet hairs of a soaked brush together. Surface tension also
determines the shape of rain drops. Experiments show that their shape is spherical for drops smaller than
Vol. V, page 308 2 mm diameter, and that larger rain drops are lens shaped, with the flat part towards the bottom. The usual
Ref. 183 tear shape is not encountered in nature; something vaguely similar to it appears during drop detachment,
but never during drop fall.
236 7 classical mechanics, force and the predictability of motion

where the more conservative estimate of 3 % is used. An opposing wind speed of −2 m/s
gives an increase in time of 0.13 s, enough to change a potential world record into an
‘only’ excellent result. (Are you able to deduce the 𝑐w value for running humans from
Challenge 425 ny the formula?)
Likewise, parachuting exists due to wind resistance. Can you determine how the speed
of a falling body, with or without a parachute, changes with time, assuming constant
Challenge 426 s shape and drag coefficient?
In contrast, static friction has different properties. It is proportional to the force press-
Ref. 185 ing the two bodies together. Why? Studying the situation in more detail, sticking friction
is found to be proportional to the actual contact area. It turns out that putting two solids
into contact is rather like turning Switzerland upside down and putting it onto Austria;
the area of contact is much smaller than that estimated macroscopically. The import-
ant point is that the area of actual contact is proportional to the normal force, i.e., the
force component that is perpendicular to the surface. The study of what happens in that
contact area is still a topic of research; researchers are investigating the issues using in-
struments such as atomic force microscopes, lateral force microscopes and triboscopes.

Motion Mountain – The Adventure of Physics
These efforts resulted in computer hard discs which last longer, as the friction between
the disc and the reading head is a central quantity in determining the lifetime.
All forms of friction are accompanied by an increase in the temperature of the mov-
ing body. The reason became clear after the discovery of atoms. Friction is not observed
in few – e.g. 2, 3, or 4 – particle systems. Friction only appears in systems with many
particles, usually millions or more. Such systems are called dissipative. Both the tem-
perature changes and friction itself are due to the motion of large numbers of micro-
scopic particles against each other. This motion is not included in the Galilean descrip-
tion. When it is included, friction and energy loss disappear, and potentials can then be

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
used throughout. Positive accelerations – of microscopic magnitude – then also appear,
and motion is found to be conserved.
In short, all motion is conservative on a microscopic scale. On a microscopic scale it
is thus possible and most practical to describe all motion without the concept of force.*
The moral of the story is twofold: First, one should use force and power only in one
situation: in the case of friction, and only when one does not want to go into the details.**
Secondly, friction is not an obstacle to predictability. Motion remains predictable.

* The first scientist who eliminated force from the description of nature was Heinrich Rudolf Hertz (b. 1857
Hamburg, d. 1894 Bonn), the famous discoverer of electromagnetic waves, in his textbook on mechanics,
Die Prinzipien der Mechanik, Barth, 1894, republished by Wissenschaftliche Buchgesellschaft, 1963. His idea
was strongly criticized at that time; only a generation later, when quantum mechanics quietly got rid of the
concept for good, did the idea become commonly accepted. (Many have speculated about the role Hertz
would have played in the development of quantum mechanics and general relativity, had he not died so
young.) In his book, Hertz also formulated the principle of the straightest path: particles follow geodesics.
This same description is one of the pillars of general relativity, as we will see later on.
** But the cost is high; in the case of human relations the evaluation should be somewhat more discerning,
Ref. 186 as research on violence has shown.
7 classical mechanics, force and the predictability of motion 237

“
Et qu’avons-nous besoin de ce moteur, quand
l’étude réfléchie de la nature nous prouve que le
mouvement perpétuel est la première de ses

”
lois ?*
Donatien de Sade Justine, ou les malheurs de la
vertu.

C omplete states – initial conditions

“ ”
Quid sit futurum cras, fuge quaerere ...**
Horace, Odi, lib. I, ode 9, v. 13.

Let us continue our exploration of the predictability of motion. We often describe the
motion of a body by specifying the time dependence of its position, for example as

𝑥(𝑡) = 𝑥0 + 𝑣0 (𝑡 − 𝑡0 ) + 12 𝑎0 (𝑡 − 𝑡0 )2 + 16 𝑗0 (𝑡 − 𝑡0 )3 + ... . (69)

The quantities with an index 0, such as the starting position 𝑥0 , the starting velocity

Motion Mountain – The Adventure of Physics
𝑣0 , etc., are called initial conditions. Initial conditions are necessary for any description
of motion. Different physical systems have different initial conditions. Initial conditions
thus specify the individuality of a given system. Initial conditions also allow us to distin-
guish the present situation of a system from that at any previous time: initial conditions
specify the changing aspects of a system. Equivalently, they summarize the past of a sys-
tem.
Page 27 Initial conditions are thus precisely the properties we have been seeking for a descrip-
tion of the state of a system. To find a complete description of states we thus need only a
complete description of initial conditions, which we can thus righty call also initial states.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
It turns out that for gravitation, as for all other microscopic interactions, there is no need
for initial acceleration 𝑎0 , initial jerk 𝑗0 , or higher-order initial quantities. In nature, acce-
leration and jerk depend only on the properties of objects and their environment; they
do not depend on the past. For example, the expression 𝑎 = 𝐺𝑀/𝑟2 of universal gravity,
giving the acceleration of a small body near a large one, does not depend on the past, but
only on the environment. The same happens for the other fundamental interactions, as
we will find out shortly.
Page 87 The complete state of a moving mass point is thus described by specifying its position
and its momentum at all instants of time. Thus we have now achieved a complete de-
scription of the intrinsic properties of point objects, namely by their mass, and of their
states of motion, namely by their momentum, energy, position and time. For extended
rigid objects we also need orientation and angular momentum. This is the full list for
rigid objects; no other state observables are needed.
Can you specify the necessary state observables in the cases of extended elastic bod-
Challenge 427 s ies and of fluids? Can you give an example of an intrinsic property that we have so far
Challenge 428 s missed?

* ‘And whatfor do we need this motor, when the reasoned study of nature proves to us that perpetual motion
is the first of its laws?’
Ref. 85 ** ‘What future will be tomorrow, never ask ...’ Horace is Quintus Horatius Flaccus (65–8 bce), the great
Roman poet.
238 7 classical mechanics, force and the predictability of motion

The set of all possible states of a system is given a special name: it is called the phase
space. We will use the concept repeatedly. Like any space, it has a number of dimensions.
Challenge 429 s Can you specify this number for a system consisting of 𝑁 point particles?
It is interesting to recall an older challenge and ask again: does the universe have initial
Challenge 430 s conditions? Does it have a phase space?
Given that we now have a description of both properties and states for point objects,
extended rigid objects and deformable bodies, can we predict all motion? Not yet. There
are situations in nature where the motion of an object depends on characteristics other
Challenge 431 s than its mass; motion can depend on its colour (can you find an example?), on its tem-
perature, and on a few other properties that we will soon discover. And for each intrinsic
property there are state observables to discover. Each additional intrinsic property is the
basis of a field of physical enquiry. Speed was the basis for mechanics, temperature is the
basis for thermodynamics, charge is the basis for electrodynamics, etc. We must there-
fore conclude that as yet we do not have a complete description of motion.

“
An optimist is somebody who thinks that the

”
future is uncertain.

Motion Mountain – The Adventure of Physics
Anonymous

Do surprises exist? Is the fu ture determined?

“
Die Ereignisse der Zukunft können wir nicht
aus den gegenwärtigen erschließen. Der Glaube

”
an den Kausalnexus ist ein Aberglaube.*
Ludwig Wittgenstein, Tractatus, 5.1361

“ ”
Freedom is the recognition of necessity.
Friedrich Engels (1820–1895)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
If, after climbing a tree, we jump down, we cannot halt the jump in the middle of the
trajectory; once the jump has begun, it is unavoidable and determined, like all passive
motion. However, when we begin to move an arm, we can stop or change its motion
from a hit to a caress. Voluntary motion does not seem unavoidable or predetermined.
Challenge 432 e Which of these two cases is the general one?
Let us start with the example that we can describe most precisely so far: the fall of
a body. Once the gravitational potential 𝜑 acting on a particle is given and taken into
account, we can use the expression

𝑎(𝑥) = −∇𝜑 = −𝐺𝑀𝑟/𝑟3 , (70)

and we can use the state at a given time, given by initial conditions such as

𝑥(𝑡0 ) and 𝑣(𝑡0 ) , (71)

to determine the motion of the particle in advance. Indeed, with these two pieces of
information, we can calculate the complete trajectory 𝑥(𝑡).
* ‘We cannot infer the events of the future from those of the present. Belief in the causal nexus is supersti-
tion.’ Our adventure, however, will confirm the everyday observation that this statement is wrong.
7 classical mechanics, force and the predictability of motion 239

An equation that has the potential to predict the course of events is called an evolution
equation. Equation (70), for example, is an evolution equation for the fall of the object.
(Note that the term ‘evolution’ has different meanings in physics and in biology.) An
evolution equation embraces the observation that not all types of change are observed
in nature, but only certain specific cases. Not all imaginable sequences of events are ob-
served, but only a limited number of them. In particular, equation (70) embraces the
idea that from one instant to the next, falling objects change their motion based on the
gravitational potential acting on them.
Evolution equations do not exist only for motion due to gravity, but for motion due to
all forces in nature. Given an evolution equation and initial state, the whole motion of a
system is thus uniquely fixed, a property of motion often called determinism. For example,
astronomers can calculate the position of planets with high precision for thousands of
years in advance.
Let us carefully distinguish determinism from several similar concepts, to avoid mis-
understandings. Motion can be deterministic and at the same time be unpredictable in
Vol. V, page 46 practice. The unpredictability of motion can have four origins:

Motion Mountain – The Adventure of Physics
1. an impracticably large number of particles involved, including situations with fric-
tion,
2. insufficient information about initial conditions, and
3. the mathematical complexity of the evolution equations,
4. strange shapes of space-time.
For example, in case of the weather the first three conditions are fulfilled at the same time.
It is hard to predict the weather over periods longer than about a week or two. (In 1942,
Hitler made once again a fool of himself across Germany by requesting a precise weather

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
forecast for the following twelve months.) Despite the difficulty of prediction, weather
change is still deterministic. As another example, near black holes all four origins apply
together. We will discuss black holes in the section on general relativity. Despite being
unpredictable, motion is deterministic near black holes.
Motion can be both deterministic and time random, i.e., with different outcomes in
Page 127 similar experiments. A roulette ball’s motion is deterministic, but it is also random.* As
Vol. IV, page 157 we will see later, quantum systems fall into this category, as do all examples of irreversible
motion, such as a drop of ink spreading out in clear water. Also the fall of a die is both
deterministic and random. In fact, studies on how to predict the result of a die throw
Ref. 188 with the help of a computer are making rapid progress; these studies also show how
to throw a die in order to increase the odds to get a specific desired result. In all such
cases the randomness and the irreproducibility are only apparent; they disappear when
the description of states and initial conditions in the microscopic domain are included.
In short, determinism does not contradict (macroscopic) irreversibility. However, on the
microscopic scale, deterministic motion is always reversible.
A final concept to be distinguished from determinism is acausality. Causality is the
requirement that a cause must precede the effect. This is trivial in Galilean physics, but
* Mathematicians have developed a large number of tests to determine whether a collection of numbers may
be called random; roulette results pass all these tests – in honest casinos only, however. Such tests typically
check the equal distribution of numbers, of pairs of numbers, of triples of numbers, etc. Other tests are the
Ref. 187 𝜒2 test, the Monte Carlo test(s), and the gorilla test.
240 7 classical mechanics, force and the predictability of motion

becomes of importance in special relativity, where causality implies that the speed of light
is a limit for the spreading of effects. Indeed, it seems impossible to have deterministic
motion (of matter and energy) which is acausal, in other words, faster than light. Can you
Challenge 433 s confirm this? This topic will be looked at more deeply in the section on special relativity.
Saying that motion is ‘deterministic’ means that it is fixed in the future and also in
the past. It is sometimes stated that predictions of future observations are the crucial test
for a successful description of nature. Owing to our often impressive ability to influence
the future, this is not necessarily a good test. Any theory must, first of all, describe past
observations correctly. It is our lack of freedom to change the past that results in our lack
of choice in the description of nature that is so central to physics. In this sense, the term
‘initial condition’ is an unfortunate choice, because in fact, initial conditions summarize
the past of a system.* The central ingredient of a deterministic description is that all
motion can be reduced to an evolution equation plus one specific state. This state can be
either initial, intermediate, or final. Deterministic motion is uniquely specified into the
past and into the future.
To get a clear concept of determinism, it is useful to remind ourselves why the concept

Motion Mountain – The Adventure of Physics
of ‘time’ is introduced in our description of the world. We introduce time because we
observe first that we are able to define sequences in observations, and second, that un-
restricted change is impossible. This is in contrast to films, where one person can walk
through a door and exit into another continent or another century. In nature we do
not observe metamorphoses, such as people changing into toasters or dogs into tooth-
brushes. We are able to introduce ‘time’ only because the sequential changes we observe
Challenge 434 s are extremely restricted. If nature were not reproducible, time could not be used. In short,
determinism expresses the observation that sequential changes are restricted to a single
possibility.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Since determinism is connected to the use of the concept of time, new questions arise
whenever the concept of time changes, as happens in special relativity, in general relativ-
ity and in theoretical high energy physics. There is a lot of fun ahead.
In summary, every description of nature that uses the concept of time, such as that
of everyday life, that of classical physics and that of quantum mechanics, is intrinsically
and inescapably deterministic, since it connects observations of the past and the future,
eliminating alternatives. In short,

⊳ The use of time implies determinism, and vice versa.

When drawing metaphysical conclusions, as is so popular nowadays when discussing
Vol. V, page 46 quantum theory, one should never forget this connection. Whoever uses clocks but
denies determinism is nurturing a split personality!** The future is determined.

* The problems with the term ‘initial conditions’ become clear near the big bang: at the big bang, the uni-
verse has no past, but it is often said that it has initial conditions. This contradiction will be explored later
in our adventure.
** That can be a lot of fun though.
7 classical mechanics, force and the predictability of motion 241

Free will

“ ”
You do have the ability to surprise yourself.
Richard Bandler and John Grinder

The idea that motion is determined often produces fear, because we are taught to asso-
ciate determinism with lack of freedom. On the other hand, we do experience freedom
in our actions and call it free will. We know that it is necessary for our creativity and for
our happiness. Therefore it seems that determinism is opposed to happiness.
But what precisely is free will? Much ink has been consumed trying to find a pre-
cise definition. One can try to define free will as the arbitrariness of the choice of initial
conditions. However, initial conditions must themselves result from the evolution equa-
tions, so that there is in fact no freedom in their choice. One can try to define free will
from the idea of unpredictability, or from similar properties, such as uncomputability.
But these definitions face the same simple problem: whatever the definition, there is no
way to prove experimentally that an action was performed freely. The possible defini-
tions are useless. In short, because free will cannot be defined, it cannot be observed.

Motion Mountain – The Adventure of Physics
(Psychologists also have a lot of additional data to support this conclusion, but that is
another topic.)
No process that is gradual – in contrast to sudden – can be due to free will; gradual
processes are described by time and are deterministic. In this sense, the question about
free will becomes one about the existence of sudden changes in nature. This will be a
recurring topic in the rest of this walk. Can nature surprise us? In everyday life, nature
does not. Sudden changes are not observed. Of course, we still have to investigate this
question in other domains, in the very small and in the very large. Indeed, we will change
our opinion several times during our adventure, but the conclusion remains.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
We note that the lack of surprises in everyday life is built deep into our nature: evolu-
tion has developed curiosity because everything that we discover is useful afterwards. If
nature continually surprised us, curiosity would make no sense.
Many observations contradict the existence of surprises: in the beginning of our walk
we defined time using the continuity of motion; later on we expressed this by saying
that time is a consequence of the conservation of energy. Conservation is the opposite of
surprise. By the way, a challenge remains: can you show that time would not be definable
Challenge 435 s even if surprises existed only rarely?
In summary, so far we have no evidence that surprises exist in nature. Time exists be-
cause nature is deterministic. Free will cannot be defined with the precision required by
physics. Given that there are no sudden changes, there is only one consistent conclusion:
free will is a feeling, in particular of independence of others, of independence from fear
and of accepting the consequences of one’s actions.* Free will is a strange name for a

* That free will is a feeling can also be confirmed by careful introspection. Indeed, the idea of free will always
arises after an action has been started. It is a beautiful experiment to sit down in a quiet environment, with
the intention to make, within an unspecified number of minutes, a small gesture, such as closing a hand. If
Challenge 436 e you carefully observe, in all detail, what happens inside yourself around the very moment of decision, you
find either a mechanism that led to the decision, or a diffuse, unclear mist. You never find free will. Such an
experiment is a beautiful way to experience deeply the wonders of the self. Experiences of this kind might
also be one of the origins of human spirituality, as they show the connection everybody has with the rest of
nature.
242 7 classical mechanics, force and the predictability of motion

Ref. 189 feeling of satisfaction. This solves the apparent paradox; free will, being a feeling, exists
as a human experience, even though all objects move without any possibility of choice.
There is no contradiction.
Ref. 190 Even if human action is determined, it is still authentic. So why is determinism so
frightening? That is a question everybody has to ask themselves. What difference does
Challenge 437 e determinism imply for your life, for the actions, the choices, the responsibilities and the
pleasures you encounter?* If you conclude that being determined is different from being
free, you should change your life! Fear of determinism usually stems from refusal to take
the world the way it is. Paradoxically, it is precisely the person who insists on the existence
of free will who is running away from responsibility.

Summary on predictability
Despite difficulties to predict specific cases, all motion we encountered so far is both
deterministic and predictable. Even friction is predictable, in principle, if we take into
account the microscopic details of matter.

Motion Mountain – The Adventure of Physics
In short, classical mechanics states that the future is determined. In fact, we will dis-
cover that all motion in nature, even in the domains of quantum theory and general
relativity, is predictable.
Motion is predictable. This is not a surprising result. If motion were not predictable,
we could not have introduced the concepts of ‘motion’ and ‘time’ in the first place. We
can only talk and think about motion because it is predictable.

From predictability to global descriptions of motion

“ ”
Πλεῖν ἀνάγκε, ζῆν οὐκ ἀνάγκη.**
Pompeius

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Physicists aim to talk about motion with the highest precision possible. Predictability is
an aspect of precision. The highest predictability – and thus the highest precision – is
possible when motion is described as globally as possible.
All over the Earth – even in Australia – people observe that stones fall ‘down’. This
ancient observation led to the discovery of universal gravity. To find it, all that was neces-
sary was to look for a description of gravity that was valid globally. The only additional
observation that needs to be recognized in order to deduce the result 𝑎 = 𝐺𝑀/𝑟2 is the
variation of gravity with height.
In short, thinking globally helps us to make our description of motion more precise
and our predictions more useful. How can we describe motion as globally as possible? It
turns out that there are six approaches to this question, each of which will be helpful on
our way to the top of Motion Mountain. We first give an overview; then we explore each
of them.
1. Action principles or variational principles, the first global approach to motion, arise

Challenge 438 s * If nature’s ‘laws’ are deterministic, are they in contrast with moral or ethical ‘laws’? Can people still be
held responsible for their actions?
** Navigare necesse, vivere non necesse. ‘To navigate is necessary, to live is not.’ Gnaeus Pompeius Magnus
Ref. 191 (106–48 bce) is cited in this way by Plutarchus (c. 45 to c. 125).
7 classical mechanics, force and the predictability of motion 243

F I G U R E 182 What shape of rail allows the F I G U R E 183 Can motion be described in a
black stone to glide most rapidly from manner common to all observers?
point A to the lower point B?

when we overcome a fundamental limitation of what we have learned so far. When
we predict the motion of a particle from its current acceleration with an evolution
Page 238 equation, we are using the most local description of motion possible. We use the acce-

Motion Mountain – The Adventure of Physics
leration of a particle at a certain place and time to determine its position and motion
just after that moment and in the immediate neighbourhood of that place. Evolution
equations thus have a mental ‘horizon’ of radius zero.
The contrast to evolution equations are variational principles. A famous example
is illustrated in Figure 182. The challenge is to find the path that allows the fastest
possible gliding motion from a high point to a distant low point. The sought path is
Challenge 439 d the brachistochrone, from ancient Greek for ‘shortest time’, This puzzle asks about
a property of motion as a whole, for all times and all positions. The global approach
required by questions such as this one will lead us to a description of motion which is

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
simple, precise and fascinating: the so-called principle of cosmic laziness, also known
as the principle of least action.
2. Relativity, the second global approach to motion, emerges when we compare the vari-
ous descriptions of the same system produced by all possible observers. For example,
the observations by somebody falling from a cliff – as shown in Figure 183 – a pas-
senger in a roller coaster, and an observer on the ground will usually differ. The re-
lationships between these observations, the so-called symmetry transformations, lead
us to a global description of motion, valid for everybody. Later, this approach will
lead us to Einstein’s special and general theory of relativity.
3. Mechanics of extended and rigid bodies, rather than mass points, is required to un-
derstand the objects, plants and animals of everyday life. For such bodies, we want
to understand how all parts of them move. As an example, the counter-intuitive res-
Challenge 440 e ult of the experiment in Figure 184 shows why this topic is worthwhile. The rapidly
rotating wheel suspended on only one end of the axis remains almost horizontal, but
slowly rotates around the rope.
In order to design machines, it is essential to understand how a group of rigid bod-
ies interact with one another. For example, take the Peaucellier-Lipkin linkage shown
Ref. 192 in Figure 185. A joint F is fixed on a wall. Two movable rods lead to two opposite
corners of a movable rhombus, whose rods connect to the other two corners C and P.
This mechanism has several astonishing properties. First of all, it implicitly defines a
244 7 classical mechanics, force and the predictability of motion

bicycle wheel
rotating rapidly
rope on rigid axis rope
a b b

F C P

a b b

F I G U R E 184 What happens when one F I G U R E 185 A famous mechanism, the
rope is cut? Peaucellier-Lipkin linkage, consists of (grey) rods
and (red) joints and allows drawing a straight
line with a compass: ﬁx point F, put a pencil into
joint P, and then move C with a compass along
a circle.

Motion Mountain – The Adventure of Physics
circle of radius 𝑅 so that one always has the relation 𝑟C = 𝑅2 /𝑟P between the distances
of joints C and P from the centre of this circle. This is called an inversion at a circle.
Challenge 441 s Can you find this special circle? Secondly, if you put a pencil in joint P, and let joint
C follow a certain circle, the pencil P draws a straight line. Can you find that circle?
Challenge 442 s The mechanism thus allows drawing a straight line with the help of a compass.
Ref. 193 A famous machine challenge is to devise a wooden carriage, with gearwheels that
connect the wheels to an arrow, with the property that, whatever path the carriage
Challenge 443 d takes, the arrow always points south (see Figure 187). The solution to this puzzle will
even be useful in helping us to understand general relativity, as we will see. Such a

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Vol. II, page 206 wagon allows measuring the curvature of a surface and of space.
Another important machine part is the differential gearbox. Without it, cars could
Challenge 444 e not follow bends on the road. Can you explain it to your friends?
Also nature uses machine parts. In 2011, screws and nuts were found in a joint
Ref. 194 of a weevil beetle, Trigonopterus oblongus. In 2013, the first example of biological
gears have been discovered: in young plant hoppers of the species Issus coleoptratus,
Ref. 195 toothed gears ensure that the two back legs jump synchronously. Figure 186 shows
some details. You might enjoy the video on this discovery available at www.youtube.
com/watch?v=Q8fyUOxD2EA.
Another interesting example of rigid motion is the way that human movements,
such as the general motions of an arm, are composed from a small number of basic
Ref. 196 motions. All these examples are from the fascinating field of engineering; unfortu-
nately, we will have little time to explore this topic in our hike.
4. The next global approach to motion is the description of non-rigid extended bod-
ies. For example, fluid mechanics studies the flow of fluids (like honey, water or air)
around solid bodies (like spoons, ships, sails or wings). The aim is to understand
how all parts of the fluid move. Fluid mechanics thus describes how insects, birds
Ref. 197 and aeroplanes fly,* why sailing-boats can sail against the wind, what happens when

* The mechanisms of insect flight are still a subject of active research. Traditionally, fluid dynamics has
7 classical mechanics, force and the predictability of motion 245

Motion Mountain – The Adventure of Physics
F I G U R E 186 The
gears found in young
plant hoppers
(© Malcolm Burrows).

W E
path

F I G U R E 187 A south-pointing carriage: whatever the path it follows, the arrow on it always points
south.

a hard-boiled egg is made to spin on a thin layer of water, or how a bottle full of wine
Challenge 445 s can be emptied in the fastest way possible.
As well as fluids, we can study the behaviour of deformable solids. This area of re-
search is called continuum mechanics. It deals with deformations and oscillations of

concentrated on large systems, like boats, ships and aeroplanes. Indeed, the smallest human-made object
that can fly in a controlled way – say, a radio-controlled plane or helicopter – is much larger and heavier
than many flying objects that evolution has engineered. It turns out that controlling the flight of small things
requires more knowledge and more tricks than controlling the flight of large things. There is more about
this topic on page 278 in Volume V.
246 7 classical mechanics, force and the predictability of motion

possible? possible?

F I G U R E 188 How and where does a falling F I G U R E 189 Why do hot-air balloons
brick chimney break? stay inﬂated? How can you measure the
weight of a bicycle rider using only a
ruler?

Motion Mountain – The Adventure of Physics
F I G U R E 190 Why do marguerites – or ox-eye daisies, Leucanthemum vulgare – usually have around 21
(left and centre) or around 34 (right) petals? (© Anonymous, Giorgio Di Iorio and Thomas Lüthi)

extended structures. It seeks to explain, for example, why bells are made in particu-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 446 s lar shapes; how large bodies – such as the falling chimneys shown in Figure 188 –
or small bodies – such as diamonds – break when under stress; and how cats can
turn themselves the right way up as they fall. During the course of our journey we
will repeatedly encounter issues from this field, which impinges even upon general
relativity and the world of elementary particles.
5. Statistical mechanics is the study of the motion of huge numbers of particles. Statist-
ical mechanics is yet another global approach to the study of motion. The concepts
needed to describe gases, such as temperature, entropy and pressure (see Figure 189),
are essential tools of this discipline. In particular, the concepts of statistical physics
help us to understand why some processes in nature do not occur backwards. These
concepts will also help us take our first steps towards the understanding of black
holes.
6. The last global approach to motion, self-organization, involves all of the above-
mentioned viewpoints at the same time. Such an approach is needed to understand
everyday experience, and life itself. Why does a flower form a specific number of
petals, as shown in Figure 190? How does an embryo differentiate in the womb? What
makes our hearts beat? How do mountains ridges and cloud patterns emerge? How
do stars and galaxies evolve? How are sea waves formed by the wind?
All these phenomena are examples of self-organization processes; life scientists
speak of growth processes. Whatever we call them, all these processes are charac-
7 classical mechanics, force and the predictability of motion 247

terized by the spontaneous appearance of patterns, shapes and cycles. Self-organized
processes are a common research theme across many disciplines, including biology,
chemistry, medicine, geology and engineering.
We will now explore the six global approaches to motion. We will begin with the first
approach, namely, the description of motion using a variational principle. This beautiful
method for describing, understanding and predicting motion was the result of several
centuries of collective effort, and is the highlight of Galilean physics. Variational prin-
ciples also provide the basis for all the other global approaches just mentioned. They are
also needed for all the further descriptions of motion that we will explore afterwards.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Chapter 8

M E A SU R I NG C HA NG E W I T H AC T ION

M
otion can be described by numbers. Take a single particle that
oves. The expression (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) describes how, during its
otion, position changes with time. The description of particle motion is com-
pleted by stating how the velocity (𝑣𝑥 (𝑡), 𝑣𝑦 (𝑡), 𝑣𝑧 (𝑡)) changes over time. Realizing that
these two expressions fully describe the behaviour of a moving point particle was a

Motion Mountain – The Adventure of Physics
milestone in the development of modern physics.
The next milestone of modern physics is achieved by answering a short but hard ques-
Page 20 tion. If motion is a type of change, as the Greek already said,

⊳ How can we measure the amount of change?

Physicists took almost two centuries of attempts to uncover the way to measure change.
In fact, change can be measured by a single number. Due to the long search, the quantity
that measures change has a strange name: it is called (physical) action,** usually abbre-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
viated 𝑆. To remember the connection of ‘action’ with change, just think about a Holly-
wood film: a lot of action means a large amount of change.
Introducing physical action as a measure of change is important because it provides
the first and also the most useful global description of motion. In fact, we already know
enough to define action straight away.
Imagine taking two snapshots of a system at different times. How could you define
the amount of change that occurred in between? When do things change a lot, and when
do they change only a little? First of all, a system with many moving parts shows a lot of

** Note that this ‘action’ is not the same as the ‘action’ appearing in statements such as ‘every action has an
equal and opposite reaction’. This other usage, coined by Newton for certain forces, has not stuck; therefore
the term has been recycled. After Newton, the term ‘action’ was first used with an intermediate meaning,
before it was finally given the modern meaning used here. This modern meaning is the only meaning used
in this text.
Another term that has been recycled is the ‘principle of least action’. In old books it used to have a
different meaning from the one in this chapter. Nowadays, it refers to what used to be called Hamilton’s
principle in the Anglo-Saxon world, even though it is (mostly) due to others, especially Leibniz. The old
names and meanings are falling into disuse and are not continued here.
Behind these shifts in terminology is the story of an intense two-centuries-long attempt to describe mo-
tion with so-called extremal or variational principles: the objective was to complete and improve the work
initiated by Leibniz. These principles are only of historical interest today because all are special cases of the
Ref. 198 principle of least action described here.
8 measuring change with action 249

F I G U R E 191 Giuseppe Lagrangia/Joseph Lagrange (1736 –1813).

Motion Mountain – The Adventure of Physics
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F I G U R E 192 Physical action
measures change: an example of
process with large action value
(© Christophe Blanc).

change. So it makes sense that the action of a system composed of independent subsys-
tems should be the sum of the actions of these subsystems.
Secondly, systems with high energy or speed, such as the explosions shown in Fig-
ure 192, show larger change than systems at lower energy or speed. Indeed, we already
Page 111 introduced energy as the quantity that measures how much a system changes over time.
Thirdly, change often – but not always – builds up over time; in other cases, recent
change can compensate for previous change, as in a pendulum, when the system can
return back to the original state. Change can thus increase or decrease with time.
Finally, for a system in which motion is stored, transformed or shifted from one subsys-
tem to another, especially when kinetic energy is stored or changed to potential energy,
change diminishes over time.
250 8 measuring change with action

TA B L E 31 Some action values for changes and processes either observed or imagined.

System and process A pproxim at e
a c t i o n va l u e
Smallest measurable action 1.1 ⋅ 10−34 Js
Light
Smallest blackening of photographic film < 10−33 Js
Photographic flash c. 10−17 Js
Electricity
Electron ejected from atom or molecule c. 10−33 Js
Current flow in lightning bolt c. 104 Js
Mechanics and materials
Tearing apart two neighbouring iron atoms c. 10−33 Js
Breaking a steel bar c. 101 Js
Tree bent by the wind from one side to the other c. 500 Js

Motion Mountain – The Adventure of Physics
Making a white rabbit vanish by ‘real’ magic c. 100 PJs
Hiding a white rabbit c. 0.1 Js
Car crash c. 2 kJs
Driving car stops within the blink of an eye c. 20 kJs
Levitating yourself within a minute by 1 m c. 40 kJs
Large earthquake c. 1 PJs
Driving car disappears within the blink of an eye c. 1 ZJs
Sunrise c. 0.1 ZJs
Chemistry

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Atom collision in liquid at room temperature c. 10−33 Js
Smelling one molecule c. 10−31 Js
Burning fuel in a cylinder in an average car engine explosion c. 104 Js
Held versus dropped glass c. 0.8 Js
Life
Air molecule hitting eardrum c. 10−32 Js
Ovule fertilization c. 10−20 Js
Cell division c. 10−15 Js
Fruit fly’s wing beat c. 10−10 Js
Flower opening in the morning c. 1 nJs
Getting a red face c. 10 mJs
Maximum brain change in a minute c. 5 Js
Person walking one body length c. 102 Js
Birth c. 2 kJs
Change due to a human life c. 1 EJs
Nuclei, stars and more
Single nuclear fusion reaction in star c. 10−15 Js
Explosion of gamma-ray burster c. 1046 Js
Universe after one second has elapsed undefinable
8 measuring change with action 251

𝐿
𝐿(𝑡) = 𝑇 − 𝑈

average 𝐿

integral
∫ 𝐿(𝑡)d𝑡

F I G U R E 193 Deﬁning a total change or action
𝑡
Δ𝑡 𝑡m as an accumulation (addition, or integral) of
𝑡i 𝑡f
small changes or actions over time (simpliﬁed
elapsed time for clarity).

Motion Mountain – The Adventure of Physics
All the mentioned properties, taken together, imply:

⊳ The natural measure of change is the average difference between kinetic and
potential energy multiplied by the elapsed time.

This quantity has all the right properties: it is the sum of the corresponding quantities for
all subsystems if these are independent; it generally increases with time; and the quantity
Challenge 447 e decreases if the system transforms motion into potential energy.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Thus the (physical) action 𝑆, measuring the change in a (physical) system, is defined
as
𝑡f 𝑡f
𝑆 = 𝐿 Δ𝑡 = 𝑇 − 𝑈 (𝑡f − 𝑡i ) = ∫ (𝑇 − 𝑈) d𝑡 = ∫ 𝐿 d𝑡 , (72)
𝑡i 𝑡i

Page 186 where 𝑇 is the kinetic energy, 𝑈 the potential energy we already know, 𝐿 is the difference
between these, and the overbar indicates a time average. The quantity 𝐿 is called the
Lagrangian (function) of the system,* describes what is being added over time, whenever
things change. The sign ∫ is a stretched ‘S’, for ‘sum’, and is pronounced ‘integral of’. In
intuitive terms it designates the operation – called integration – of adding up the values
of a varying quantity in infinitesimal time steps d𝑡. The initial and the final times are
written below and above the integration sign, respectively. Figure 193 illustrates the idea:
the integral is simply the size of the dark area below the curve 𝐿(𝑡).

* It is named for Giuseppe Lodovico Lagrangia (b. 1736 Torino, d. 1813 Paris), better known as Joseph Louis
Lagrange. He was the most important mathematician of his time; he started his career in Turin, then worked
for 20 years in Berlin, and finally for 26 years in Paris. Among other things he worked on number theory and
analytical mechanics, where he developed most of the mathematical tools used nowadays for calculations
in classical mechanics and classical gravitation. He applied them successfully to many motions in the solar
system.
252 8 measuring change with action

Challenge 448 s Mathematically, the integral of the Lagrangian, i.e., of the curve 𝐿(𝑡), is defined as

𝑡f f
∫ 𝐿(𝑡) d𝑡 = lim ∑ 𝐿(𝑡m)Δ𝑡 = 𝐿 ⋅ (𝑡f − 𝑡i ) . (73)
𝑡i Δ𝑡→0
m=i

In other words, the integral is the limit, as the time slices get smaller, of the sum of the
areas of the individual rectangular strips that approximate the function. Since the ∑ sign
also means a sum, and since an infinitesimal Δ𝑡 is written d𝑡, we can understand the nota-
tion used for integration. Integration is a sum over slices. The notation was developed by
Gottfried Wilhelm Leibniz to make exactly this point. Physically speaking, the integral
of the Lagrangian measures the total effect that 𝐿 builds up over time. Indeed, action is
called ‘effect’ in some languages, such as German. The effect that builds up is the total
change in the system. In short,

⊳ The integral of the Lagrangian, the action, measures the total change that

Motion Mountain – The Adventure of Physics
occurs in a system.

Physical action is total change. Action, or change, is the integral of the Lagrangian over
time. The unit of action, and thus of change, is the unit of energy, the Joule, times the
unit of time, the second.

⊳ Change is measured in Js.

A large value means a big change. Table 31 shows some action values observed in nature.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
To understand the definition of action in more detail, we start with the simplest possible
case: a system for which the potential energy vanishes, such as a particle moving freely.
Obviously, the higher the kinetic energy is, the more change there is in a given time. Also,
if we observe the particle at two instants, the more distant they are the larger the change.
The change of a free particle accumulates with time. This is as expected.
Next, we explore a single particle moving in a potential. For example, a falling stone
loses potential energy in exchange for a gain in kinetic energy. The more kinetic energy is
stored into potential energy, the less change there is. Hence the minus sign in the defini-
tion of 𝐿. If we explore a particle that is first thrown up in the air and then falls, the curve
for 𝐿(𝑡) first is below the times axis, then above. We note that the definition of integra-
tion makes us count the grey surface below the time axis negatively. Change can thus be
negative, and be compensated by subsequent change, as expected.
To measure change for a system made of several independent components, we simply
add all the kinetic energies and subtract all the potential energies. This technique allows
us to define action values for gases, liquids and solid matter. In short, action is an additive
quantity. Even if the components interact, we still get a sensible result.
In summary, physical action measures, in a single number, the change observed in a
system between two instants of time. Action, or change, is measured in Js. Physical action
quantifies the change due to a physical process. This is valid for all observations, i.e., for
all processes and all systems: an explosion of a firecracker, a caress of a loved one or a
colour change of computer display. We will discover later that describing change with a
8 measuring change with action 253

F I G U R E 194 The minimum of a curve has vanishing slope.

single number is also possible in relativity and quantum theory: any change going on in
Page 20 any system of nature, be it transport, transformation or growth, can be measured with a
single number.

Motion Mountain – The Adventure of Physics
The principle of least action

“
The optimist thinks this is the best of all

”
possible worlds, and the pessimist knows it.
Robert Oppenheimer

We now have a precise measure of change. This, as it turns out, allows a simple, global and
powerful description of motion. In nature, the change happening between two instants
is always the smallest possible.

⊳ In nature, action is minimal.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
This is the essence of the famous principle of least action. It is valid for every example of
motion.* Of all possible motions, nature always chooses for which the change is minimal.
Let us study a few examples.
The simple case of a free particle, when no potentials are involved, the principle of
least action implies that the particle moves in a straight line with constant velocity. All
Challenge 449 e other paths would lead to larger actions. Can you verify this?
When gravity is present, a thrown stone flies along a parabola – or more precisely,
along an ellipse – because any other path, say one in which the stone makes a loop in the
Challenge 450 e air, would imply a larger action. Again you might want to verify this for yourself.
All observations support the simple and basic statement: things always move in a way
that produces the smallest possible value for the action. This statement applies to the full
path and to any of its segments. Bertrand Russell called it the ‘law of cosmic laziness’.
We could also call it the principle of maximal efficiency of nature.
It is customary to express the idea of minimal change in a different way. The action
varies when the path is varied. The actual path is the one with the smallest action. You will

* In fact, in some macroscopic situations the action can be a saddle point, so that the snobbish form of the
Ref. 199 principle is that the action is ‘stationary’. In contrast to what is often heard, the action is never a maximum.
Moreover, for motion on small (infinitesimal) scales, the action is always a minimum. The mathematical
condition of vanishing variation, given below, encompasses all these details.
254 8 measuring change with action

recall from school that at a minimum the derivative of a quantity vanishes: a minimum
has a horizontal slope. This relation is shown in Figure 194. In the present case, we do
not vary a quantity, but a complete path; hence we do not speak of a derivative or slope,
but of a variation. It is customary to write the variation of action as 𝛿𝑆. The principle of
least action thus states:

⊳ The actual trajectory between specified end points satisfies 𝛿𝑆 = 0.

Mathematicians call this a variational principle. Note that the end points have to be spe-
cified: we have to compare motions with the same initial and final situations.
Before discussing the principle of least action further, we check that it is indeed equi-
valent to the evolution equation.* To do this, we can use a standard procedure, part of

* For those interested, here are a few comments on the equivalence of Lagrangians and evolution equations.
First of all, Lagrangians do not exist for non-conservative, or dissipative systems. We saw that there is no
Page 234 potential for any motion involving friction (and more than one dimension); therefore there is no action in

Motion Mountain – The Adventure of Physics
these cases. One approach to overcome this limitation is to use a generalized formulation of the principle
of least action. Whenever there is no potential, we can express the work variation 𝛿𝑊 between different
trajectories 𝑥𝑖 as
𝛿𝑊 = ∑ 𝑚𝑖 𝑥̈𝑖 𝛿𝑥𝑖 . (74)
𝑖

Motion is then described in the following way:
𝑡f
⊳ The actual trajectory satisfies ∫ (𝛿𝑇 + 𝛿𝑊)d𝑡 = 0 provided 𝛿𝑥(𝑡i ) = 𝛿𝑥(𝑡f ) = 0 . (75)
𝑡i

The quantity being varied has no name; it represents a generalized notion of change. You might want to
Challenge 451 e check that it leads to the correct evolution equations. Thus, although proper Lagrangian descriptions exist

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
only for conservative systems, for dissipative systems the principle can be generalized and remains useful.
Many physicists will prefer another approach. What a mathematician calls a generalization is a special
case for a physicist: the principle (75) hides the fact that all results from the usual principle of minimal
action, if we include the complete microscopic details. There is no friction in the microscopic domain.
Friction is an approximate, macroscopic concept.
Nevertheless, more mathematical viewpoints are useful. For example, they lead to interesting limitations
for the use of Lagrangians. These limitations, which apply only if the world is viewed as purely classical –
which it isn’t – were discovered about a hundred years ago. In those times, computers were not available,
and the exploration of new calculation techniques was important. Here is a summary.
The coordinates used in connection with Lagrangians are not necessarily the Cartesian ones. Generalized
coordinates are especially useful when there are constraints on the motion. This is the case for a pendulum,
where the weight always has to be at the same distance from the suspension, or for an ice skater, where the
Ref. 200 skate has to move in the direction in which it is pointing. Generalized coordinates may even be mixtures of
positions and momenta. They can be divided into a few general types.
Generalized coordinates are called holonomic–scleronomic if they are related to Cartesian coordinates in
a fixed way, independently of time: physical systems described by such coordinates include the pendulum
and a particle in a potential. Coordinates are called holonomic–rheonomic if the dependence involves time.
An example of a rheonomic system would be a pendulum whose length depends on time. The two terms
Page 392 rheonomic and scleronomic are due to Ludwig Boltzmann. These two cases, which concern systems that are
only described by their geometry, are grouped together as holonomic systems. The term is due to Heinrich
Vol. III, page 99 Hertz.
The more general situation is called anholonomic, or nonholonomic. Lagrangians work well only for holo-
nomic systems. Unfortunately, the meaning of the term ‘nonholonomic’ has changed. Nowadays, the term
is also used for certain rheonomic systems. The modern use calls nonholonomic any system which involves
8 measuring change with action 255

the so-called calculus of variations. The condition 𝛿𝑆 = 0 implies that the action, i.e., the
area under the curve in Figure 193, is a minimum. A little bit of thinking shows that if the
Lagrangian is of the form 𝐿(𝑥n, 𝑣n ) = 𝑇(𝑣n ) − 𝑈(𝑥n), then the minimum area is achieved
Challenge 452 e when
d ∂𝑇 ∂𝑈
( )=− (76)
d𝑡 ∂𝑣n ∂𝑥n

where n counts all coordinates of all particles.* For a single particle, these Lagrange’s
Challenge 453 e equations of motion reduce to
𝑚𝑎 = −∇𝑈 . (78)

This is the evolution equation: it says that the acceleration times the mass of a particle
is the gradient of the potential energy 𝑈. The principle of least action thus implies the
Challenge 454 s equation of motion. (Can you show the converse, which is also correct?)
In other words, all systems evolve in such a way that the change or action is as small as
possible. Nature is economical. Nature is maximally efficient. Or: Nature is lazy. Nature

Motion Mountain – The Adventure of Physics
is thus the opposite of a Hollywood thriller, in which the action is maximized; nature is
more like a wise old man who keeps his actions to a minimum.
The principle of minimal action states that the actual trajectory is the one for which
the average of the Lagrangian over the whole trajectory is minimal (see Figure 193).
Challenge 455 e Nature is a Dr. Dolittle. Can you verify this? This viewpoint allows one to deduce Lag-
range’s equations (76) directly.
The principle of least action distinguishes the actual trajectory from all other ima-
ginable ones. This observation lead Leibniz to his famous interpretation that the actual
world is the ‘best of all possible worlds.’** We may dismiss this as metaphysical specula-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
tion, but we should still be able to feel the fascination of the issue. Leibniz was so excited
about the principle of least action because it was the first time that actual observations
were distinguished from all other imaginable possibilities. For the first time, the search
for reasons why things are the way they are became a part of physical investigation. Could
the world be different from what it is? In the principle of least action, we have a hint of

velocities. Therefore, an ice skater or a rolling disc is often called a nonholonomic system. Care is thus
necessary to decide what is meant by nonholonomic in any particular context.
Even though the use of Lagrangians, and of action, has its limitations, these need not bother us at the
microscopic level, because microscopic systems are always conservative, holonomic and scleronomic. At
the fundamental level, evolution equations and Lagrangians are indeed equivalent.
* The most general form for a Lagrangian 𝐿(𝑞n , 𝑞ṅ , 𝑡), using generalized holonomic coordinates 𝑞n , leads to
Lagrange equations of the form
d ∂𝐿 ∂𝐿
( )= . (77)
d𝑡 ∂𝑞ṅ ∂𝑞n
In order to deduce these equations, we also need the relation 𝛿𝑞 ̇ = d/d𝑡(𝛿𝑞). This relation is valid only for
holonomic coordinates introduced in the previous footnote and explains their importance.
We remark that the Lagrangian for a moving system is not unique; however, the study of how the various
Ref. 201 Lagrangians for a given moving system are related is not part of this walk.
By the way, the letter 𝑞 for position and 𝑝 for momentum were introduced in physics by the mathem-
atician Carl Jacobi (b. 1804 Potsdam, d. 1851 Berlin).
** This idea was ridiculed by the influential philosopher Voltaire (b. 1694 Paris, d. 1778 Paris) in his lucid
writings, notably in the brilliant book Candide, written in 1759, and still widely available.
256 8 measuring change with action

a negative answer. Leibniz also deduced from the result that gods cannot choose their
Challenge 456 s actions. (What do you think?)

L agrangians and motion

“ ”
Never confuse movement with action.
Ref. 202 Ernest Hemingway

Systems evolve by minimizing change. Change, or action, is the time integral of the Lag-
rangian. As a way to describe motion, the Lagrangian has several advantages over the
evolution equation. First of all, the Lagrangian is usually more compact than writing the
corresponding evolution equations. For example, only one Lagrangian is needed for one
system, however many particles it includes. One makes fewer mistakes, especially sign
mistakes, as one rapidly learns when performing calculations. Just try to write down the
evolution equations for a chain of masses connected by springs; then compare the effort
Challenge 457 e with a derivation using a Lagrangian. (The system is often studied because it behaves in
many aspects like a chain of atoms.) We will encounter another example shortly: David

Motion Mountain – The Adventure of Physics
Hilbert took only a few weeks to deduce the equations of motion of general relativity us-
ing a Lagrangian, whereas Albert Einstein had worked for ten years searching for them
directly.
In addition, the description with a Lagrangian is valid with any set of coordinates de-
scribing the objects of investigation. The coordinates do not have to be Cartesian; they
can be chosen as we prefer: cylindrical, spherical, hyperbolic, etc. These so-called gen-
eralized coordinates allow one to rapidly calculate the behaviour of many mechanical
systems that are in practice too complicated to be described with Cartesian coordinates.
For example, for programming the motion of robot arms, the angles of the joints provide

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
a clearer description than Cartesian coordinates of the ends of the arms. Angles are non-
Cartesian coordinates. They simplify calculations considerably: the task of finding the
most economical way to move the hand of a robot from one point to another is solved
much more easily with angular variables.
More importantly, the Lagrangian allows us to quickly deduce the essential proper-
ties of a system, namely, its symmetries and its conserved quantities. We will develop this
Page 276 important idea shortly, and use it regularly throughout our walk.
Finally, the Lagrangian formulation can be generalized to encompass all types of in-
teractions. Since the concepts of kinetic and potential energy are general, the principle
of least action can be used in electricity, magnetism and optics as well as mechanics. The
principle of least action is central to general relativity and to quantum theory, and allows
us to easily relate both fields to classical mechanics.
As the principle of least action became well known, people applied it to an ever-increa-
Ref. 198 sing number of problems. Today, Lagrangians are used in everything from the study of
elementary particle collisions to the programming of robot motion in artificial intelli-
gence. (Table 32 shows a few examples.) However, we should not forget that despite its
remarkable simplicity and usefulness, the Lagrangian formulation is equivalent to the
Challenge 458 s evolution equations. It is neither more general nor more specific. In particular, it is not
an explanation for any type of motion, but only a different view of it. In fact, the search
for a new physical ‘law’ of motion is just the search for a new Lagrangian. This makes
8 measuring change with action 257

TA B L E 32 Some Lagrangians.

System Lagrangian Q ua nt i t i e s

Free, non-relativistic 𝐿 = 12 𝑚𝑣2 mass 𝑚, speed 𝑣 = d𝑥/d𝑡
mass point
Particle in potential 𝐿 = 12 𝑚𝑣2 − 𝑚𝜑(𝑥) gravitational potential 𝜑
Mass on spring 𝐿 = 12 𝑚𝑣2 − 12 𝑘𝑥2 elongation 𝑥, spring
constant 𝑘
Mass on frictionless 𝐿 = 12 𝑚𝑣2 − 𝑘(𝑥2 + 𝑦2 ) spring constant 𝑘,
table attached to spring coordinates 𝑥, 𝑦
Chain of masses and 𝐿 = 12 𝑚 ∑ 𝑣𝑖2 − 12 𝑚𝜔2 ∑𝑖,𝑗 (𝑥𝑖 − 𝑥𝑗 )2 coordinates 𝑥𝑖 , lattice
springs (simple model of frequency 𝜔
atoms in a linear crystal)
Free, relativistic mass 𝐿 = −𝑐2 𝑚√1 − 𝑣2 /𝑐2 mass 𝑚, speed 𝑣, speed of
point light 𝑐

Motion Mountain – The Adventure of Physics
sense, as the description of nature always requires the description of change. Change in
nature is always described by actions and Lagrangians.
The principle of least action states that the action is minimal when the end points of
Ref. 203 the motion, and in particular the time between them, are fixed. It is less well known that
the reciprocal principle also holds: if the action value – the change value – is kept fixed,
Challenge 459 ny the elapsed time for the actual motion is maximal. Can you show this?
Even though the principle of least action is not an explanation of motion, the principle
somehow calls for such an explanation. We need some patience, though. Why nature fol-
lows the principle of least action, and how it does so, will become clear when we explore

Why is motion so often b ounded?
Looking around ourselves on Earth or in the sky, we find that matter is not evenly distrib-
uted. Matter tends to be near other matter: it is lumped together in aggregates. Figure 195
Ref. 204 shows a typical example. Some major examples of aggregates are listed in Figure 196 and
Table 33. All aggregates have mass and size. In the mass–size diagram of Figure 196, both
scales are logarithmic. We note three straight lines: a line 𝑚 ∼ 𝑙 extending from the
Planck mass* upwards, via black holes, to the universe itself; a line 𝑚 ∼ 1/𝑙 extending
from the Planck mass downwards, to the lightest possible aggregate; and the usual matter
line with 𝑚 ∼ 𝑙3 , extending from atoms upwards, via everyday objects, the Earth to the
Sun. The first of the lines, the black hole limit, is explained by general relativity; the last
two, the aggregate limit and the common matter line, by quantum theory.**
The aggregates outside the common matter line also show that the stronger the inter-
action that keeps the components together, the smaller the aggregate. But why is matter
mainly found in lumps?

* The Planck mass is given by 𝑚Pl = √ℏ𝑐/𝐺 = 21.767(16) μg.
** Figure 196 suggests that domains beyond physics exist; we will discover later on that this is not the case,
as mass and size are not definable in those domains.
258 8 measuring change with action

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 195 Motion in the universe is bounded. (© Mike Hankey)

First of all, aggregates form because of the existence of attractive interactions between
objects. Secondly, they form because of friction: when two components approach, an
aggregate can only be formed if the released energy can be changed into heat. Thirdly,
aggregates have a finite size because of repulsive effects that prevent the components from
collapsing completely. Together, these three factors ensure that in the universe, bound
motion is much more common than unbound, ‘free’ motion.
Only three types of attraction lead to aggregates: gravity, the attraction of electric
charges, and the strong nuclear interaction. Similarly, only three types of repulsion are
observed: rotation, pressure, and the Pauli exclusion principle (which we will encounter
Vol. IV, page 136 later on). Of the nine possible combinations of attraction and repulsion, not all appear
in nature. Can you find out which ones are missing from Figure 196 and Table 33, and
Challenge 460 s why?
Together, attraction, friction and repulsion imply that change and action are minim-
8 measuring change with action 259

universe
mass
[kg]

galaxy
1040 black
holes star cluster

lim ce:
Sun

Beyond nature and science: beyond Planck length limit

ol ien
it
k h sc
Earth

e
neutron

ac d
bl an
1020

Beyond nature and science: undefined
star

e e
th t u r
nd na
mountain

yo nd

ine
be eyo
B

ter l
mat
human
100

mon
Planck mass

com
cell

Motion Mountain – The Adventure of Physics
heavy DNA
10-20 nucleus
uranium
muon hydrogen
proton
electron
Aggregates
m

10-40 neutrino
icr
os
co
pi
ca
gg

Elementary
re
ga

particles

10-60 imaginable
it

aggregate

10-40 10-20 100 1020 size [m]
F I G U R E 196 Elementary particles and aggregates found in nature.

ized when objects come and stay together. The principle of least action thus implies the
stability of aggregates. By the way, the formation history also explains why so many ag-
Challenge 461 s gregates rotate. Can you tell why?
But why does friction exist at all? And why do attractive and repulsive interactions
exist? And why is it – as it would appear from the above – that in some distant past
matter was not found in lumps? In order to answer these questions, we must first study
another global property of motion: symmetry.

TA B L E 33 Some major aggregates observed in nature.

A g g r e g at e Size Obs. Constituents
(diameter) num.

Gravitationally bound aggregates
260 8 measuring change with action

A g g r e g at e Size Obs. Constituents
(diameter) num.

Matter across universe c. 100 Ym 1 superclusters of galaxies, hydrogen
and helium atoms
Quasar 1012 to 1014 m 20 ⋅ 106 baryons and leptons
Supercluster of galaxies c. 3 Ym 107 galaxy groups and clusters
9
Galaxy cluster c. 60 Zm 25 ⋅ 10 10 to 50 galaxies
Galaxy group or cluster c. 240 Zm 50 to over 2000 galaxies
Our local galaxy group 50 Zm 1 c. 40 galaxies
General galaxy 0.5 to 2 Zm 3.5 ⋅ 1012 1010 to 3 ⋅ 1011 stars, dust and gas
clouds, probably star systems
Our galaxy 1.0(0.1) Zm 1 1011 stars, dust and gas clouds, solar
systems
5
Interstellar clouds up to 15 Em ≫ 10 hydrogen, ice and dust
Solar System 𝑎 unknown > 400 star, planets

Motion Mountain – The Adventure of Physics
Our Solar System 30 Pm 1 Sun, planets (Pluto’s orbit’s
diameter: 11.8 Tm), moons,
planetoids, comets, asteroids, dust,
gas
Oort cloud 6 to 30 Pm 1 comets, dust
Kuiper belt 60 Tm 1 planetoids, comets, dust
Star 𝑏 10 km to 100 Gm 1022±1 ionized gas: protons, neutrons,
electrons, neutrinos, photons
Our star, the Sun 1.39 Gm

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Planet 𝑎 (Jupiter, Earth) 143 Mm, 12.8 Mm 8+ > solids, liquids, gases; in particular,
400 heavy atoms
Planetoids (Varuna, etc) 50 to 1 000 km > 100 solids
(est. 109 )
Moons 10 to 1 000 km > 50 solids
Neutron stars 10 km > 1000 mainly neutrons
Electromagnetically bound aggregates 𝑐
Dwarf planets, minor 1 m to 2400 km > 106 (109 estimated) solids, usually
planets, asteroids 𝑑 monolithic
Comets 10 cm to 50 km > 109 (1012 possible) ice and dust
Mountains, solids, liquids, 1 nm to > 100 km n.a. molecules, atoms
gases, cheese
Animals, plants, kefir 5 μm to 1 km 1026±2 organs, cells
brain, human 0.2 m 1010 neurons and other cell types
Cells: 1031±1 organelles, membranes, molecules
smallest (Nanoarchaeum c. 400 nm molecules
equitans)
amoeba c. 600 μm molecules
8 measuring change with action 261

A g g r e g at e Size Obs. Constituents
(diameter) num.

largest (whale nerve, c. 30 m molecules
single-celled plants)
Molecules: 1078±2 atoms
H2 c. 50 pm 1072±2 atoms
DNA (human) 2 m (total per cell) 1021 atoms
Atoms, ions 30 pm to 300 pm 1080±2 electrons and nuclei
Aggregates bound by the weak interaction 𝑐
None
Aggregates bound by the strong interaction 𝑐
Nucleus 0.9 to > 7 f m 1079±2 nucleons
Nucleon (proton, neutron) 0.9 f m 1080±2 quarks
Mesons c. 1 f m n.a. quarks

Motion Mountain – The Adventure of Physics
Neutron stars: see further up

𝑎. Only in 1994 was the first evidence found for objects circling stars other than our Sun; of over 5000
extrasolar planets found so far, most are found around F, G and K stars, including neutron stars. For example,
Ref. 205 three objects circle the pulsar PSR 1257+12, and a matter ring circles the star β Pictoris. The objects seem to
be dark stars, brown dwarfs or large gas planets like Jupiter. Due to the limitations of observation systems,
none of the systems found so far form solar systems of the type we live in. In fact, only a few Earth-like
planets have been found so far.
𝑏. The Sun is among the brightest 7 % of stars. Of all stars, 80 %, are red M dwarfs, 8 % are orange K dwarfs,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
and 5 % are white D dwarfs: these are all faint. Almost all stars visible in the night sky belong to the bright
7 %. Some of these are from the rare blue O class or blue B class (such as Spica, Regulus and Rigel); 0.7 %
consist of the bright, white A class (such as Sirius, Vega and Altair); 2 % are of the yellow–white F class (such
as Canopus, Procyon and Polaris); 3.5 % are of the yellow G class (like Alpha Centauri, Capella or the Sun).
Exceptions include the few visible K giants, such as Arcturus and Aldebaran, and the rare M supergiants,
Vol. II, page 250 such as Betelgeuse and Antares. More on stars later on.
Vol. V, page 342 𝑐. For more details on microscopic aggregates, see the table of composites.
Ref. 206 𝑑. It is estimated that there are up to 1020 small Solar System bodies (asteroids, meteoroids, planetoids or
minor planets) that are heavier than 100 kg. Incidentally, no asteroids between Mercury and the Sun – the
hypothetical Vulcanoids – have been found so far.

Curiosities and fun challenges ab ou t L agrangians
The principle of least action as a mathematical description is due to Leibniz. He under-
stood its validity in 1707. It was then rediscovered and named by Maupertuis in 1746,
Page 136 who wrote:
Lorsqu’il arrive quelque changement dans la Nature, la quantité d’action né-
cessaire pour ce changement est la plus petite qu’il soit possible.*

* ‘When some change occurs in Nature, the quantity of action necessary for this change is the smallest that
is possible.’
262 8 measuring change with action

Samuel König, the first scientist to state publicly and correctly in 1751 that the principle
was due to Leibniz, and not to Maupertuis, was expelled from the Prussian Academy of
Sciences for stating so. This was due to an intrigue of Maupertuis, who was the president
of the academy at the time. The intrigue also made sure that the bizarre term ‘action’
was retained. Despite this disgraceful story, Leibniz’ principle quickly caught on, and
was then used and popularized by Euler, Lagrange and finally by Hamilton.
∗∗
The basic idea of the principle of least action, that nature is as lazy as possible, is also
called lex parismoniae. This general idea is already expressed by Ptolemy, and later by
Fermat, Malebranche, and ’s Gravesande. But Leibniz was the first to understand its
validity and mathematical usefulness for the description of all motion.
∗∗
When Lagrange published his book Mécanique analytique, in 1788, it formed one of the
high points in the history of mechanics and established the use of variational principles.

Motion Mountain – The Adventure of Physics
He was proud of having written a systematic exposition of mechanics without a single
figure. Obviously the book was difficult to read and was not a sales success at all. There-
fore his methods took another generation to come into general use.
∗∗
Given that action is the basic quantity describing motion, we can define energy as action
per unit time, and momentum as action per unit distance. The energy of a system thus
describes how much it changes over time, and the momentum describes how much it
Challenge 462 s changes over distance. What are angular momentum and rotational energy?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
In Galilean physics, the Lagrangian is the difference between kinetic and potential en-
ergy. Later on, this definition will be generalized in a way that sharpens our understand-
ing of this distinction: the Lagrangian becomes the difference between a term for free
particles and a term due to their interactions. In other words, every particle motion is
a continuous compromise between what the particle would do if it were free and what
other particles want it to do. In this respect, particles behave a lot like humans beings.
∗∗
‘In nature, effects of telekinesis or prayer are impossible, as in most cases the change
inside the brain is much smaller than the change claimed in the outside world.’ Is this
Challenge 463 s argument correct?
∗∗
Challenge 464 ny How is action measured? What is the best device or method to measure action?
∗∗
Challenge 465 s Explain: why is 𝑇 + 𝑈 constant, whereas 𝑇 − 𝑈 is minimal?
∗∗
8 measuring change with action 263

𝛼 air

water

𝛽
F I G U R E 197 Refraction of light is due to travel-time optimization.

In nature, the sum 𝑇 + 𝑈 of kinetic and potential energy is constant during motion (for
closed systems), whereas the action is minimal. Is it possible to deduce, by combining
Challenge 466 s these two facts, that systems tend to a state with minimum potential energy?

Motion Mountain – The Adventure of Physics
∗∗
Another minimization principle can be used to understand the construction of animal
Ref. 207 bodies, especially their size and the proportions of their inner structures. For example,
the heart pulse and breathing frequency both vary with animal mass 𝑚 as 𝑚−1/4 , and
the dissipated power varies as 𝑚3/4 . It turns out that such exponents result from three
properties of living beings. First, they transport energy and material through the organ-
ism via a branched network of vessels: a few large ones, and increasingly many smaller
ones. Secondly, the vessels all have the same minimum size. And thirdly, the networks
are optimized in order to minimize the energy needed for transport. Together, these rela-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
tions explain many additional scaling rules; they might also explain why animal lifespan
scales as 𝑚−1/4 , or why most mammals have roughly the same number of heart beats in
Page 125 a lifetime.
A competing explanation, using a different minimization principle, states that quarter
powers arise in any network built in order that the flow arrives to the destination by the
Ref. 208 most direct path.
∗∗
The minimization principle for the motion of light is even more beautiful: light always
takes the path that requires the shortest travel time. It was known long ago that this idea
describes exactly how light changes direction when it moves from air to water, and effect
illustrated in Figure 197. In water, light moves more slowly; the speed ratio between air
and water is called the refractive index of water. The refractive index, usually abbreviated
𝑛, is material-dependent. The value for water is about 1.3. This speed ratio, together with
the minimum-time principle, leads to the ‘law’ of refraction, a simple relation between
Challenge 467 s the sines of the two angles. Can you deduce it?
∗∗
Can you confirm that all the mentioned minimization principles – that for the growth
of trees, that for the networks inside animals, that for the motion of light – are special
Challenge 468 s cases of the principle of least action? In fact, this is the case for all known minimization
264 8 measuring change with action

principles in nature. Each of them, like the principle of least action, is a principle of least
change.
∗∗
In Galilean physics, the value of the action depends on the speed of the observer, but not
on his position or orientation. But the action, when properly defined, should not depend
on the observer. All observers should agree on the value of the observed change. Only
special relativity will fulfil the requirement that action be independent of the observer’s
Challenge 469 s speed. How will the relativistic action be defined?
∗∗
What is the amount of change accumulated in the universe since the big bang? Measuring
all the change that is going on in the universe presupposes that the universe is a physical
Challenge 470 s system. Is this the case?
∗∗

Motion Mountain – The Adventure of Physics
One motion for which action is particularly well minimized in nature is dear to us:
Ref. 209 walking. Extensive research efforts try to design robots which copy the energy saving
functioning and control of human legs. For an example, see the website by Tao Geng at
cswww.essex.ac.uk/tgeng/research.html.
∗∗
Challenge 471 d Can you prove the following integration challenge?
𝜑
π 𝜑
∫ sec 𝑡 d𝑡 = ln tan ( + ) (79)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
0 4 2
∗∗
What is the shape of the ideal halfpipe for skateboarding? What does ‘ideal’ imply?
Challenge 472 s Which requirement leads to a cycloid? Which requirement speaks against a cycloid?
∗∗
Page 125 As mentioned above, animal death is a physical process and occurs when an animal has
consumed or metabolized around 1 GJ/kg. Show that the total action of an animal scales
Challenge 473 e as 𝑀5/4 .

Summary on action
Systems move by minimizing change. Change, or action, is the time average of kinetic
energy minus potential energy. The statement ‘motion minimizes change’ expresses mo-
tion’s predictability and its continuity. The statement also implies that all motion is as
simple as possible.
Systems move by minimizing change. Equivalently, systems move by maximizing the
elapsed time between two situations. Both statements show that nature is lazy.
Systems move by minimizing change. In the next chapters, we show that this state-
ment implies the observer-invariance, conservation, mirror-invariance, reversibility and
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
265
8 measuring change with action

relativity of everyday motion.
Chapter 9

MOT ION A N D SYM M ET RY

“ ”
Am Anfang war die Symmetrie.**
Ref. 210 Werner Heisenberg

T
he second way to describe motion globally is to describe it in such a way
hat all observers agree. Now, whenever an observation stays exactly

Motion Mountain – The Adventure of Physics
he same when switching from one observer to another, we call the observation
invariant or absolute or symmetric. Whenever an observation changes when switching
Page 242 from one observer to another, we call it relative. To explore relativity thus means to
explore invariance and symmetry.

⊳ Symmetry is invariance after change.

Change of observer, or change of point of view, is one such possible change; another pos-
sibility can be some change operated on the system under observation itself. For example,
a forget-me-not flower, shown in Figure 198, is symmetrical because it looks the same

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
after turning around it, or after turning it, by 72 degrees; many fruit tree flowers have the
same symmetry. One also says that under certain changes of viewpoint the flower has
an invariant property, namely its shape. If many such viewpoints are possible, one talks
about a high symmetry, otherwise a low symmetry. For example, a four-leaf clover has a
higher symmetry than a three-leaf one. In physics, the viewpoints are often called frames
of reference.
Whenever we speak about symmetry in flowers, in everyday life, in architecture or in
the arts we usually mean mirror symmetry, rotational symmetry or some combination.
These are geometric symmetries. Like all symmetries, geometric symmetries imply in-
variance under specific change operations. The complete list of geometric symmetries is
Ref. 211 known for a long time. Table 34 gives an overview of the basic types. Figure 199 and Fig-
ure 200 give some important examples. Additional geometric symmetries include colour
symmetries, where colours are exchanged, and spin groups, where symmetrical objects do
not contain only points but also spins, with their special behaviour under rotations. Also
combinations with scale symmetry, as they appear in fractals, and variations on curved
backgrounds are extensions of the basic table.

Challenge 474 e ** ‘In the beginning, there was symmetry.’ Do you agree with this statement? It has led many researchers
astray during the search for the unification of physics. Probably, Heisenberg meant to say that in the be-
ginning, there was simplicity. However, there are many conceptual and mathematical differences between
symmetry and simplicity.
9 motion and symmetry 267

F I G U R E 198 Forget-me-not, also called
Myosotis (Boraginaceae), has ﬁve-fold
symmetry (© Markku Savela).

TA B L E 34 The classiﬁcation and the number of simple geometric symmetries.

Motion Mountain – The Adventure of Physics
D i m e n s i o nR e p e t i t i o n Tr a n s l at i o n s
types
0 1 2 3
point line plane s pa c e
groups groups groups groups
1 1 row 2 2 n.a. n.a.
2 5 nets or plane lattice 2 (cyclic, 7 friezes 17 wall-papers n.a.
types (square, oblique, dihedral) or

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hexagonal, 10 rosette
rectangular, centred groups
rectangular) (C1 , C2 , C3 , C4 ,
C6 , D1 , D2 , D3 ,
D4 , D6 )
3 14 (Bravais) lattices (3 32 crystal 75 rods 80 layers 230 crystal
cubic, 2 tetragonal, 4 groups, also structures,
orthorhombic, 1 called crystal- also called
hexagonal, 1 trigonal, lographic space groups,
2 monoclinic, 1 point groups Fedorov
triclinic type) groups or
crystallo-
graphic
groups

A high symmetry means that many possible changes leave an observation invariant.
At first sight, not many objects or observations in nature seem to be symmetrical: after
all, in the nature around us, geometric symmetry is more the exception than the rule.
But this is a fallacy. On the contrary, we can deduce that nature as a whole is symmetric
Challenge 475 s from the simple fact that we have the ability to talk about it! Moreover, the symmetry of
nature is considerably higher than that of a forget-me-not or of any other symmetry from
268 9 motion and symmetry

The 17 wallpaper patterns and a way to identify them quickly.
Is the maximum rotation order 1, 2, 3, 4 or 6?
Is there a mirror (m)? Is there an indecomposable glide reﬂection (g)?
Is there a rotation axis on a mirror? Is there a rotation axis not on a mirror?

oo pg K ** pm A *o cm M

0 2222
p1
p2
T
S2222

y
*632
n n 22o
y
g? n
p6m g? pgg

D632 y P22
n g?

Motion Mountain – The Adventure of Physics
m?
y
y n
1 *2222
632 m? m?
6 max 2 n pmm
n rotation y
p6 g? D2222
order
y
3 4
S632 22*
y an axis on
m? m? n pmg
n a mirror?
an axis not y
D22
y on a mirror? n y
3*3
2*22
an axis not
p31m on a mirror? cmm

*333 442
p3m1 p4

D333 *442 S442
333 p3 S333 4*2 p4g D42 p4m D442

Every pattern is identiﬁed according to three systems of notation:

442 The Conway-Thurston notation.
p4 The International Union of Crystallography notation.
S442 The Montesinos notation, as in his book
“Classical Tesselations and Three Manifolds”

F I G U R E 199 The full list of possible symmetries of wallpaper patterns, the so-called wallpaper groups,
their usual names, and a way to distinguish them (© Dror Bar-Natan).
9 motion and symmetry 269

Crystal system Crystall class or crystal group

Triclinic system
(three axes,
none at right angles)
C1 Ci

Monoclinic system
(two axes at
right angles, a C2 Cs or C1h C2h
third not)

Orthorhombic system
(three unequal axes
at right angles)
D2 C2v D2h

Motion Mountain – The Adventure of Physics
Tetragonal system
(three axes at right
angles, one
unequal)
C4 S4 C4h D4 C4v D2d D4h

Trigonal system
(three equal axes
at 120 degrees, a

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fourth at right angles
with threefold C3 S6 D3 C3v D3d
symmetry)

Hexagonal system
(three equal axes
at 120 degrees, a
fourth at right
angles with sixfold C6 C3h C6h D6 C6v D3h D6h
symmetry)

Cubic or isometric
system (three equal
axes at right
angles)
T Th O Td Oh

F I G U R E 200 The full list of possible symmetries of units cells in crystals, the crystallographic point
groups or crystal groups or crystal classes (© Jonathan Goss, after Neil Ashcroft and David Mermin).
270 9 motion and symmetry

Table 34. A consequence of this high symmetry is, among others, the famous expression
𝐸0 = 𝑐2 𝑚.

Why can we think and talk ab ou t the world?

“
The hidden harmony is stronger than the

”
apparent.
Ref. 212 Heraclitus of Ephesus, about 500 bce

Why can we understand somebody when he is talking about the world, even though
we are not in his shoes? We can for two reasons: because most things look similar from
different viewpoints, and because most of us have already had similar experiences be-
forehand.
‘Similar’ means that what we and what others observe somehow correspond. In other
words, many aspects of observations do not depend on viewpoint. For example, the num-
ber of petals of a flower has the same value for all observers. We can therefore say that this
quantity has the highest possible symmetry. We will see below that mass is another such

Motion Mountain – The Adventure of Physics
example. Observables with the highest possible symmetry are called scalars in physics.
Other aspects change from observer to observer. For example, the apparent size varies
with the distance of observation. However, the actual size is observer-independent. In
general terms,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
⊳ Any type of viewpoint-independence is a form of symmetry.

The observation that two people looking at the same thing from different viewpoints can
understand each other proves that nature is symmetric. We start to explore the details of
this symmetry in this section and we will continue during most of the rest of our hike.
In the world around us, we note another general property: not only does the same
phenomenon look similar to different observers, but different phenomena look similar
to the same observer. For example, we know that if fire burns the finger in the kitchen, it
will do so outside the house as well, and also in other places and at other times. Nature
shows reproducibility. Nature shows no surprises. In fact, our memory and our thinking
Challenge 476 s are only possible because of this basic property of nature. (Can you confirm this?) As
we will see, reproducibility leads to additional strong restrictions on the description of
nature.
Without viewpoint-independence and reproducibility, talking to others or to one-
self would be impossible. Even more importantly, we will discover that viewpoint-
independence and reproducibility do more than determine the possibility of talking to
each other: they also fix much (but not all) of the content of what we can say to each
other. In other words, we will see that most of our description of nature follows logically,
almost without choice, from the simple fact that we can talk about nature to our friends.
9 motion and symmetry 271

Viewpoints

“
Toleranz ... ist der Verdacht der andere könnte

”
Recht haben.*
Kurt Tucholsky (b. 1890 Berlin,
d. 1935 Göteborg), German writer

“
Toleranz – eine Stärke, die man vor allem dem

”
politischen Gegner wünscht.**
Wolfram Weidner (b. 1925) German journalist

When a young human starts to meet other people in childhood, he quickly finds out that
certain experiences are shared, while others, such as dreams, are not. Learning to make
this distinction is one of the adventures of human life. In these pages, we concentrate on
a section of the first type of experiences: physical observations. However, even between
these, distinctions are to be made. In daily life we are used to assuming that weights,
volumes, lengths and time intervals are independent of the viewpoint of the observer.
We can talk about these observed quantities to anybody, and there are no disagreements

Motion Mountain – The Adventure of Physics
over their values, provided they have been measured correctly. However, other quantities
do depend on the observer. Imagine talking to a friend after he jumped from one of the
trees along our path, while he is still falling downwards. He will say that the forest floor
is approaching with high speed, whereas the observer below will maintain that the floor
is stationary. Obviously, the difference between the statements is due to their different
viewpoints. The velocity of an object (in this example that of the forest floor or of the
friend himself) is thus a less symmetric property than weight or size. Not all observers
agree on the value. of velocity, nor even on its direction.
In the case of viewpoint-dependent observations, understanding each other is still
possible with the help of a little effort: each observer can imagine observing from the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
point of view of the other, and check whether the imagined result agrees with the state-
ment of the other.*** If the statement thus imagined and the actual statement of the
other observer agree, the observations are consistent, and the difference in statements is
due only to the different viewpoints; otherwise, the difference is fundamental, and they
cannot agree or talk. Using this approach, you can even argue whether human feelings,
Challenge 477 s judgements, or tastes arise from fundamental differences or not.
The distinction between viewpoint-independent – or invariant – quantities and
viewpoint-dependent – or relative –quantities is an essential one. Invariant quantities,
such as mass or shape, describe intrinsic properties, and relative quantities, depending
on the observer, make up the state of the system. Therefore, in order to find a complete
description of the state of a physical system, we must answer the following questions:
— Which viewpoints are possible?
— How are descriptions transformed from one viewpoint to another?
— Which observables do these symmetries admit?

* ‘Tolerance ... is the suspicion that the other might be right.’
** ‘Tolerance – a strength one mainly wishes to political opponents.’
*** Humans develop the ability to imagine that others can be in situations different from their own at the
Ref. 213 age of about four years. Therefore, before the age of four, humans are unable to conceive special relativity;
afterwards, they can.
272 9 motion and symmetry

— What do these results tell us about motion?
So far, in our exploration of motion we have first of all studied viewpoints that differ
in location, in orientation, in time and, most importantly, in motion. With respect to
each other, observers can be at rest, can be rotated, can move with constant speed or can
even accelerate. These ‘concrete’ changes of viewpoint are those we will study first. In
this case, the requirement of consistency of observations made by different observers is
Page 156 called the principle of relativity. The symmetries associated with this type of invariance
Page 281 are also called external symmetries. They are listed in Table 36.
A second class of fundamental changes of viewpoint concerns ‘abstract’ changes.
Viewpoints can differ by the mathematical description used: such changes are called
Vol. III, page 85 changes of gauge. They will be introduced first in the section on electrodynamics. Again,
it is required that all statements be consistent across different mathematical descriptions.
This requirement of consistency is called the principle of gauge invariance. The associated
symmetries are called internal symmetries.
The third class of changes, whose importance may not be evident from everyday life,

Motion Mountain – The Adventure of Physics
is that of the behaviour of a system under the exchange of its parts. The associated in-
variance is called permutation symmetry. It is a discrete symmetry, and we will encounter
Vol. IV, page 112 it as a fundamental principle when we explore quantum theory.
The three consistency requirements just described are called ‘principles’ because
these basic statements are so strong that they almost completely determine the ‘laws’ of
physics – i.e., the description of motion – as we will see shortly. Later on we will discover
that looking for a complete description of the state of objects will also yield a complete
description of their intrinsic properties. But enough of introduction: let us come to the
heart of the topic.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Symmetries and groups
Because we are looking for a description of motion that is complete, we need to under-
stand and describe the full set of symmetries of nature. But what is symmetry?
A system is said to be symmetric or to possess a symmetry if it appears identical when
observed from different viewpoints. We also say that the system possesses an invariance
under change from one viewpoint to the other. Viewpoint changes are equivalent to sym-
metry operations or transformations of a system. A symmetry is thus a set of transform-
ations that leaves a system invariant. However, a symmetry is more than a set: the suc-
cessive application of two symmetry operations is another symmetry operation. In other
terms, a symmetry is a set 𝐺 = {𝑎, 𝑏, 𝑐, ...} of elements, the transformations, together
with a binary operation ∘ called concatenation or multiplication and pronounced ‘after’
or ‘times’, in which the following properties hold for all elements 𝑎, 𝑏 and 𝑐:

associativity, i.e., (𝑎 ∘ 𝑏) ∘ 𝑐 = 𝑎 ∘ (𝑏 ∘ 𝑐)
a neutral element 𝑒 exists such that 𝑒 ∘ 𝑎 = 𝑎 ∘ 𝑒 = 𝑎
an inverse element 𝑎−1 exists such that 𝑎−1 ∘ 𝑎 = 𝑎 ∘ 𝑎−1 = 𝑒 . (80)

Any set that fulfils these three defining properties, or axioms, is called a (mathematical)
group. Historically, the notion of group was the first example of a mathematical struc-
9 motion and symmetry 273

Motion Mountain – The Adventure of Physics
F I G U R E 201 A ﬂower of Crassula ovata showing three ﬁvefold multiplets: petals, stems and buds (© J.J.
Harrison)

ture which was defined in a completely abstract manner.* Can you give an example of a
Challenge 478 s group taken from daily life? Groups appear frequently in physics and mathematics, be-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 214 cause symmetries are almost everywhere, as we will see.** Can you list the symmetry
Challenge 480 s operations of the pattern of Figure 202?

Multiplets
Looking at a symmetric and composed system such as the ones shown in Figure 201 or
Challenge 481 e Figure 202, we notice that each of its parts, for example each red patch, belongs to a set
of similar objects, called a multiplet.

⊳ Each part or component of a symmetric system can be classified according
to what type of multiplet it belongs to.
* The term ‘group’ is due to Evariste Galois (b. 1811 Bourg-la-Reine, d. 1832 Paris), its structure to Augustin-
Louis Cauchy (b. 1789 Paris, d. 1857 Sceaux) and the axiomatic definition to Arthur Cayley (b. 1821 Rich-
mond upon Thames, d. 1895 Cambridge).
** In principle, mathematical groups need not be symmetry groups; but it can be proven that all groups can
be seen as transformation groups on some suitably defined mathematical space, so that in mathematics we
can use the terms ‘symmetry group’ and ‘group’ interchangeably.
A group is called Abelian if its concatenation operation is commutative, i.e., if 𝑎 ∘ 𝑏 = 𝑏 ∘ 𝑎 for all pairs of
elements 𝑎 and 𝑏. In this case, the concatenation is sometimes called addition. Do rotations form an Abelian
Challenge 479 e group?
A subset 𝐺1 ⊂ 𝐺 of a group 𝐺 can itself be a group; one then calls it a subgroup and often says sloppily
that 𝐺 is larger than 𝐺1 or that 𝐺 is a higher symmetry group than 𝐺1 .
274 9 motion and symmetry

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Copyright © 1990 Christoph Schiller

F I G U R E 202 A Hispano–Arabic ornament from the Governor’s Palace in Sevilla (© Christoph Schiller).

For some of the coloured patches in Figure 202 we need four objects to make up a full
multiplet, whereas for others we need two, or only one, as in the case of the central star.
Taken as a whole, each multiplet has (at least) the symmetry properties of the whole
system.
Therefore we have two challenges to solve. First of all, we need to find all symmetries
of nature. Secondly, throughout our adventure, we need to determine the full multiplet
for every part of nature that we observe. Above all, we will need to determine the mul-
tiplets for the smallest parts found in nature, the elementary particles.

⊳ A multiplet is a set of parts or components that transform into each other
under all symmetry transformations.
9 motion and symmetry 275

R epresentations
Mathematicians often call abstract multiplets representations. By specifying to which
multiplet or representation a part or component belongs, we describe in which way the
component is part of the whole system. Let us see how this classification is achieved.
In mathematical language, symmetry transformations are often described by
matrices. For example, in the plane, a reflection along the first diagonal is represen-
ted by the matrix
0 1
𝐷(refl) = ( ) , (81)
1 0

since every point (𝑥, 𝑦) becomes transformed to (𝑦, 𝑥) when multiplied by the matrix
Challenge 482 e 𝐷(refl). Therefore, for a mathematician a representation of a symmetry group 𝐺 is an
assignment of a matrix 𝐷(𝑎) to each group element 𝑎 such that the representation of the
concatenation of two elements 𝑎 and 𝑏 is the product of the representations 𝐷 of the
elements:
𝐷(𝑎 ∘ 𝑏) = 𝐷(𝑎)𝐷(𝑏) . (82)

Motion Mountain – The Adventure of Physics
For example, the matrix of equation (81), together with the corresponding matrices for
all the other symmetry operations, have this property.*
For every symmetry group, the construction and classification of all possible repres-
entations is an important task. It corresponds to the classification of all possible mul-
tiplets a symmetric system can be made of. Therefore, if we understand the classification
of all multiplets and parts which can appear in Figure 202, we will also understand how to
classify all possible parts of which an object or an example of motion can be composed!
A representation 𝐷 is called unitary if all matrices 𝐷(𝑎) are unitary.** All representa-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
tions appearing in physics, with only a handful of exceptions, are unitary: this term is the
* There are some obvious, but important, side conditions for a representation: the matrices 𝐷(𝑎) must be
invertible, or non-singular, and the identity operation of 𝐺 must be mapped to the unit matrix. In even more
compact language one says that a representation is a homomorphism from 𝐺 into the group of non-singular
or invertible matrices. A matrix 𝐷 is invertible if its determinant det 𝐷 is not zero.
In general, if a mapping 𝑓 from a group 𝐺 to another 𝐺󸀠 satisfies
𝑓(𝑎 ∘𝐺 𝑏) = 𝑓(𝑎) ∘𝐺󸀠 𝑓(𝑏) , (83)
the mapping 𝑓 is called an homomorphism. A homomorphism 𝑓 that is one-to-one (injective) and onto
(surjective) is called an isomorphism. If a representation is also injective, it is called faithful, true or proper.
In the same way as groups, more complex mathematical structures such as rings, fields and associative
algebras may also be represented by suitable classes of matrices. A representation of the field of complex
Vol. IV, page 223 numbers is given later on.
** The transpose 𝐴𝑇 of a matrix 𝐴 is defined element-by-element by (𝐴𝑇 )ik = 𝐴 ki . The complex conjugate
𝐴∗ of a matrix 𝐴 is defined by (𝐴∗ )ik = (𝐴 ik )∗ . The adjoint 𝐴† of a matrix 𝐴 is defined by 𝐴† = (𝐴𝑇 )∗ .
A matrix is called symmetric if 𝐴𝑇 = 𝐴, orthogonal if 𝐴𝑇 = 𝐴−1 , Hermitean or self-adjoint (the two are
synonymous in all physical applications) if 𝐴† = 𝐴 (Hermitean matrices have real eigenvalues), and unitary
if 𝐴† = 𝐴−1 . Unitary matrices have eigenvalues of norm one. Multiplication by a unitary matrix is a one-to-
one mapping; since the time evolution of physical systems is a mapping from one time to another, evolution
is always described by a unitary matrix.
An antisymmetric or skew-symmetric matrix is defined by 𝐴𝑇 = −𝐴, an anti-Hermitean matrix by 𝐴† =
−𝐴 and an anti-unitary matrix by 𝐴† = −𝐴−1 . All the corresponding mappings are one-to-one.
A matrix is singular, and the corresponding vector transformation is not one-to-one, if det 𝐴 = 0.
276 9 motion and symmetry

most restrictive because it specifies that the corresponding transformations are one-to-
one and invertible, which means that one observer never sees more or less than another.
Obviously, if an observer can talk to a second one, the second one can also talk to the
first. Unitarity is a natural property of representations in natural systems.
The final important property of a multiplet, or representation, concerns its structure.
If a multiplet can be seen as composed of sub-multiplets, it is called reducible, else irre-
ducible; the same is said about representations. The irreducible representations obviously
cannot be decomposed any further. For example, the (almost perfect) symmetry group of
Figure 202, commonly called D4 , has eight elements. It has the general, faithful, unitary
Challenge 483 e and irreducible matrix representation

cos 𝑛π/2 − sin 𝑛π/2 −1 0 1 0 0 1 0 −1
( ) 𝑛 = 0..3, ( ),( ),( ),( ). (84)
sin 𝑛π/2 cos 𝑛π/2 0 1 0 −1 1 0 −1 0

The representation is an octet. The complete list of possible irreducible representations of
Challenge 484 e the group D4 also includes singlets, doublets and quartets. Can you find them all? These

Motion Mountain – The Adventure of Physics
representations allow the classification of all the white and black ribbons that appear in
the figure, as well as all the coloured patches. The most symmetric elements are singlets,
and the least symmetric ones are members of the quartets. The complete system is always
a singlet as well.
With these concepts we are now ready to talk about motion and moving systems with
improved precision.

The symmetries and vo cabulary of motion
Every day we experience that we are able to talk to each other about motion. It must

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
therefore be possible to find an invariant quantity describing it. We already know it: it is
the action, the measure of change. For example, lighting a match is a change. The mag-
nitude of the change is the same whether the match is lit here or there, in one direction
or another, today or tomorrow. Indeed, the (Galilean) action is a number whose value is
the same for each observer at rest, independent of his orientation or the time at which
he makes his observation.
In the case of the Arabic pattern of Figure 202, the symmetry allows us to deduce the
list of multiplets, or representations, that can be its building blocks. This approach must
be possible for a moving system as well. Table 35 shows how. In the case of the Arabic
pattern, from the various possible observation viewpoints, we deduced the classification
of the ribbons into singlets, doublets, etc. For a moving system, the building blocks, cor-
responding to the ribbons, are the (physical) observables. Since we observe that nature is
symmetric under many different changes of viewpoint, we can classify all observables.
To do so, we first need to take the list of all viewpoint transformations and then deduce
the list of all their representations.
Our everyday life shows that the world stays unchanged after changes in position,
orientation and instant of observation. We also speak of space translation invariance,
rotation invariance and time translation invariance. These transformations are different
from those of the Arabic pattern in two respects: they are continuous and they are un-
bounded. As a result, their representations will generally be continuously variable and
9 motion and symmetry 277

TA B L E 35 Correspondences between the symmetries of an ornament, a ﬂower and motion.

System H i s pa n o – A r - Flower Motion
abic
pat t e r n
Structure and set of ribbons and set of petals, stem motion path and
components patches observables
System pattern symmetry flower symmetry symmetry of Lagrangian
symmetry
Mathematical D4 C5 in Galilean relativity:
description of the position, orientation,
symmetry group instant and velocity changes
Invariants number of multiplet petal number number of coordinates,
elements magnitude of scalars,
vectors and tensors

Motion Mountain – The Adventure of Physics
Representations multiplet types of multiplet types of tensors, including scalars
of the elements components and vectors
components
Most symmetric singlet part with circular scalar
representation symmetry
Simplest faithful quartet quintet vector
representation
Least symmetric quartet quintet no limit (tensor of infinite

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
representation rank)

without bounds: they will be quantities, or magnitudes. In other words,

⊳ Because the continuity of observation change, observables must be con-
structed with (real) numbers.

In this way, we have deduced why numbers are necessary for any description of motion.*
Since observers can differ in orientation, representations will be mathematical objects
possessing a direction. To cut a long story short, the symmetry under change of obser-
vation position, orientation or instant leads to the result that all observables are either
‘scalars’, ‘vectors’ or higher-order ‘tensors.’**

⊳ A scalar is an observable quantity which stays the same for all observers.

A scalar corresponds to a singlet. Examples are the mass or the charge of an object, the
distance between two points, the distance of the horizon, and many others. The possible
* Only scalars, in contrast to vectors and higher-order tensors, may also be quantities that only take a dis-
Challenge 485 e crete set of values, such as +1 or −1 only. In short, only scalars may be discrete observables.
** Later on, spinors will be added to, and complete, this list.
278 9 motion and symmetry

values of a scaler are (usually) continuous, unbounded and without direction. Other ex-
amples of scalars are the potential at a point and the temperature at a point. Velocity is
obviously not a scalar; nor is the coordinate of a point. Can you find more examples and
Challenge 486 s counter-examples?
Energy is a puzzling observable. It is a scalar if only changes of place, orientation and
instant of observation are considered. But energy is not a scalar if changes of observer
speed are included. Nobody ever searched for a generalization of energy that is a scalar
also for moving observers. Only Albert Einstein discovered it, completely by accident.
Vol. II, page 65 More about this issue will be told shortly.

⊳ Any quantity which has a magnitude and a direction and which ‘stays the
same’ with respect to the environment when changing viewpoint is a vector.

For example, the arrow between two fixed points on the floor is a vector. Its length is
the same for all observers; its direction changes from observer to observer, but not with
respect to its environment. On the other hand, the arrow between a tree and the place

Motion Mountain – The Adventure of Physics
where a rainbow touches the Earth is not a vector, since that place does not stay fixed
when the observer changes.
Mathematicians say that vectors are directed entities staying invariant under coordin-
ate transformations. Velocities of objects, accelerations and field strength are examples
Challenge 487 e of vectors. (Can you confirm this?) The magnitude of a vector is a scalar: it is the same
for any observer. By the way, a famous and baffling result of nineteenth-century exper-
iments is that the velocity of a light beam is not a vector like the velocity of a car; the
velocity of a light beam is not a vector for Galilean transformations.* This mystery will
Vol. II, page 15 be solved shortly.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Tensors are generalized vectors. As an example, take the moment of inertia of an
Ref. 215 object. It specifies the dependence of the angular momentum on the angular velocity.
Page 117 For any object, doubling the magnitude of angular velocity doubles the magnitude of
angular momentum; however, the two vectors are not parallel to each other if the object
Page 164 is not a sphere. In general, if any two vector quantities are proportional, in the sense that
doubling the magnitude of one vector doubles the magnitude of the other, but without
the two vectors being parallel to each other, then the proportionality ‘factor’ is a (second
order) tensor. Like all proportionality factors, tensors have a magnitude. In addition,
tensors have a direction and a shape: they describe the connection between the vectors
they relate. Just as vectors are the simplest quantities with a magnitude and a direction,
so tensors are the simplest quantities with a magnitude, a direction and a shape, i.e., a
direction depending on a second, chosen direction. Just as vectors can be visualized as
oriented arrows, symmetric tensors – but not non-symmetric ones – can be visualized as
Challenge 489 s oriented ellipsoids.** Can you name another example of a tensor?

* Galilean transformations are changes of viewpoints from one observer to a second one, moving with re-
spect to the first. ‘Galilean transformation’ is just a term for what happens in everyday life, where velocities
add and time is the same for everybody. The term, introduced in 1908 by Philipp Frank, is mostly used as
a contrast to the Lorentz transformation that is so common in special relativity.
** A rank-𝑛 tensor is the proportionality factor between a rank-1 tensor – i.e., a vector – and a rank-(𝑛 − 1)
tensor. Vectors and scalars are rank 1 and rank 0 tensors. Scalars can be pictured as spheres, vectors as
arrows, and symmetric rank-2 tensors as ellipsoids. A general, non-symmetric rank-2 tensor can be split
9 motion and symmetry 279

Let us get back to the description of motion. Table 35 shows that in physical sys-
tems – like in a Hispano-Arabic ornament – we always have to distinguish between the
symmetry of the whole Lagrangian – corresponding to the symmetry of the complete
ornament – and the representation of the observables – corresponding to the ribbon
multiplets. Since the action must be a scalar, and since all observables must be tensors,
Lagrangians contain sums and products of tensors only in combinations forming scal-
ars. Lagrangians thus contain only scalar products or generalizations thereof. In short,
Lagrangians always look like

𝐿 = 𝛼 𝑎𝑖 𝑏𝑖 + 𝛽 𝑐𝑗𝑘 𝑑𝑗𝑘 + 𝛾 𝑒𝑙𝑚𝑛 𝑓𝑙𝑚𝑛 + ... (85)

where the indices attached to the variables 𝑎, 𝑏, 𝑐 etc. always come in matching pairs to be
summed over. (Therefore summation signs are usually simply left out.) The Greek letters
represent constants. For example, the action of a free point particle in Galilean physics
was given as
𝑚
𝑆 = ∫ 𝐿 d𝑡 = ∫ 𝑣2 d𝑡

Motion Mountain – The Adventure of Physics
(86)
2

which is indeed of the form just mentioned. We will encounter many other cases during
our study of motion.*
Galileo already understood that motion is also invariant under changes of viewpoints

uniquely into a symmetric and an antisymmetric tensor. An antisymmetric rank-2 tensor corresponds to a
polar vector. Tensors of higher rank correspond to more and more complex shapes.
A vector has the same length and direction for every observer; a tensor (of rank 2) has the same determ-
inant, the same trace, and the same sum of diagonal subdeterminants for all observers.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
A vector is described mathematically by a list of components; a tensor (of rank 2) is described by a matrix
of components. The rank or order of a tensor thus gives the number of indices the observable has. Can you
Challenge 488 e show this?
* By the way, is the usual list of possible observation viewpoints – namely different positions, different
observation instants, different orientations, and different velocities – also complete for the action (86)? Sur-
Ref. 216 prisingly, the answer is no. One of the first who noted this fact was Niederer, in 1972. Studying the quantum
theory of point particles, he found that even the action of a Galilean free point particle is invariant under
some additional transformations. If the two observers use the coordinates (𝑡, 𝑥) and (𝜏, 𝜉), the action (86)
Challenge 490 ny is invariant under the transformations

𝑟𝑥 + 𝑥0 + 𝑣𝑡 𝛼𝑡 + 𝛽
𝜉= and 𝜏= with 𝑟𝑇 𝑟 = 1 and 𝛼𝛿 − 𝛽𝛾 = 1 . (87)
𝛾𝑡 + 𝛿 𝛾𝑡 + 𝛿

where 𝑟 describes the rotation from the orientation of one observer to the other, 𝑣 the velocity between the
two observers, and 𝑥0 the vector between the two origins at time zero. This group contains two important
special cases of transformations:

The connected, static Galilei group 𝜉 = 𝑟𝑥 + 𝑥0 + 𝑣𝑡 and 𝜏=𝑡
𝑥 𝛼𝑡 + 𝛽
The transformation group SL(2,R) 𝜉 = and 𝜏= (88)
𝛾𝑡 + 𝛿 𝛾𝑡 + 𝛿

The latter, three-parameter group includes spatial inversion, dilations, time translation and a set of time-
dependent transformations such as 𝜉 = 𝑥/𝑡, 𝜏 = 1/𝑡 called expansions. Dilations and expansions are rarely
mentioned, as they are symmetries of point particles only and do not apply to everyday objects and systems.
They will return to be of importance later on, however.
280 9 motion and symmetry

Page 156 with different velocities. However, the action just given does not reflect this. It took some
years to find out the correct generalization: it is given by the theory of special relativity.
But before we study it, we need to finish the present topic.

R eproducibility, conservation and Noether ’ s theorem

“
I will leave my mass, charge and momentum to

”
science.
Graffito

The reproducibility of observations, i.e., the symmetry under change of instant of time
or ‘time translation invariance’, is a case of viewpoint-independence. (That is not obvi-
Challenge 491 ny ous; can you find its irreducible representations?) The connection has several important
consequences. We have seen that symmetry implies invariance. It turns out that for con-
tinuous symmetries, such as time translation symmetry, this statement can be made more
precise:

Motion Mountain – The Adventure of Physics
⊳ For any continuous symmetry of the Lagrangian there is an associated con-
served constant of motion and vice versa.

The exact formulation of this connection is the theorem of Emmy Noether.* She found
the result in 1915 when helping Albert Einstein and David Hilbert, who were both strug-
gling and competing at constructing general relativity. However, the result applies to any
Ref. 217 type of Lagrangian.
Noether investigated continuous symmetries depending on a continuous parameter
𝑏. A viewpoint transformation is a symmetry if the action 𝑆 does not depend on the value
of 𝑏. For example, changing position as

leaves the action
𝑆0 = ∫ 𝑇(𝑣) − 𝑈(𝑥) d𝑡 (90)

invariant, since 𝑆(𝑏) = 𝑆0 . This situation implies that

∂𝑇
= 𝑝 = const . (91)
∂𝑣
In short, symmetry under change of position implies conservation of momentum. The
converse is also true.
Challenge 492 e In the case of symmetry under shift of observation instant, we find

𝑇 + 𝑈 = const . (92)
* Emmy Noether (b. 1882 Erlangen, d. 1935 Bryn Mawr), mathematician. The theorem is only a sideline in
her career which she dedicated mostly to number theory. The theorem also applies to gauge symmetries,
where it states that to every gauge symmetry corresponds an identity of the equation of motion, and vice
versa.
9 motion and symmetry 281

In other words, time translation invariance implies constant energy. Again, the converse
is also correct.
The conserved quantity for a continuous symmetry is sometimes called the Noether
charge, because the term charge is used in theoretical physics to designate conserved
extensive observables. So, energy and momentum are Noether charges. ‘Electric charge’,
‘gravitational charge’ (i.e., mass) and ‘topological charge’ are other common examples.
Challenge 493 s What is the conserved charge for rotation invariance?
We note that the expression ‘energy is conserved’ has several meanings. First of all, it
means that the energy of a single free particle is constant in time. Secondly, it means that
the total energy of any number of independent particles is constant. Finally, it means that
the energy of a system of particles, i.e., including their interactions, is constant in time.
Collisions are examples of the latter case. Noether’s theorem makes all of these points at
Challenge 494 e the same time, as you can verify using the corresponding Lagrangians.
But Noether’s theorem also makes, or rather repeats, an even stronger statement: if
energy were not conserved, time could not be defined. The whole description of nature
requires the existence of conserved quantities, as we noticed when we introduced the

Motion Mountain – The Adventure of Physics
Page 27 concepts of object, state and environment. For example, we defined objects as permanent
entities, that is, as entities characterized by conserved quantities. We also saw that the
Page 238 introduction of time is possible only because in nature there are ‘no surprises’. Noether’s
theorem describes exactly what such a ‘surprise’ would have to be: the non-conservation
of energy. However, energy jumps have never been observed – not even at the quantum
level.
Since symmetries are so important for the description of nature, Table 36 gives an
overview of all the symmetries of nature that we will encounter. Their main properties
are also listed. Except for those marked as ‘approximate’, an experimental proof of in-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
correctness of any of them would be a big surprise indeed – and guarantee eternal fame.
Various speculations about additional symmetries exist; so far, all these speculations and
quests for even more eternal fame have turned out to be mistaken. The list of symmet-
ries is also the full list of universal statements, i.e., of statements about all observations,
that scientists make. For example, when it is said that ‘‘all stones fall down’’ the state-
ment implies the existence of time and space translation invariance. For philosophers
interested in logical induction, the list is thus important also from this point of view.

TA B L E 36 The known symmetries of nature, with their properties; also the complete list of logical
inductions used in physics.

Geometric or space-time, external, symmetries
Time and space R × R3 space, not scalars, momentum yes/yes allow
translation [4 par.] time compact vectors, and energy everyday
282 9 motion and symmetry

TA B L E 36 (Continued) The known symmetries of nature, with their properties; also the complete list of
logical inductions used in physics.

Symmetry Type S pa c e G r o u p Pos - Con- Va - Main
[num- of ac-topo- sible served cuum/ effect
ber of tion logy rep- qua nt - m at -
pa r a - r e s e nt- ity/ ter is
met- ations charge sym-
ers] metric
Rotation SO(3) space 𝑆2 tensors angular yes/yes communi-
[3 par.] momentum cation
Galilei boost R3 [3 par.] space, not scalars, velocity of yes/for relativity
time compact vectors, centre of low of motion
tensors mass speeds
Lorentz homogen- space- not tensors, energy- yes/yes constant
eous Lie time compact spinors momentum light speed
SO(3,1) 𝑇𝜇𝜈

Motion Mountain – The Adventure of Physics
[6 par.]
Poincaré inhomo- space- not tensors, energy- yes/yes
ISL(2,C) geneous time compact spinors momentum
Lie 𝑇𝜇𝜈
[10 par.]
Dilation R+ [1 par.] space- ray 𝑛-dimen. none yes/no massless
invariance time continuum particles
Special R4 [4 par.] space- R4 𝑛-dimen. none yes/no massless
conformal time continuum particles
invariance

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Conformal [15 par.] space- involved massless none yes/no light cone
invariance time tensors, invariance
spinors
Dynamic, interaction-dependent symmetries: gravity
1/𝑟2 gravity SO(4) config. as SO(4) vector pair perihelion yes/yes closed
[6 par.] space direction orbits
Diffeomorphism [∞ par.] space- involved space- local yes/no perihelion
invariance time times energy– shift
momentum
Dynamic, classical and quantum-mechanical motion symmetries
Parity (‘spatial’) discrete Hilbert discrete even, odd P-parity yes/no mirror
inversion P or phase world
space exists
Motion (‘time’) discrete Hilbert discrete even, odd T-parity yes/no reversibil-
inversion T or phase ity
space
9 motion and symmetry 283

TA B L E 36 (Continued) The known symmetries of nature, with their properties; also the complete list of
logical inductions used in physics.

Symmetry Type S pa c e G r o u p Pos - Con- Va - Main
[num- of ac-topo- sible served cuum/ effect
ber of tion logy rep- qua nt - m at -
pa r a - r e s e nt- ity/ ter is
met- ations charge sym-
ers] metric
Charge global, Hilbert discrete even, odd C-parity yes/no anti-
conjugation C antilinear, or phase particles
anti- space exist
Hermitean
CPT discrete Hilbert discrete even CPT-parity yes/yes makes field
or phase theory
space possible

Motion Mountain – The Adventure of Physics
Dynamic, interaction-dependent, gauge symmetries
Electromagnetic [∞ par.] space of unim- unim- electric yes/yes massless
classical gauge fields portant portant charge light
invariance
Electromagnetic Abelian Hilbert circle S1 fields electric yes/yes massless
q.m. gauge inv. Lie U(1) space charge photon
[1 par.]
Electromagnetic Abelian space of circle S1 abstract abstract yes/no none
duality Lie U(1) fields
[1 par.]

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Weak gauge non- Hilbert as 𝑆𝑈(3) particles weak no/
Abelian space charge approx.
Lie SU(2)
[3 par.]
Colour gauge non- Hilbert as 𝑆𝑈(3) coloured colour yes/yes massless
Abelian space quarks gluons
Lie SU(3)
[8 par.]
Chiral discrete fermions discrete left, right helicity approxi- ‘massless’
symmetry mately fermions𝑎
Permutation symmetries
Particle discrete Fock discrete fermions none n.a./yes Gibbs’
exchange space and paradox
etc. bosons

Vol. III, page 320 For details about the connection between symmetry and induction, see later on. The explanation
of the terms in the table will be completed in the rest of the walk. The real numbers are denoted
as 𝑅.
𝑎. Only approximate; ‘massless’ means that 𝑚 ≪ 𝑚Pl , i.e., that 𝑚 ≪ 22 μg.
284 9 motion and symmetry

Parit y inversion and motion reversal
The symmetries in Table 36 include two so-called discrete symmetries that are important
for the discussion of motion.
The first symmetry is parity invariance for objects or processes under spatial inversion.
The symmetry is also called mirror invariance or right-left symmetry. Both objects and
processes can be mirror symmetric. A single glove or a pair of scissors are not mirror-
symmetric. How far can you throw a stone with your other hand? Most people have a
preferred hand, and the differences are quite pronounced. Does nature have such a right-
left preference? In everyday life, the answer is clear: everything that exists or happens in
one way can also exist or happen in its mirrored way.
Numerous precision experiments have tested mirror invariance; they show that

⊳ Every process due to gravitation, electricity or magnetism can also happen
in a mirrored way.

Motion Mountain – The Adventure of Physics
There are no exceptions. For example, there are people with the heart on the right side;
there are snails with left-handed houses; there are planets that rotate the other way. Astro-
nomy and everyday life – which are governed by gravity and electromagnetic processes
– are mirror-invariant. We also say that gravitation and electromagnetism are parity in-
variant. Later we will discover that certain rare processes not due to gravity or electro-
Vol. V, page 245 magnetism, but to the weak nuclear interaction, violate parity.
Mirror symmetry has two representations: ‘+ or singlet’, such as mirror-symmetric
objects, and ‘− or doublet’, such as handed objects. Because of mirror symmetry, scalar
quantities can thus be divided into true scalars, like temperature, and pseudo-scalars,
like magnetic flux or magnetic charge. True scalars do not change sign under mirror

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
reflection, whereas pseudo-scalars do. In the same way, one distinguishes true vectors, or
polar vectors, such as velocity, from pseudo-vectors, or axial vectors, like angular velocity,
angular momentum, torque, vorticity and the magnetic field. (Can you find an example
Challenge 495 ny of a pseudo-tensor?)
The other discrete symmetry is motion reversal. It is sometimes also called, falsely,
‘time reversal’. Can phenomena happen backwards? Does reverse motion trace out the
forward path? This question is not easy. Exploring motion due to gravitation shows that
such motion can always also happen in the reverse direction. (Also for motion reversal,
observables belong either to a + or to a − representation.) In the case of motion due to
electricity and magnetism, such as the behaviour of atoms in gases, solids and liquids,
the question is more involved. Can broken objects be made to heal? We will discuss the
issue in the section on thermodynamics, but we will reach the same conclusion, despite
the appearance of the contrary:

⊳ Every motion due to gravitation, electricity or magnetism can also happen
in the reverse direction.

Motion reversion is a symmetry for all processes due to gravitation and electromagnetic
interaction. Everyday motion is reversible. And again, certain even rarer nuclear pro-
cesses will provide exceptions.
9 motion and symmetry 285

Interaction symmetries
In nature, when we observe a system, we can often neglect the environment. Many pro-
cesses occur independently of what happens around them. This independence is a phys-
ical symmetry. Given the independence of observations from the details occurring in
the environment, we deduce that interactions between systems and the environment de-
crease with distance. In particular, we deduce that gravitational attraction, electric attrac-
tion and repulsion, and also magnetic attraction and repulsion must vanish for distant
Challenge 496 e bodies. Can you confirm this?
Gauge symmetry is also an interaction symmetry. We will encounter them in our
exploration of quantum physics. In a sense, these symmetries are more specific cases of
the general decrease of interactions with distance.

Curiosities and fun challenges ab ou t symmetry
Right-left symmetry is an important property in everyday life; for example, humans
prefer faces with a high degree of right-left symmetry. Humans also prefer that objects on

Motion Mountain – The Adventure of Physics
the walls have shapes that are right-left symmetric. It turns out that the eye and the brain
has symmetry detectors built in. They detect deviations from perfect right-left symmetry.
∗∗
What is the path followed by four turtles starting on the four angles of a square, if each
of them continuously walks, at constant speed, towards the next one? How long is the
Challenge 497 s distance they travel?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 498 s What is the symmetry of a simple oscillation? And of a wave?
∗∗
Challenge 499 s For what systems is motion reversal a symmetry transformation?
∗∗
Challenge 500 s What is the symmetry of a continuous rotation?
∗∗
A sphere has a tensor for the moment of inertia that is diagonal with three equal numbers.
The same is true for a cube. Can you distinguish spheres and cubes by their rotation
Challenge 501 s behaviour?
∗∗
Challenge 502 s Is there a motion in nature whose symmetry is perfect?
∗∗
Can you show that in two dimensions, finite objects can have only rotation and reflec-
tion symmetry, in contrast to infinite objects, which can have also translation and glide-
reflection symmetry? Can you prove that for finite objects in two dimensions, if no ro-
tation symmetry is present, there is only one reflection symmetry? And that all possible
286 9 motion and symmetry

rotations are always about the same centre? Can you deduce from this that at least one
Challenge 503 e point is unchanged in all symmetrical finite two-dimensional objects?
∗∗
Challenge 504 s Which object of everyday life, common in the 20th century, had sevenfold symmetry?
∗∗
Here is little puzzle about the lack of symmetry. A general acute triangle is defined as a
triangle whose angles differ from a right angle and from each other by at least 15 degrees.
Challenge 505 e Show that there is only one such general triangle and find its angles.
∗∗
Can you show that in three dimensions, finite objects can have only rotation, reflection,
inversion and rotatory inversion symmetry, in contrast to infinite objects, which can have
also translation, glide-reflection, and screw rotation symmetry? Can you prove that for
finite objects in three dimensions, if no rotation symmetry is present, there is only one

Motion Mountain – The Adventure of Physics
reflection plane? And that for all inversions or rotatory inversions the centre must lie on
a rotation axis or on a reflection plane? Can you deduce from this that at least one point
Challenge 506 e is unchanged in all symmetrical finite three-dimensional objects?

Summary on symmetry
Symmetry is invariance to change. The simplest symmetries are geometrical: the point
symmetries of flowers or the translation symmetries of infinite crystals are examples. All
the possible changes that leave a system invariant – i.e., all possible symmetry transform-
ations of a system – form a mathematical group. Apart from the geometrical symmetry

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
groups, several additional symmetry groups appear for motion itself.
Motion is universal. Any universality statement implies a symmetry. The reproducib-
ility and predictability of nature implies a number of fundamental continuous symmet-
ries: since we can talk about nature we can deduce that above all, nature is symmetrical
under time and space translations and rotations. Space-time symmetries form a group.
More precisely, they form a continuous symmetry group.
From nature’s continuous symmetries, using Noether’s theorem, we can deduce con-
served ‘charges’. These are energy, linear momentum and angular momentum. They are
described by real numbers. In other words, the definition of mass, space and time, to-
gether with their symmetry properties, is equivalent to the conservation of energy and
momenta. Conservation and symmetry are two ways to express the same property of
nature. To put it simply, our ability to talk about nature means that energy, linear mo-
mentum and angular momentum are conserved and described by numbers.
Additionally, there are two fundamental discrete symmetries about motion: first,
everyday observations are found to be mirror symmetric; secondly, many simple motion
examples are found to be symmetric under motion reversal.
Finally, the isolability of systems from their surroundings implies that interactions
must have no effect at large distances. The full list of nature’s symmetries also includes
gauge symmetry, particle exchange symmetry and certain vacuum symmetries.
All aspects of motion, like all components of a symmetric system, can be classified
9 motion and symmetry 287

by their symmetry behaviour, i.e., by the multiplet or the representation to which they
belong. As a result, observables are either scalars, vectors, spinors or tensors.
An fruitful way to formulate the patterns and ‘laws’ of nature – i.e., the Lagrangian
of a physical system – has been the search for the complete set of nature’s symmetries
first. For example, this is helpful for oscillations, waves, relativity, quantum physics and
quantum electrodynamics. We will use the method throughout our walk; in the last part
of our adventure we will discover some symmetries which are even more mind-boggling
than those of relativity and those of interactions. In the next section, though, we will
move on to the next approach for a global description of motion.

Motion Mountain – The Adventure of Physics
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C h a p t e r 10

SI M PL E MOT ION S OF E X T E N DE D
B ODI E S – O S C I L L AT ION S A N D
WAV E S

T
he observation of change is a fundamental aspect of nature. Among all
hese observations, periodic change is frequent around us. Indeed,
hroughout everyday life be observe oscillations and waves: Talking, singing, hear-
ing and seeing would be impossible without them. Exploring oscillations and waves, the
next global approach to motion in our adventure, is both useful and beautiful.

Motion Mountain – The Adventure of Physics
Oscillations
Page 248 Oscillations are recurring changes, i.e., cyclic or periodic changes. Above, we defined ac-
tion, and thus change, as the integral of the Lagrangian, and we defined the Lagrangian
as the difference between kinetic and potential energy. All oscillating systems periodic-
ally exchange one kind of energy with the other. One of the simplest oscillating systems
in nature is a mass 𝑚 attached to a (linear) spring. The Lagrangian for the mass position
𝑥 is given by
𝐿 = 12 𝑚𝑣2 − 12 𝑘𝑥2 , (93)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
where 𝑘 is a quantity characterizing the spring, the so-called spring constant. The Lag-
rangian is due to Robert Hooke, in the seventeenth century. Can you confirm the expres-
Challenge 507 e sion?
The motion that results from this Lagrangian is periodic, and illustrated in Figure 203.
The Lagrangian (93) thus describes the oscillation of the spring length over time. The
motion is exactly the same as that of a long pendulum at small amplitude. The motion
is called harmonic motion, because an object vibrating rapidly in this way produces a
completely pure – or harmonic – musical sound. (The musical instrument producing
the purest harmonic waves is the transverse flute. This instrument thus gives the best
idea of how harmonic motion ‘sounds’.)
The graph of this harmonic oscillation, also called linear oscillation, shown in Fig-
ure 203, is called a sine curve; it can be seen as the basic building block of all oscillations.
All other, anharmonic oscillations in nature can be composed from harmonic ones, i.e.,
Page 292 from sine curves, as we shall see shortly. Any quantity 𝑥(𝑡) that oscillates harmonically
is described by its amplitude 𝐴, its angular frequency 𝜔 and its phase 𝜑:

𝑥(𝑡) = 𝐴 sin(𝜔𝑡 + 𝜑) . (94)

The amplitude and the phase depend on the way the oscillation is started. In contrast,
oscillations and waves 289

A harmonically its position over time the corresponding
oscillating phase
object position
period T

amplitude A
phase φ
time

period T

An anharmonically
oscillating
object position
period T

amplitude A

Motion Mountain – The Adventure of Physics
time

F I G U R E 203 Above: the simplest oscillation, the linear or harmonic oscillation: how position changes
over time, and how it is related to rotation. Below: an example of anharmonic oscillation.

the angular frequency 𝜔 is an intrinsic property of the system. Can you show that for the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 508 s mass attached to the spring, we have 𝜔 = 2π𝑓 = 2π/𝑇 = √𝑘/𝑚 ?
Every harmonic oscillation is thus described by just three quantities: the amplitude,
the period (the inverse of the frequency) and the phase. The phase, illustrated in Fig-
ure 205, distinguishes oscillations of the same amplitude and period 𝑇; the phase defines
at what time the oscillation starts.
Some observed oscillation frequencies are listed in Table 37. Figure 203 shows how
a harmonic oscillation is related to an imaginary rotation. As a result, the phase is best
described by an angle value between 0 and 2π.

Damping
Every oscillating motion continuously transforms kinetic energy into potential energy
and vice versa. This is the case for the tides, the pendulum, or any radio receiver. But
many oscillations also diminish in time: they are damped. Systems with large damping,
such as the shock absorbers in cars, are used to avoid oscillations. Systems with small
damping are useful for making precise and long-running clocks. The simplest meas-
ure of damping is the number of oscillations a system takes to reduce its amplitude to
1/𝑒 ≈ 1/2.718 times the original value. This characteristic number is the so-called Q-
factor, named after the abbreviation of ‘quality factor’. A poor Q-factor is 1 or less, and
an extremely good one is 100 000 or more. (Can you write down a simple Lagrangian for
Challenge 509 ny a damped oscillation with a given Q-factor?) In nature, damped oscillations do not usu-
290 10 simple motions of extended bodies

TA B L E 37 Some sound frequency values found in nature.

O b s e r va t i o n Frequency

Sound frequencies in gas, emitted by black holes c. 1 fHz
Precision in measured vibration frequencies of the Sun down to 2 nHz
Vibration frequencies of the Sun down to c. 300 nHz
Vibration frequencies that disturb gravitational radiation down to 3 μHz
detection
Lowest vibration frequency of the Earth Ref. 218 309 μHz
Resonance frequency of stomach and internal organs 1 to 10 Hz
(giving the ‘sound in the belly’ experience)
Common music tempo 2 Hz
Frequency used for communication by farting fish c. 10 Hz
Sound produced by loudspeaker sets (horn, electro- c. 18 Hz to over 150 kHz
magetic, piezoelectric, electret, plasma, laser)
Sound audible to young humans 20 Hz to 20 kHz

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Hum of electrical appliances, depending on country 50 or 60 Hz
Fundamental voice frequency of speaking adult human 85 Hz to 180 Hz
male
Fundamental voice frequency of speaking adult human 165 Hz to 255 Hz
female
Official value, or standard pitch, of musical note ‘A’ or 440 Hz
‘la’, following ISO 16 (and of the telephone line signal in
many countries)
Common values of musical note ‘A’ or ‘la’ used by or- 442 to 451 Hz

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
chestras
Wing beat of tiniest flying insects c. 1000 Hz
Fundamental sound frequency produced by the feathers 1 to 1.5 kHz
of the club-winged manakin, Machaeropterus deliciosus
Fundamental sound frequency of crickets 2 kHz to 9 kHz
Quartz oscillator frequencies 20 kHz up to 350 MHz
Sonar used by bats up to over 100 kHz
Sonar used by dolphins up to 150 kHz
Sound frequency used in ultrasound imaging 2 to 20 MHz
Phonon (sound) frequencies measured in single crystals up to 20 THz and more
oscillations and waves 291

F I G U R E 204 The interior of a commercial quartz
oscillator, a few millimetres in size, driven at high
amplitude. (QuickTime ﬁlm © Microcrystal)

ally keep constant frequency; however, for the simple pendulum this remains the case
to a high degree of accuracy. The reason is that for a pendulum, the frequency does not

Motion Mountain – The Adventure of Physics
depend significantly on the amplitude (as long as the amplitude is smaller than about
20°). This is one reason why pendulums are used as oscillators in mechanical clocks.
Obviously, for a good clock, the driving oscillation must not only show small damp-
ing, but must also be independent of temperature and be insensitive to other external
influences. An important development of the twentieth century was the introduction of
quartz crystals as oscillators. Technical quartzes are crystals of the size of a few grains of
sand; they can be made to oscillate by applying an electric signal. They have little temper-
ature dependence and a large Q-factor, and therefore low energy consumption, so that
precise clocks can now run on small batteries. The inside of a quartz oscillator is shown
in Figure 204.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
R esonance
In most physical systems that are brought to oscillate by an external source, the resulting
amplitude depends on the frequency. The selected frequencies for which the amplitude is
maximal are called resonance frequencies or simply resonances. For example, the quartz
oscillator of Figure 204, or the usual vibration frequencies of guitar strings or bells –
shown in Figure 206 – are resonance frequencies.
Usually, the oscillations at which a system will oscillate when triggered by a short
hit will occur at resonance frequencies. Most musical instruments are examples. Almost
all systems have several resonance frequencies; flutes, strings and bells are well-known
examples.
In contrast to music or electronics, in many other situations resonance needs to be
avoided. In buildings, earthquakes can trigger resonances; in bridges, the wind can trig-
ger resonant oscillations; similarly, in many machines, resonances need to be dampened
or blocked in order to avoid the large amplitude of a resonance destroying the system.
In modern high-quality cars, the resonances of each part and of each structure are cal-
culated and, if necessary, adjusted in such a way that no annoying vibrations disturb the
driver or the passenger.
All systems that oscillate also emit waves. In fact, resonance only occurs because all os-
cillations are in fact localized waves. Indeed, oscillations only appear in extended systems;
292 10 simple motions of extended bodies

A harmonic wave
displacement
crest
or peak wavelength λ
or maximum

amplitude A

node node node
space
wavelength λ

trough
or minimum

An example of anharmonic signal

Motion Mountain – The Adventure of Physics
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F I G U R E 205 Top: the main properties of a harmonic wave, or sine wave. Bottom: A general periodic
signal, or anharmonic wave – here a black square wave – can be decomposed uniquely into simplest, or
harmonic waves. The ﬁrst three components (green, blue and red) and also their intermediate sum
(black dotted line) are shown. This is called a Fourier decomposition and the general method to do this
Fourier analysis. (© Wikimedia) Not shown: the unique decomposition into harmonic waves is even
possible for non-periodic signals.

Hum Prime Tierce

F I G U R E 206 The measured fundamental vibration patterns of a bell. Bells – like every other source of
oscillations, be it an atom, a molecule, a music instrument or the human voice – show that all
oscillations in nature are due to waves. (© H. Spiess & al.).
oscillations and waves 293

Motion Mountain – The Adventure of Physics
F I G U R E 207 The centre of the grooves in an old vinyl record show the amplitude of the sound
pressure, averaged over the two stereo channels (scanning electron microscope image by © Chris
Supranowitz/University of Rochester).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
and oscillations are only simplified descriptions of the repetitive motion of an extended
system. The complete and general repetitive motion of an extended system is the wave.

Waves: general and harmonic
Waves are travelling imbalances, or, equivalently, travelling oscillations of a substrate.
Waves travel, though the substrate does not. Every wave can be seen as a superposition
of harmonic waves. Every sound effect can be thought of as being composed of harmonic
waves. Harmonic waves, also called sine waves or linear waves, are the building blocks of
which all internal motions of an extended body are constructed, as shown in Figure 205.
Can you describe the difference in wave shape between a pure harmonic tone, a musical
Challenge 510 e sound, a noise and an explosion?
Every harmonic wave is characterized by an oscillation frequency 𝑓, a propagation or
phase velocity 𝑐, a wavelength 𝜆, an amplitude 𝐴 and a phase 𝜑, as can be deduced from
Figure 205. Low-amplitude water waves are examples of harmonic waves – in contrast to
water waves of large amplitude. In a harmonic wave, every position by itself performs a
harmonic oscillation. The phase of a wave specifies the position of the wave (or a crest)
at a given time. It is an angle between 0 and 2π.
The phase velocity 𝑐 is the speed with which a wave maximum moves. A few examples
294 10 simple motions of extended bodies

TA B L E 38 Some wave velocities.

Wa v e Ve l o c i t y

Tsunami around 0.2 km/s
Sound in most gases 0.3 ± 0.1 km/s
Sound in air at 273 K 0.331 km/s
Sound in air at 293 K 0.343 km/s
Sound in helium at 293 K 0.983 km/s
Sound in most liquids 1.2 ± 0.2 km/s
Seismic waves 1 to 14 km/s
Sound in water at 273 K 1.402 km/s
Sound in water at 293 K 1.482 km/s
Sound in sea water at 298 K 1.531 km/s
Sound in gold 4.5 km/s
Sound in steel 5.8 to 5.960 km/s

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Sound in granite 5.8 km/s
Sound in glass (longitudinal) 4 to 5.9 km/s
Sound in beryllium (longitudinal) 12.8 km/s
Sound in boron up to 15 km/s
Sound in diamond up to 18 km/s
Sound in fullerene (C60 ) up to 26 km/s
Plasma wave velocity in InGaAs 600 km/s
Light in vacuum 2.998 ⋅ 108 m/s

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
are listed in Table 38. Can you show that frequency and wavelength in a wave are related
Challenge 511 e by 𝑓𝜆 = 𝑐?
Waves appear inside all extended bodies, be they solids, liquids, gases or plasmas. In-
side fluid bodies, waves are longitudinal, meaning that the wave motion is in the same
direction as the wave oscillation. Sound in air is an example of a longitudinal wave. Inside
solid bodies, waves can also be transverse; in that case the wave oscillation is perpendic-
ular to the travelling direction.
Waves appear also on interfaces between bodies: water–air interfaces are a well-known
case. Even a saltwater–freshwater interface, so-called dead water, shows waves: they
can appear even if the upper surface of the water is immobile. Any flight in an aero-
plane provides an opportunity to study the regular cloud arrangements on the interface
between warm and cold air layers in the atmosphere. Seismic waves travelling along the
boundary between the sea floor and the sea water are also well-known. Low-amplitude
water waves are transverse; however, general surface waves are usually neither longitud-
inal nor transverse, but of a mixed type.
To get a first idea about waves, we have a look at water waves.
oscillations and waves 295

Water surface:

At a depth of half
the ave
w len
gth, the amplitude is negligible

F I G U R E 208 The formation of the shape of deep gravity waves, on and under water, from the circular
motion of the water particles. Note the non-sinusoidal shape of the wave.

Motion Mountain – The Adventure of Physics
Water waves
Water waves on water surfaces show a large range of fascinating phenomena. First of all,
there are two different types of surface water waves. In the first type, the force that re-
stores the plane surface is the surface tension of the wave. These so-called surface tension
waves play a role on scales up to a few centimetres. In the second, larger type of waves,
the restoring force is gravity, and one speaks of gravity waves.* The difference is easily
Page 293 noted by watching them: surface tension waves have a sinusoidal shape, whereas gravity

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
waves have a shape with sharper maxima and broader troughs. This occurs because of
the special way the water moves in such a wave. As shown in Figure 208, the surface wa-
ter for a (short) gravity water wave moves in circles; this leads to the typical wave shape
with short sharp crests and long shallow troughs: the waves are not up–down symmetric.
Under the crests, the water particles move in the direction of the wave motion; under the
troughs, the water particles move against the wave motion. As long as there is no wind
and the floor below the water is horizontal, gravity waves are symmetric under front-to-
back reflection. If the amplitude is very high, or if the wind is too strong, waves break,
because a cusp angle of more than 120° is not possible. Such waves have no front-to-back
symmetry.
In addition, water waves need to be distinguished according to the depth of the water,
when compared to their wavelength. One speaks of short or deep water waves, when the
depth of the water is so high that the floor below plays no role. In the opposite case one
speaks of long or shallow water waves. The transitional region between the two cases are
waves whose wavelength is between twice and twenty times the water depth. It turns out
that all deep water waves and all ripples are dispersive, i.e., their speed depends on their
frequency; only shallow gravity water waves are non-dispersive.
Water waves can be generated by wind and storms, by earthquakes, by the Sun and

* Meteorologists also know of a third type of water wave: there are large wavelength waves whose restoring
force is the Coriolis force.
296 10 simple motions of extended bodies

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F I G U R E 209 Three of the four main types of water waves. Top: a shallow water gravity wave,
non-sinusoidal. Bottom left: a deep water ripple – a sinusoidal surface tension wave. The not-shown
shallow water ripples look the same. Bottom right: a deep water gravity wave, here a boat wake, again
non-sinusoidal. (© Eric Willis, Wikimedia, allyhook)

Moon, and by any other effect that displaces water. The spectrum of water waves reaches
from periods shorter than 100 ms to periods longer than 24 h. An overview is given
in Table 39. The table also includes the lesser known infra-gravity waves, ultra-gravity
waves,, tides and trans-tidal waves.
The classification of periodic water waves according to their restoring force and to the
influence of the floor give four limit cases; they are shown in Figure 210. Each of the four
limit cases is interesting.
Experiments and theory show that the phase speed of gravity waves, the lower two
cases in Figure 210, depends on the wavelength 𝜆 and on the depth of the water 𝑑 in the
following way:
𝑔𝜆 2π𝑑
𝑐 = √ tanh , (95)
2π 𝜆

where 𝑔 is the acceleration due to gravity (and an amplitude much smaller than the
oscillations and waves 297

TA B L E 39 Spectrum of water waves.

Type Period Propertie s Genera-
tion
Surface tension < 0.1 s wavelength below a few cm wind with more
waves/capillary than 1 m/s, other
waves/ripples disturbances,
temperature
Ultra-gravity waves 0.1 to 1 s restoring forces are surface wind, other dis-
tension and gravity turbances
Ordinary gravity 1 to 30 s amplitude up to many wind
waves meters, restoring force is
gravity
Infra-gravity 30 s to 5 min amplitude up to 30 cm, wind, gravity
waves/surf beat related to seiches, restoring waves
force is gravity
Long-period waves 5 min to 12 h amplitude typically below storms, earth-

Motion Mountain – The Adventure of Physics
10 cm in deep water, up quakes, air
to 40 m in shallow water, pressure changes
restoring force is gravity and
Coriolis effect
Ordinary tides 12 h to 24 h amplitude depends on loca- Moon, Sun
tion, restoring force is gravity
and Coriolis effect
Trans-tidal waves above 24 h amplitude depends on loca- Moon, Sun,
tion, restoring force is gravity storms, seasons,

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and Coriolis effect climate change

wavelength is assumed*). The formula shows two limiting regimes.
First, so-called deep water or short gravity waves appear when the water depth is larger
than about half the wavelength. The usual sea wave is a deep water gravity wave, and
so are the wakes generated by ships. Deep water gravity waves generated by wind are
called sea they are generated by local winds, and swell if they are generated by distant
winds. The typical phase speed of a gravity wave is of the order of the wind speed that
generates it. For deep water waves, the phase velocity is related to the wavelength by
𝑐 ≈ √𝑔𝜆/2π ; the phase velocity is thus wavelength-dependent. In fact, all deep waves
Page 301 are dispersive. Shorter deep gravity waves are thus slower. The group velocity is half the
phase velocity. Therefore, as surfers know, waves on a shore that are due to a distant storm
arrive separately: first the long period waves, then the short period waves. The general
effects of dispersion on wave groups are shown in Figure 211.
The typical wake generated by a ship is made of waves that have the phase velocity of
the ship. These waves form a wave group, and it travels with half that speed. Therefore,
from a ship’s point of view, the wake trails the ship. Wakes are behind the ship because

* The expression for the phase velocity can be derived by solving for the motion of the liquid in the linear
regime, but this leads us too far from our walk.
298 10 simple motions of extended bodies

Water wave capillary waves or surface tension waves or ripples
dispersion sinusoidal, of small amplitude, dispersive
GAM

ripples in shallow water ripples in deep water
2 4
𝜔 = 𝛾𝑑𝑘 /𝜌 CC 𝜔2 = 𝛾𝑘3 /𝜌
104

102
shallow water 𝜔2 = (𝑔𝑘 + 𝛾𝑘3 /𝜌) tanh 𝑘𝑑 deep water
or long waves 𝑑𝑘 or short waves
10-4 10-2 1 102 104 dispersive
2 2 2
𝜔 = 𝑘 (𝑔𝑑 + 𝛾𝑑𝑘 /𝜌) 10-2 𝜔2 = 𝑔𝑘 + 𝛾𝑘3 /𝜌

10-4
𝜔2 = 𝑔𝑑𝑘2 𝜔2 = 𝑔𝑘 tanh 𝑘𝑑 𝜔2 = 𝑔𝑘

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tides, tsunamis wakes, sea, swell, etc.
non-dispersive
gravity waves
non-sinusoidal, amplitude large, but
still much smaller than wavelength
(thus without cnoidal waves, Stokes waves or solitons)

Water wave wavelength
dispersion (m) tide

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106
tsunami 𝑑𝑘 = 1
104
storm waves
storm waves on open sea
2
at shore 10

depth (m)
10-4 10-2 1 102 104
G2AM
ripples 10-2 ripples
on thin puddle on pond
10-4

F I G U R E 210 The different types of periodic water waves with low and moderate amplitude, visualized
in two different diagrams using the depth 𝑑, the wave number 𝑘 = 2π/𝜆 and the surface tension
𝛾 =72 mPa.

their group velocity is lower than their phase velocity. We will explore wakes in more
Page 328 detail below.
The second limiting regime are shallow water or long gravity waves. They appear when
oscillations and waves 299

F I G U R E 211 A visualisation of group velocity
(blue) and phase velocity (red) for different types
of wave dispersion. (QuickTime ﬁlm © ISVR,
University of Southampton)

Motion Mountain – The Adventure of Physics
the depth is less than 1/20th of the wavelength; in this case, the phase velocity is 𝑐 ≈ √𝑔𝑑 ,
there is no dispersion, and the group velocity is the same as the phase velocity. In shallow
water waves, water particles move on very flat elliptic paths.
The tide is an example of a shallow gravity wave. The most impressive shallow gravity
waves are tsunamis, the large waves triggered by submarine earthquakes. (The Japanese
term is composed of tsu, meaning harbour, and nami, meaning wave.) Since tsunamis are
shallow waves, they show no, or little, dispersion and thus travel over long distances; they
can go round the Earth several times. Typical oscillation times of tsunamis are between
6 and 60 minutes, giving wavelengths between 70 and 700 km and speeds in the open sea
of 200 to 250 m/s, similar to that of a jet plane. Their amplitude on the open sea is often

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Challenge 512 e
of the order of 10 cm; however, the amplitude scales with depth 𝑑 as 1/𝑑4 and heights
up to 40 m have been measured at the shore. This was the order of magnitude of the
large and disastrous tsunami observed in the Indian Ocean on 26 December 2004 and
the one in Japan in 2011 that destroyed several nuclear power plants. Tsunamis can also
be used to determine the ocean depth by measuring the speed of tsunamis. This allowed
researchers to deduce, long before sonar and other high-tech systems were available, that
the North Pacific has a depth of around 4 to 4.5 km.
The upper two limit regimes in Figure 210 are the surface tension waves, also called
capillary waves or ripples. The value of the surface tension of water is 72 mPa. The first
of these limit regimes are ripples on deep water. Their phase velocity is 𝑐 = √𝛾𝑘/𝜌 . As
mentioned, all deep waves are dispersive. Indeed, the group velocity of ripples on deep
Challenge 513 e water is 3/2 times the phase velocity. Therefore ripples steam ahead of a boat, whereas
wakes trail behind.
Challenge 514 e Deep ripples have a minimum speed. . The minimum speed of short ripples is the
reason for what we see when throw a pebble in a lake. A typical pebble creates ripples with
a wavelength of about 1 cm. For water waves in this region, there is a minimum group ve-
Challenge 515 e locity of 17.7 cm/s and a minimum phase velocity of around 23 cm/s. When a pebble falls
into the water, it creates waves of various wavelengths; those with a wavelength around
1 cm are the slowest ones and are seen most clearly. The existence of a minimum phase
velocity for ripples also means that insects walking on water generate no waves if they
300 10 simple motions of extended bodies

move more slowly than the minimum phase velocity; thus they feel little drag and can
walk easily.
The final limit case of waves are ripples on shallow water. An example are the waves
emitted by raindrops falling in a shallow puddle, say with a depth of 1 mm or less. The
Challenge 516 e phase velocity is 𝑐 = √𝛾𝑑𝑘2 /𝜌 ; the group velocity has twice that value. Ripples on shal-
low water are thus dispersive, as are ripples in deep water.
Figure 210 shows the four types of water surface waves discussed so far. The general
dispersion relation for all these waves is

𝜔2 = (𝑔𝑘 + 𝛾𝑘3 /𝜌) tanh 𝑘𝑑 . (96)

Several additional types of water waves also exist, such as seiches, i.e., standing waves in
lakes, internal waves in stratified water systems, and solitons, i.e., non-periodic travelling
Page 316 waves. We will explore the last case later on.

Waves and their motion

Motion Mountain – The Adventure of Physics
Waves move. Therefore, any study of motion must include the study of wave motion.
We know from experience that waves can hit or even damage targets; thus every wave
carries energy and momentum, even though (on average) no matter moves along the
wave propagation direction. The energy 𝐸 of a wave is the sum of its kinetic and potential
energy. The kinetic energy (density) depends on the temporal change of the displacement
𝑢 at a given spot: rapidly changing waves carry a larger kinetic energy. The potential
energy (density) depends on the gradient of the displacement, i.e., on its spatial change:
steep waves carry a larger potential energy than shallow ones. (Can you explain why
Challenge 517 s the potential energy does not depend on the displacement itself?) For harmonic, i.e.,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
sinusoidal, waves propagating along the direction 𝑧, each type of energy is proportional
Ref. 219 to the square of its respective displacement change:

∂𝑢 2 ∂𝑢 2
𝐸∼( ) + 𝑣2 ( ) . (97)
∂𝑡 ∂𝑧

Challenge 518 ny How is the energy density related to the frequency?
The momentum of a wave is directed along the direction of wave propagation. The
momentum value depends on both the temporal and the spatial change of displacement
𝑢. For harmonic waves, the momentum (density) 𝑃 is proportional to the product of
these two quantities:
∂𝑢 ∂𝑢
𝑃𝑧 ∼ . (98)
∂𝑡 ∂𝑧
When two linear wave trains collide or interfere, the total momentum is conserved
throughout the collision. An important consequence of momentum conservation is that
waves that are reflected by an obstacle do so with an outgoing angle equal to minus the
Challenge 519 s infalling angle. What happens to the phase?
In summary, waves, like moving bodies, carry energy and momentum. In simple
terms, if you shout against a wall, the wall is hit by the sound waves. This hit, for example,
oscillations and waves 301

can start avalanches on snowy mountain slopes. In the same way, waves, like bodies, can
carry also angular momentum. (What type of wave is necessary for this to be possible?)
Challenge 520 s However, the motion of waves differs from the motion of bodies. Six main properties
distinguish the motion of waves from the motion of bodies.
1. Waves can add up or cancel each other out; thus they can interpenetrate each other.
These effects, superposition and interference, are strongly tied to the linearity of many
wave types.
2. Waves, such as sound, can go around corners. This is called diffraction. Diffraction is
a consequence of interference, and it is thus not, strictly speaking, a separate effect.
3. Waves change direction when they change medium. This is called refraction. Refrac-
tion is central to producing images.
4. Waves can have a frequency-dependent propagation speed. This is called dispersion.
It produces rainbows and the grumble of distant thunders.
5. Often, the wave amplitude decreases over time: waves show damping. Damping pro-
duces pauses between words.

Motion Mountain – The Adventure of Physics
6. Transverse waves in three dimensions can oscillate in different directions: they show
polarization. Polarization is important for antennas and photographs.
In our everyday life, in contrast to moving waves, moving material bodies, such as stones,
do not show any of these six effects. The six wave effects appear because wave motion
is the motion of extended entities. The famous debate whether electrons are waves or
particles thus requires us to check whether the six effects specific to waves can be ob-
served for electrons or not. This exploration is part of quantum physics. Before we start
it, can you give an example of an observation that automatically implies that the specific
Challenge 521 s motion cannot be a wave?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
As a result of having a frequency 𝑓 and a propagation or phase velocity 𝑐, all sine
waves are characterized by the distance 𝜆 between two neighbouring wave crests: this
distance is called the wavelength 𝜆. All waves obey the basic relation

𝜆𝑓 = 𝑐 . (99)

In many cases the phase velocity 𝑐 depends on the wavelength of the wave. For example,
this is the case for many water waves. This change of speed with wavelength is called
dispersion. In contrast, the speed of sound in air does not depend on the wavelength
(to a high degree of accuracy). Sound in air shows (almost) no dispersion. Indeed, if
there were dispersion for sound, we could not understand each other’s speech at larger
distances.
Now comes a surprise. Waves can also exist in empty space. Light, radio waves and
gravitational waves are examples. The exploration of electromagnetism and relativity will
tell us more about their specific properties. Here is an appetizer. Light is a wave. Usu-
ally, we do not experience light as a wave, because its wavelength is only around one
two-thousandth of a millimetre. But light shows all six effects typical of wave motion.
Vol. III, page 97 A rainbow, for example, can only be understood fully when the last five wave effects are
taken into account. Diffraction and interference of light can even be observed with your
Challenge 522 s fingers only. Can you tell how?
302 10 simple motions of extended bodies

Interference (temporal aspect)

Interference (spatial aspect)

Polarisation

Motion Mountain – The Adventure of Physics
Diffraction

Refraction

Dispersion

F I G U R E 212 The six main properties of the motion of waves. (© Wikimedia)

For gravitational waves, superposition, damping and polarization have been observed.
The other three wave effects are still unobserved.
oscillations and waves 303

F I G U R E 213 Interference of two circular or spherical waves emitted in phase: a snapshot of the
amplitude (left), most useful to describe observations of water waves, and the distribution of the

Motion Mountain – The Adventure of Physics
time-averaged intensity (right), most useful to describe interference of light waves (© Rüdiger
Paschotta).

Like every anharmonic oscillation, every anharmonic wave can be decomposed into
Page 292 sine waves. Figure 205 gives examples. If the various sine waves contained in a disturb-
ance propagate differently, the original wave will change in shape while it travels. That
is the reason why an echo does not sound exactly like the original sound; for the same
reason, a nearby thunder and a far-away one sound different. These are effects of the –

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
rather weak – dispersion of sound waves.
All systems which oscillate also emit waves. Any radio or TV receiver contains oscillat-
ors. As a result, any such receiver is also a (weak) transmitter; indeed, in some countries,
the authorities search for people who listen to radio without permission by listening to
the radio waves emitted by these devices. Also, inside the human ear, numerous tiny
structures, the hair cells, oscillate. As a result, the ear must also emit sound. This pre-
diction, made in 1948 by Tommy Gold, was finally confirmed in 1979 by David Kemp.
These so-called otoacoustic emissions can be detected with sensitive microphones; they
are presently being studied in order to unravel the still unknown workings of the ear and
Ref. 220 in order to diagnose various ear illnesses without the need for surgery.
Since any travelling disturbance can be decomposed into sine waves, the term ‘wave’
is used by physicists for all travelling disturbances, whether they look like sine waves or
not. In fact, the disturbances do not even have to be travelling. Take a standing wave: is it a
wave or an oscillation? Standing waves do not travel; they are oscillations. But a standing
wave can be seen as the superposition of two waves travelling in opposite directions.
In fact, in nature, any object that we call ‘oscillating’ or ‘vibrating’ is extended, and its
Challenge 523 e oscillation or vibration is always a standing wave (can you confirm this?); so we can say
that in nature, all oscillations are special forms of waves.
The most important travelling disturbances are those that are localized. Figure 205
Page 292 shows an example of a localized wave, also called a wave group or pulse, together with
its decomposition into harmonic waves. Wave groups are used to talk and as signals for
304 10 simple motions of extended bodies

secondary
waves

primary envelope of
wave secondary waves
at time 𝑡1 at time 𝑡2
F I G U R E 214 Wave propagation as a
consequence of Huygens’ principle.

Motion Mountain – The Adventure of Physics
communication.

Why can we talk to each other? – Huygens ’ principle
The properties of our environment often disclose their full importance only when we
ask simple questions. Why can we use the radio? Why can we talk on mobile phones?
Why can we listen to each other? It turns out that a central part of the answer to these
questions is that the space we live has an odd numbers of dimensions.
In spaces of even dimension, it is impossible to talk, because messages do not stop – or

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
stop very slowly. This is an important result which is easily checked by throwing a stone
into a lake: even after the stone has disappeared, waves are still emitted from the point
at which it entered the water. Yet, when we stop talking, no waves are emitted any more
from our mouth. In short, waves in two and in three dimensions behave differently.
In three dimensions, it is possible to describe the propagation of a wave in the follow-
ing way: Every point on a wave front (of light or of sound) can be regarded as the source
of secondary waves; the surface that is formed by the envelope of all the secondary waves
determines the future position of the wave front. The idea is illustrated in Figure 214.
This idea can be used to describe, without mathematics, the propagation of waves, their
reflection, their refraction, and, with an extension due to Augustin Fresnel, their diffrac-
Challenge 524 e tion. (Try!)
This idea of secondary waves was first proposed by Christiaan Huygens in 1678 and
is called Huygens’ principle. Almost two hundred years later, Gustav Kirchhoff showed
that the principle is a consequence of the wave equation in three dimensions, and thus,
in the case of light, a consequence of Maxwell’s field equations.
But the description of wave fronts as envelopes of secondary waves has an import-
ant limitation. It is not correct in two dimensions (even though Figure 214 is two-
dimensional!). In particular, it does not apply to water waves. Water wave propagation
cannot be calculated in this way in an exact manner. (It is only possible in two dimen-
sions if the situation is limited to a wave of a single frequency.) It turns out that for water
waves, secondary waves do not only depend on the wave front of the primary waves,
oscillations and waves 305

F I G U R E 215 An impossible
gravity water wave: the centre
is never completely ﬂat.

but depend also on their interior. The reason is that in two (and other even) dimensions,
waves of different frequency necessarily have different speeds. (Tsunamis are not counter-
examples; they non-dispersive only as a limit case.) And indeed, a stone falling into water
is comparable to a banging sound: it generates waves of many frequencies. In contrast,
in three (and larger odd) dimensions, waves of all frequencies have the same speed.
Mathematically, we can say that Huygens’ principle holds if the wave equation is
solved by a circular wave leaving no amplitude behind it. Mathematicians translate this
by requiring that the evolving delta function 𝛿(𝑐2 𝑡2 − 𝑟2 ) satisfies the wave equation, i.e.,
that ∂2𝑡 𝛿 = 𝑐2 Δ𝛿. The delta function is that strange ‘function’ which is zero everywhere

Motion Mountain – The Adventure of Physics
except at the origin, where it is infinite. A few more properties describe the precise way
in which this happens.* In short, the delta function is an idealized, infinitely short and
infinitely loud bang. It turns out that the delta function is a solution of the wave equation
only if the space dimension is odd and at least three. In other words, while a spherical
wave pulse is possible, a circular pulse is not: there is no way to keep the centre of an ex-
panding wave quiet. (See Figure 215.) That is exactly what the stone experiment shows.
You can try to produce a circular pulse (a wave that has only a few crests) the next time
Challenge 525 e you are in the bathroom or near a lake: you will not succeed.
In summary, the reason a room gets dark when we switch off the light, is that we live

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in a space with a number of dimensions which is odd and larger than one.

Wave equations**
Waves are fascinating phenomena. Equally fascinating is their mathematical description.
The amplitude 𝐴(𝑥, 𝑡) of a linear wave in one, two or three dimensions, the simplest
of all wave types, results from

∂2 𝐴(𝑥, 𝑡)
= 𝑣2 ∇2 𝐴(𝑥, 𝑡) . (100)
∂𝑡2
The equation says that the acceleration of the amplitude (the left-hand term) is given by
the square gradient, i.e., by the spatial variation, multiplied by the squared of the phase
velocity 𝑣. In many cases, the amplitude is a vector, but the equation remains the same.
More correctly, the amplitude of a wave is what physicists call a field, because it is a num-
ber (or vector, or tensor) that depends on space and time.
Equation (100) is a wave equation. Mathematically, it is a linear partial differential
equation. It is called linear because it is linear in the amplitude 𝐴. Therefore, its solu-

* The main property is ∫𝛿(𝑥) d𝑥 = 1. In mathematically precise terms, the delta ‘function’ is a distribution.
** This section can be skipped at first reading.
306 10 simple motions of extended bodies

TA B L E 40 Some signals.

System Signal Speed Sensor

Matter signals
Human voltage pulses in nerves up to 120 m/s brain,
muscles
hormones in blood stream up to 0.3 m/s molecules on
cell
membranes
immune system signals up to 0.3 m/s molecules on
cell
membranes
singing 340 m/s ear
Elephant, insects soil trembling c. 2 km/s feet
Whale singing, sonar 1500 m/s ear
Dog chemical tracks 1 m/s nose

Motion Mountain – The Adventure of Physics
Butterfly chemical mating signal carried by up to 10 m/s antennae
the wind
Tree chemical signal of attack carried by up to 10 m/s leaves
the air from one tree to the next
Erratic block carried by glacier up to foot
0.1 μm/s
Post paper letters transported by trucks, up to 300 m/s mailbox
ships and planes
Electromagnetic fields

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Humans yawning 300 Mm/s eye
Electric eel voltage pulse up to nerves
300 Mm/s
Insects, fish, light pulse sequence up to eye
molluscs 300 Mm/s
Flag signalling orientation of flags 300 Mm/s eye
Radio transmissions electromagnetic field strength up to radio
300 Mm/s
Nuclear signals
Supernovas neutrino pulses close to specific
300 Mm/s chemical and
radiation
detectors
Nuclear reactions glueballs, if they exist close to custom
300 Mm/s particle
detectors

Challenge 526 e tions are sine and cosine waves of the type 𝐴 = sin(𝑥 − 𝑣𝑡 + 𝜑). Linear wave equations
follow from the elastic behaviour of some medium. The linearity also implies that the
sum of two waves is also a possible wave; this so-called superposition principle is valid
oscillations and waves 307

for all linear wave equations (and a few rare, but important non-linear wave equations).
If linearity applies, any general wave can be decomposed into an infinite sum of sine and
cosine waves. This discovery is due to Joseph Fourier (b. 1768, Auxerre, d. 1830, Paris).
The wave equation (100) is also homogeneous, meaning that there is no term inde-
pendent of 𝐴, and thus, no energy source that drives the waves. Fourier decomposi-
tion also helps to understand and solve inhomogeneous wave equations, thus externally
driven elastic media.
In several dimensions, the shape of the wavefront is also of interest. In two dimen-
Challenge 527 e sions, the simplest cases described by equation (100) are linear and circular waves. In
Challenge 528 e three dimensions, the simplest cases described by equation (100) are plane and spherical
waves.
In appropriate situations – thus when the elastic medium is finite and is excited in
Challenge 529 e specific ways – equation (100) also leads to standing waves. Standing waves are super-
positions of waves travelling in opposite directions.
Sound in gases, sound in liquids and solids, earthquakes, light in vacuum, certain
water waves of small amplitude, and various other cases of waves with small amplitude

Motion Mountain – The Adventure of Physics
are described by linear wave equations.
Mathematically, all wave equations, whether linear or not, are hyperbolic partial dif-
ferential equations. This just means that the spatial second derivative has opposite sign
to the temporal second derivative. By far the most interesting wave equations are non-
linear. The most famous non-linear equation is the Korteweg–de Vries equation, which is
the one-dimensional wave equation

∂𝑡 𝐴 + 𝐴∂𝑥 𝐴 + 𝑏∂𝑥𝑥𝑥 𝐴 = 0 . (101)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
It was discovered only very late that this evolution equation for the field 𝐴(𝑥, 𝑡) can be
Page 316 solved with paper and pencil.
Other non-linear wave equations describe specific situations. The Boussinesq equa-
tion, the sine–Gordon equation and many other wave equations have sparked a vast re-
search field in mathematical and experimental physics. Non-linear partial differential
Page 415 equations are also essential in the study of self-organization.

Why are music and singing voices so beau tiful?
Music works because it connects to emotions. And it does so, among others, by remind-
ing us of the sounds (and emotions connected to them) that we experienced before birth.
Percussion instruments remind us of the heartbeat of our mother and ourselves, cord
and wind instruments remind us of all the voices we heard back then. Musical instru-
ments are especially beautiful if they are driven and modulated by the body and the art of
the player. All classical instruments are optimized to allow this modulation and the ex-
pression of emotions. The connection between the musician and the instrument is most
intense for the human voice; the next approximation are the wind instruments.
Every musical instrument, the human voice included, consists of four elements: an en-
ergy source, an oscillating sound source, one or more resonators, and a radiating surface
or orifice. In the human voice, the energy source is formed by the muscles of the thorax
and belly, the sound source are vocal folds – also called vocal cords – the resonator is the
308 10 simple motions of extended bodies

Motion Mountain – The Adventure of Physics
F I G U R E 216 The human
larynx is that part of the
anatomy that contains the
sound source of speech, the
vocal folds (© Wikimedia).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
vocal tract, and the mouth and nose form the orifice.
The breath of the singer or of the wind instrument player provides the energy for
sound generation and gives the input for the feedback loop that sets the pitch. While
singing, the air passes the vocal folds. The rapid air flow reduces the air pressure, which
attracts the cords to each other and thus reduces the cross section for the airflow. (This
Page 361 pressure reduction is described by the Bernoulli equation, as explained below.) As a result
of the smaller cross-section, the airflow is reduced, the pressure rises again, and the vocal
cords open up again. This leads to larger airflow, and the circle starts again. The change
between larger and smaller cord distance repeats so rapidly that sound is produced; the
sound is then amplified in the mouth by the resonances that depend on the shape of
Ref. 221 the oral cavity. Using modern vocabulary, singing a steady note is a specific case of self-
Page 415 organization, namely an example of a limit cycle.
But how can a small instrument like the vocal tract achieve sounds more intense than
that of a trombone, which is several metres long when unwound? How can the voice
cover a range of 80 dB in intensity? How can the voice achieve up to five, even eight
octaves in fundamental frequency with just two vocal folds? And how can the human
voice produce its unmatched variation in timbre? Many details of these questions are
still the subject of research, though the general connections are now known.
The human vocal folds are on average the size of a thumb’s nail; but they can vary
in length and tension. The vocal folds have three components. Above all, they contain a
oscillations and waves 309

ligament that can sustain large changes in tension or stress and forms the basic structure;
such a ligament is needed to achieve a wide range of frequencies. Secondly, 90 % of the
vocal folds are made of muscles, so that the stress and thus the frequency range can be in-
creased even further. Finally, the cords are covered by a mucosa, a fluid-containing mem-
brane that is optimized to enter in oscillation, through surface waves, when air passes by.
This strongly non-linear system achieves, in exceptional singers, up to 5 octaves of fun-
damental pitch range.
Also the resonators of the human voice are exceptional. Despite the small size avail-
able, the non-linear properties of the resonators in the vocal tract – especially the effect
called inertive reactance – allow to produce high intensity sound. This complex system,
together with intense training, produces the frequencies, timbres and musical sequences
that we enjoy in operas, in jazz, and in all other vocal performances. In fact, several res-
ults from research into the human voice – which were also deduced with the help of
magnetic resonance imaging – are now regularly used to train and teach singers, in par-
ticular on when to use open mouth and when to use closed-mouth singing, or when to
lower the larynx.

Motion Mountain – The Adventure of Physics
Singing is thus beautiful also because it is a non-linear effect. In fact, all instruments
Ref. 222 are non-linear oscillators. In reed instruments, such as the clarinet, the reed has the role
of the vocal cords, and the pipe has the role of the resonator, and the mechanisms shift
the opening that has the role of the mouth and lips. In brass instruments, such as the
trombone, the lips play the role of the reed. In airflow instruments, such as the flute,
the feedback loop is due to another effect: at the sound-producing edge, the airflow is
deflected by the sound itself.
The second reason that music is beautiful is due to the way the frequencies of the
notes are selected. Certain frequencies sound agreeable to the ear when they are played

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
together or closely after each other; others produce a sense of tension. Already the an-
cient Greeks had discovered that these sensations depend exclusively on the ratio of the
frequencies, or as musicians say, on the interval between the pitches.
More specifically, a frequency ratio of 2 – musicians call the interval an octave – is the
most agreeable consonance. A ratio of 3/2 (called perfect fifth) is the next most agreeable,
followed by the ratio 4/3 (a perfect fourth), the ratio 5/4 (a major third) and the ratio 6/5
(a [third, minor]minor third). The choice of the first third in a scale has an important
effect on the average emotions expressed by the music and is therefore also taken over
in the name of the scale. Songs in C major generally have a more happy tune, whereas
songs in A minor tend to sound sadder.
The least agreeable frequency ratios, the dissonances, are the tritone (7/5, also called
augmented fourth or diminished fifth or false quint) and, to a lesser extent, the major and
minor seventh (15/8 and 9/5). The tritone is used for the siren in German red cross vans.
Long sequences of dissonances have the effect to induce trance; they are common in
Balinese music and in jazz.
After centuries of experimenting, these results lead to a standardized arrangement of
the notes and their frequencies that is shown in Figure 217. The arrangement, called the
equal intonation or well-tempered intonation, contains approximations to all the men-
tioned intervals; the approximations have the advantage that they allow the transpos-
ition of music to lower or higher notes. This is not possible with the ideal, so-called just
Ref. 223 intonation.
310 10 simple motions of extended bodies

Equal-tempered frequency ratio 1 1.059 1.122 1.189 1.260 1.335 1.414 1.498 1.587 1.682 1.782 1.888 2

Just intonation frequency ratio 1 9/8 6/5 5/4 4/3 none 3/2 8/5 5/3 15/8 2

Appears as harmonic nr. 1,2,4,8 9 5, 10 3, 6 c. 7 15 1,2,4,8

Italian and international Do Do # Re Re # Mi Fa Fa # Sol Sol # La La # Si Do
solfège names Si # Re b Mi b Fa b Mi # Sol b La b Si b Do b Si #

French names Ut Ut # Re Re # Mi Fa Fa # Sol Sol # La La # Si Ut

Motion Mountain – The Adventure of Physics
Si # Re b Mi b Fa b Mi # Sol b La b Si b Ut b Si #

German names C Cis D Dis E F Fis G Gis A Ais H C
His Des Es Fes Eis Ges As B Ces His

English names C C# D D# E F F# G G# A A# B C
B# Db Eb Fb E# Gb Ab Bb Cb B#

Interval name, starting uni- min. maj. min. maj. min. maj. min. maj.
from Do / Ut / C son 2nd 2nd 3rd 3rd 4th triton 5th 6th 6th 7th 7th octave

F I G U R E 217 The twelve notes used in music and their frequency ratios.

The next time you sing a song that you like, you might try to determine whether you
Challenge 530 e use just or equal intonation – or a different intonation altogether. Different people have
different tastes and habits.

Measuring sound
At every point in space, a sound wave in air produces two effects: a pressure change
and a speed change. Figure 218 shows both of them: pressure changes induce changes in
the density of the molecules, whereas velocity changes act on the average speed of the
molecules.
The local sound pressure is measured with a microphone or an ear. The local molecu-
lar speed is measured with a microanemometer or acoustic particle velocity sensor. No
such device existed until 1994, when Hans Elias de Bree invented a way to build one. As
shown in Figure 219, such a microanemometer is not easy to manufacture. Two tiny plat-
inum wires are heated to 220°C; their temperature difference depends on the air speed
and can be measured by comparing their electrical resistances. Due to the tiny dimen-
sions, a frequency range up to about 20 kHz is possible. By putting three such devices at
oscillations and waves 311

F I G U R E 218 A schematic
visualisation of the motion of
molecules in a sound wave in air
(QuickTime ﬁlm © ISVR, University
of Southampton)

Motion Mountain – The Adventure of Physics
TA B L E 41 Selected sound intensities.

O b s e r va t i o n Sound intensity

Sound threshold at 1 kHz 0 dB or 1 pW
Human speech 25 to 35 dB
Subway entering a subway station 100 dB
Ultrasound imaging of babies over 100 dB

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Conventional pain threshold 120 dB or 1 W
Rock concert with 400 000 W loudspeakers 135 to 145 dB
Fireworks up to 150 dB
Gunfire up to 155 dB
Missile launch up to 170 dB
Blue whale singing up to 175 dB
Volcanic eruptions, earthquakes, conventional bomb up to 210 dB
Large meteoroid impact, large nuclear bomb over 300 dB

right angles to each other it is possible to localize the direction of a noise source. This
is useful for the repair and development of cars, or to check trains and machinery. By
arranging many devices on a square grid one can even build an ‘acoustic camera’. It can
even be used to pinpoint aircraft and drone positions, a kind of ‘acoustic radar’. Because
microanemometers act as extremely small and extremely directional microphones, and
because they also work under water, the military and the spies are keen on them.
By the way: how does the speed of molecules due to sound compare to the speed of
Challenge 531 ny molecules due to the air temperature?
312 10 simple motions of extended bodies

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 219 Top: two tiny heated platinum wires allow building a microanemometer. It can help to
pinpoint exactly where a rattling noise comes from (© Microﬂown Technologies).
oscillations and waves 313

F I G U R E 220 A modern ultrasound imaging system, and a common, but harmful ultrasound image of a
foetus (© General Electric, Wikimedia).

Motion Mountain – The Adventure of Physics
Is ultrasound imaging safe for babies?
Ultrasound is used in medicine to explore the interior of human bodies. The technique,
called ultrasound imaging, is helpful, convenient and widespread, as shown in Figure 220.
However, it has a disadvantage. Studies at the Mayo Clinic in Minnesota have found that
pulsed ultrasound, in contrast to continuous ultrasound, produces extremely high levels
Ref. 224 of audible sound inside the body. (Some sound intensities are listed in Table 41.)
Pulsed ultrasound is used in ultrasound imaging, and in some, but not all, foetal
heartbeat monitors. Such machines thus produce high levels of sound in the audible

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
range. This seems paradoxical; if you go to a gynaecologist and put the ultrasound head
on your ear or head, you will only hear a very faint noise. In fact, it is this low intensity
that tricks everybody to think that the noise level heard by babies is low. The noise level
is only low because the human ear is full of air. In contrast, in a foetus, the ear is filled
with liquid. This fact changes the propagation of sound completely: the sound generated
by imaging machines is now fully focused and directly stimulates the inner ear. The total
effect is similar to what happens if you put your finger in your ear: this can be very loud
to yourself, but nobody else can hear what happens.
Ref. 224 Recent research has shown that sound levels of over 100 dB, corresponding to a sub-
way train entering the station, are generated by ultrasound imaging systems. Indeed,
every gynaecologist will confirm that imaging disturbs the foetus. Questioned about this
issue, several makers of ultrasound imaging devices confirmed that “a sound output of
only 5 mW is used”. That is ‘only’ the acoustic power of an oboe at full power! Since
many ultrasound examinations take ten minutes and more, damage to the ear of the
Ref. 225 foetus cannot be excluded. It is not sensible to expose a baby to this level of noise without
good reason.
In short, ultrasound should be used for pregnant mothers only in case of necessity.
Ultrasound is not safe for the ears of foetuses. (Statements by medical ultrasound soci-
eties saying the contrary are wrong.) In all other situations, ultrasound imaging is safe.
It should be noted, however, that another potential problem of ultrasound imaging, the
Ref. 226 issue of tissue damage through cavitation, has not been explored in full detail yet.
314 10 simple motions of extended bodies

Signals
A signal is the transport of information. Every signal, including those from Table 40, is
the motion of energy. Signals can be either objects or waves. A thrown stone can be a
signal, as can a whistle. Waves are a more practical form of communication because they
do not require the transport of matter: it is easier to use electricity in a telephone wire
to transport a statement than to send a messenger. Indeed, most modern technological
advances can be traced to the separation between signal and matter transport. Instead
of transporting an orchestra to transmit music, we can send radio signals. Instead of
sending paper letters, we write email messages. Instead of going to the library, we browse
the internet.
The greatest advances in communication have resulted from the use of signals for the
transport of large amounts of energy. That is what electric cables do: they transport en-
ergy without transporting any (noticeable) matter. We do not need to attach our kitchen
machines to the power station: we can get the energy via a copper wire.
For all these reasons, the term ‘signal’ is often meant to imply waves only. Voice,

Motion Mountain – The Adventure of Physics
sound, electric signals, radio and light signals are the most common examples of wave
signals.
Signals are characterized by their speed and their information content. Both quantities
turn out to be limited. The limit on speed is the central topic of the theory of special
Vol. II, page 15 relativity.
A simple limit on the information content of wave signals can be expressed when
noting that the information flow is given by the detailed shape of the signal. The shape
is characterized by a frequency (or wavelength) and a position in time (or space). For
every signal – and every wave – there is a relation between the time-of-arrival error Δ𝑡
and the angular frequency error Δ𝜔:

This time–frequency indeterminacy relation expresses that, in a signal, it is impossible
to specify both the time of arrival and the frequency with full precision. The two errors
are (within a numerical factor) the inverse of each other. (We can also say that the time-
bandwidth product is always larger than 1/4π.) The limitation appears because on the
one hand we need a wave as similar as possible to a long sine wave in order to precisely
determine the frequency, but on the other hand we need a signal as short as possible to
precisely determine its time of arrival. The contrast in the two requirements leads to the
limit. The indeterminacy relation is thus a feature of every wave phenomenon. You might
Challenge 532 e want to test this relation with any wave in your environment.
Similarly, there is a relation between the position error Δ𝑥 and the wave vector error
Δ𝑘 = 2π/Δ𝜆 of a signal:
Δ𝑥 Δ𝑘 ⩾ 12 . (103)

Like the previous case, also this indeterminacy relation expresses that it is impossible
to specify both the position of a signal and its wavelength with full precision. Also this
position–wave-vector indeterminacy relation is a feature of any wave phenomenon.
Every indeterminacy relation is the consequence of a smallest entity. In the case of
oscillations and waves 315

Motion Mountain – The Adventure of Physics
F I G U R E 221 The electrical signals calculated (above) and measured (below) in a nerve, following
Hodgkin and Huxley.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
waves, the smallest entity of the phenomenon is the period (or cycle, as it used to be
called). Whenever there is a smallest unit in a natural phenomenon, an indeterminacy
relation results. We will encounter other indeterminacy relations both in relativity and
in quantum theory. As we will find out, they are due to smallest entities as well.
Whenever signals are sent, their content can be lost. Each of the six characteristics of
waves listed on page 301 can lead to content degradation. Can you provide an example
Challenge 533 s for each case? The energy, the momentum and all other conserved properties of signals
are never lost, of course. The disappearance of signals is akin to the disappearance of
motion. When motion disappears by friction, it only seems to disappear, and is in fact
transformed into heat. Similarly, when a signal disappears, it only seems to disappear, and
is in fact transformed into noise. (Physical) noise is a collection of numerous disordered
signals, in the same way that heat is a collection of numerous disordered movements.
All signal propagation is described by a wave equation. A famous example is the set
Ref. 227 of equations found by Hodgkin and Huxley. It is a realistic approximation for the be-
haviour of electrical potential in nerves. Using facts about the behaviour of potassium
and sodium ions, they found an elaborate wave equation that describes the voltage 𝑉
in nerves, and thus the way the signals are propagated along them. The equation de-
scribes the characteristic voltage spikes measured in nerves, shown in Figure 221. The
figure clearly shows that these waves differ from sine waves: they are not harmonic. An-
harmonicity is one result of non-linearity. But non-linearity can lead to even stronger
effects.
316 10 simple motions of extended bodies

F I G U R E 222 A solitary
water wave followed by
a motor boat,
reconstructing the
discovery by Scott
Russel (© Dugald
Duncan).

Motion Mountain – The Adventure of Physics
S olitary waves and solitons
In August 1834, the Scottish engineer John Scott Russell (b. 1808 Glasgow,
d. 1882 London) recorded a strange observation in a water canal in the countryside
near Edinburgh. When a boat pulled through the channel was suddenly stopped, a
strange water wave departed from it. It consisted of a single crest, about 10 m long and
0.5 m high, moving at about 4 m/s. He followed that crest, shown in a reconstruction

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in Figure 222, with his horse for several kilometres: the wave died out only very slowly.
Russell did not observe any dispersion, as is usual in deep water waves: the width of the
crest remained constant. Russell then started producing such waves in his laboratory
Ref. 228 and extensively studied their properties.
Russell showed that the speed depended on the amplitude, in contrast to linear, har-
monic waves. He also found that the depth 𝑑 of the water canal was an important para-
meter. In fact, the speed 𝑣, the amplitude 𝐴 and the width 𝐿 of these single-crested waves
are related by
𝐴 4𝑑3
𝑣 = √𝑔𝑑 (1 + ) and 𝐿 = √ . (104)
2𝑑 3𝐴

As shown by these expressions, and noted by Russell, high waves are narrow and fast,
whereas shallow waves are slow and wide. The shape of the waves is fixed during their
motion. Today, these and all other stable waves with a single crest are called solitary
waves. They appear only where the dispersion and the non-linearity of the system exactly
compensate for each other. Russell also noted that the solitary waves in water channels
can cross each other unchanged, even when travelling in opposite directions; solitary
waves with this property are called solitons. In short, solitons are stable against encoun-
ters, as shown in Figure 223, whereas solitary waves in general are not.
Page 307 Sixty years later, in 1895, Korteweg and de Vries found out that solitary waves in water
oscillations and waves 317

F I G U R E 223 Solitons are stable against encounters. (QuickTime ﬁlm © Jarmo Hietarinta)

Motion Mountain – The Adventure of Physics
channels have a shape described by

𝑥 − 𝑣𝑡 2
𝑢(𝑥, 𝑡) = 𝐴 sech2 where sech 𝑥 = , (105)
𝐿 e𝑥 + e−𝑥
and that the relation found by Russell was due to the wave equation

1 ∂𝑢 3 ∂𝑢 𝑑2 ∂3 𝑢

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
+ (1 + 𝑢) + =0. (106)
√𝑔𝑑 ∂𝑡 2𝑑 ∂𝑥 6 ∂𝑥3

This equation for the elongation 𝑢 is now called the Korteweg–de Vries equation in their
honour.* The surprising stability of the solitary solutions is due to the opposite effect of
the two terms that distinguish the equation from linear wave equations: for the solitary
solutions, the non-linear term precisely compensates for the dispersion induced by the
third-derivative term.
For many decades such solitary waves were seen as mathematical and physical curios-
ities. The reason was simple: nobody could solve the equations. All this changed almost
a hundred years later, when it became clear that the Korteweg–de Vries equation is a
universal model for weakly non-linear waves in the weak dispersion regime, and thus of
basic importance. This conclusion was triggered by Kruskal and Zabusky, who in 1965
proved mathematically that the solutions (105) are unchanged in collisions. This discov-
Ref. 231 ery prompted them to introduce the term soliton. These solutions do indeed interpen-
etrate one another without changing velocity or shape: a collision only produces a small
positional shift for each pulse.

* The equation can be simplified by transforming the variable 𝑢; most concisely, it can be rewritten as 𝑢𝑡 +
𝑢𝑥𝑥𝑥 = 6𝑢𝑢𝑥 . As long as the solutions are sech functions, this and other transformed versions of the equation
are known by the same name.
318 10 simple motions of extended bodies

Solitary waves play a role in many examples of fluid flows. They are found in ocean
currents; and even the red spot on Jupiter, which was a steady feature of Jupiter photo-
graphs for many centuries, is an example.
Solitary waves also appear when extremely high-intensity sound is generated in solids.
Ref. 232 In these cases, they can lead to sound pulses of only a few nanometres in length. Solitary
light pulses are also used inside certain optical communication fibres, where the lack
of dispersion allows higher data transmission rates than are achievable with usual light
Ref. 228 pulses.
Towards the end of the twentieth century, mathematicians discovered that solitons
obey a non-linear generalization of the superposition principle. (It is due to the
Darboux–Backlund transformations and the structure of the Sato Grassmannian.) The
mathematics of solitons is extremely interesting. The progress in mathematics triggered
a second wave of interest in the mathematics of solitons arose, when quantum theorists
became interested in them. The reason is simple: a soliton is a ‘middle thing’ between a
Ref. 229 particle and a wave; it has features of both concepts. For this reason, solitons were often
seen – incorrectly though – as candidates for the description of elementary particles.

Motion Mountain – The Adventure of Physics
Curiosities and fun challenges ab ou t waves and oscillation

“
Society is a wave. The wave moves onward, but

”
the water of which it is composed does not.
Ralph Waldo Emerson, Self-Reliance.

Sounds can be beautiful. If you enjoy the sound in the Pisa Baptistery, near the leaning
tower, or like to feel the sound production in singing sand, or about The Whispering
Gallery in St Paul‘s Cathedral, read the book by Trevor C ox, Sonic Wonderland. It
opens the door to a new world.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
When the frequency of a tone is doubled, one says that the tone is higher by an octave.
Two tones that differ by an octave, when played together, sound pleasant to the ear. Two
other agreeable frequency ratios – or ‘intervals’, as musicians say – are quarts and quints.
Challenge 534 e What are the corresponding frequency ratios? The answer to this question was one of
the oldest discoveries in physics and perception research; it is attributed to Pythagoras,
around 500 b ce.
∗∗
Many teachers, during their course, inhale helium in class. When they talk, they have a
very high voice. The high pitch is due to the high value of the speed of sound in helium.
A similar trick is also possible with SF6 ; it leads to a very deep voice. What is less well
known is that these experiments are dangerous! Helium is not a poison. But due to its
small atomic radius it diffuses rapidly. Various people got an embolism from performing
the experiment, and a few died. Never do it yourself; watch a youtube video instead.
∗∗
When a child on a swing is pushed by an adult, the build-up of amplitude is due to
(direct) resonance. When, on the other hand, the child itself sets the swing into motion,
oscillations and waves 319

mass

rod

loose joint

vertically
driven,
oscillating
base F I G U R E 224 A vertically driven inverse pendulum
is stable in the upright position at certain
combinations of frequency and amplitude.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 225 A particularly slow
wave: a ﬁeld of ski moguls
(© Andreas Hallerbach).

it uses twice the natural frequency of the swing; this effect is called parametric resonance.
An astonishing effect of parametric resonance appears when an upside-down pen-
dulum is attached to a vibrating base. Figure 224 shows the set-up; due to the joint, the
mass is free to fall down on either side. Such a vertically driven inverse pendulum, some-
times also called a Kapitza pendulum, will remain firmly upright if the driving frequency
of the joint is well chosen. For one of the many videos of the phenomenon, see www.
youtube.com/watch?v=is_ejYsvAjY. Parametric resonance appears in many settings, in-
cluding the sky. The Trojan asteroids are kept in orbit by parametric resonance.
∗∗
Also the bumps of skiing slopes, the so-called ski moguls, are waves: they move. Ski
Ref. 230 moguls are essential in many winter Olympic disciplines. Observation shows that ski
moguls have a wavelength of typically 5 to 6 m and that they move with an average
speed of 8 cm/day. Surprisingly, the speed is directed upwards, towards the top of the
320 10 simple motions of extended bodies

Challenge 535 s skiing slope. Can you explain why this is so? In fact, ski moguls are also an example of
Page 415 self-organization; this topic will be covered in more detail below.
∗∗
An orchestra is playing music in a large hall. At a distance of 30 m, somebody is listening
to the music. At a distance of 3000 km, another person is listening to the music via the
Challenge 536 s radio. Who hears the music first?
∗∗
What is the period of a simple pendulum, i.e., a mass 𝑚 attached to a massless string of
Challenge 537 s length 𝑙? What is the period if the string is much longer than the radius of the Earth?
∗∗
What path is followed by a body that moves without friction on a plane, but that is at-
Challenge 538 s tached by a spring to a fixed point on the plane?

Motion Mountain – The Adventure of Physics
∗∗
The blue whale, Balaenoptera musculus, is the loudest animal found in nature: its voice
can be heard at a distance of hundreds of kilometres.
∗∗
The exploration of sound in the sea, from the communication of whales to the sonar of
dolphins, is a world of its own. As a start, explore the excellent www.dosits.org website.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
An interesting device that shows how rotation and oscillation are linked is the alarm
Challenge 539 e siren. Find out how it works, and build one yourself.
∗∗
Jonathan Swift’s Lilliputians are one twelfth of the size of humans. Show that the fre-
quency of their voices must therefore be 144 times higher as that of humans, and thus be
inaudible. Gulliver could not have heard what Lilliputians were saying. The same, most
probably, would be true for Brobdingnagians, who were ten times taller than humans.
Challenge 540 s Their sentences would also be a hundred times slower.
∗∗
Light is a wave, as we will discover later on. As a result, light reaching the Earth from
space is refracted when it enters the atmosphere. Can you confirm that as a result, stars
Challenge 541 e appear somewhat higher in the night sky than they really are?
∗∗
What are the highest sea waves? This question has been researched systematically only
Ref. 233 recently, using satellites. The surprising result is that sea waves with a height of 25 m
and more are common: there are a few such waves on the oceans at any given time. This
result confirms the rare stories of experienced ship captains and explains many otherwise
unexplained ship sinkings.
oscillations and waves 321

air

air water

coin

F I G U R E 226 Shadows show
the refraction of light.

Surfers may thus get many chances to ride 30 m waves. (The present record is just be-
low this height.) But maybe the most impressive waves to surf are those of the Pororoca,
a series of 4 m waves that move from the sea into the Amazon River every spring, against
the flow of the river. These waves can be surfed for tens of kilometres.

Motion Mountain – The Adventure of Physics
∗∗
Interestingly, every water surface has waves, even if it seems completely flat. As a con-
sequence of the finite temperature of water, its surface always has some roughness: there
are thermal capillary waves. For water, with a surface tension of 72 mPa, the typical
roughness at usual conditions is 0.2 nm. These thermal capillary waves, predicted since
Ref. 234 many centuries, have been observed only recently.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
All waves are damped, eventually. This effect is often frequency-dependent. Can you
Challenge 542 s provide a confirmation of this dependence in the case of sound in air?
∗∗
When you make a hole with a needle in black paper, the hole can be used as a magnifying
Challenge 543 e lens. (Try it.) Diffraction is responsible for the lens effect. By the way, the diffraction of
light by holes was noted already by Francesco Grimaldi in the seventeenth century; he
correctly deduced that light is a wave. His observations were later discussed by Newton,
who wrongly dismissed them, because they did not fit his beliefs.
∗∗
Put an empty cup near a lamp, in such a way that the bottom of the cup remains in the
Challenge 544 e shadow. When you fill the cup with water, some of the bottom will be lit, because of the
refraction of the light from the lamp, as shown in Figure 226. The same effect allows us
to build lenses. Refraction is thus at the basis of many optical instruments, such as the
Vol. III, page 163 telescope or the microscope.
∗∗
Challenge 545 s Are water waves transverse or longitudinal?
∗∗
322 10 simple motions of extended bodies

The speed of water waves limits the speed of ships. A surface ship cannot travel (much)
faster than about 𝑣crit = √0.16𝑔𝑙 , where 𝑔 = 9.8 m/s2 , 𝑙 is the boat’s length, and 0.16 is
a number determined experimentally, called the critical Froude number. This relation is
valid for all vessels, from large tankers (𝑙 = 100 m gives 𝑣crit = 13 m/s) down to ducks (𝑙 =
0.3 m gives 𝑣crit = 0.7 m/s). The critical speed is that of a wave with the same wavelength
as the ship. In fact, moving a ship at higher speeds than the critical value is possible, but
requires much more energy. (A higher speed is also possible if the ship surfs on a wave.)
Challenge 546 s How far away is the crawl olympic swimming record from the critical value?
Most water animals and ships are faster when they swim below the surface – where
the limit due to surface waves does not exist – than when they swim on the surface. For
example, ducks can swim three times as fast under water than on the surface.
∗∗
The group velocity of water waves (in deep water) is less than the velocity of the indi-
vidual wave crests, the so-called phase velocity. As a result, when a group of wave crests
travels, within the group the crests move from the back to the front: they appear at the

Motion Mountain – The Adventure of Physics
back, travel forward and then die out at the front. The group velocity of water waves is
lower than its phase velocity.
∗∗
We can hear the distant sea or a distant highway more clearly in the evening than in
the morning. This is an effect of refraction. Sound speed increases with temperature.
In the evening, the ground cools more quickly than the air above. As a result, sound
leaving the ground and travelling upwards is refracted downwards, leading to the long
hearing distance typical of evenings. In the morning, usually the air is cold above and

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
warm below. Sound is refracted upwards, and distant sound does not reach a listener
on the ground. Refraction thus implies that mornings are quiet, and that we can hear
more distant sounds in the evenings. Elephants use the sound situation during evenings
to communicate over distances of more than 10 km. (They also use sound waves in the
ground to communicate, but that is another story.)
∗∗
Refraction also implies that there is a sound channel in the ocean, and one in the at-
mosphere. Sound speed increases with temperature, and increases with pressure. At an
ocean depth of 1 km, or at an atmospheric height of 13 to 17 km (that is at the top of
the tallest cumulonimbus clouds or equivalently, at the middle of the ozone layer) sound
has minimal speed. As a result, sound that starts from that level and tries to leave is
channelled back to it. Whales use this sound channel to communicate with each other
Challenge 547 e with beautiful songs; one can find recordings of these songs on the internet. The military
successfully uses microphones placed at the sound channel in the ocean to locate sub-
marines, and microphones on balloons in the atmospheric channel to listen for nuclear
Ref. 235 explosions.
In fact, sound experiments conducted by the military are the main reason why whales
are deafened and lose their orientation, stranding on the shores. Similar experiments in
the air with high-altitude balloons are often mistaken for flying saucers, as in the famous
Roswell incident.
oscillations and waves 323

F I G U R E 227 An
artiﬁcial rogue wave –
scaled down – created
in a water tank
(QuickTime ﬁlm
© Amin Chabchoub).

Motion Mountain – The Adventure of Physics
∗∗
Ref. 236 Also small animals communicate by sound waves. In 2003, it was found that herring
communicate using noises they produce when farting. When they pass wind, the gas
creates a ticking sound whose frequency spectrum reaches up to 20 kHz. One can even
listen to recordings of this sound on the internet. The details of the communication, such
as the differences between males and females, are still being investigated. It is possible
that the sounds may also be used by predators to detect herring, and they might even be
used by future fishing vessels.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Do plants produce sounds? Yes. Many plants, including pine trees and other trees, are
known to produce low-power ultrasound when transporting sap; other plants, like corn,
produce low-power audible sound in their roots.
Are plants sensitive to sound? Yes, they are. A large number of plants, including to-
mato plants, release pollen only when stimulated with a specific sound frequency emitted
by their pollinators; in other plants, including corn again, roots seem to grow in the dir-
ection of certain sound sources.
But do plants communicate using sound? There is a remote possibility that root sys-
tems influence each other in this way; research is still ongoing. So far, no evidence was
Challenge 548 ny found. Maybe you can make a discovery, and jump-start the field of phytoacoustics?
∗∗
On windy seas, the white wave crests have several important effects. The noise stems
from tiny exploding and imploding water bubbles. The noise of waves on the open sea
is thus the superposition of many small explosions. At the same time, white crests are
the events where the seas absorb carbon dioxide from the atmosphere, and thus reduce
global warming.
∗∗
So-called rogue waves – also called monster waves or freak waves– are single waves on
324 10 simple motions of extended bodies

the open sea with a height of over 30 m that suddenly appear among much lower waves,
have been a puzzling phenomenon for decades. For a long time it was not clear whether
they really occurred. Only scattered reports by captains and mysteriously sunken ships
pointed to their existence. Finally, from 1995 onwards, measurements started to confirm
their existence. One reason for the scepticism was that the mechanism of their forma-
tion remained unclear. But experiments from 2010 onwards widened the understanding
of non-linear water waves. These experiments first confirmed that under idealized condi-
tions, water waves also show so-called breather solutions, or non-linear focussing. Finally,
Ref. 237 in 2014, Chabchoub and Fink managed to show, with a clever experimental technique
based on time reversal, that non-linear focussing – including rogue waves – can appear
in irregular water waves of much smaller amplitude. As Amin Chabchoub explains, the
video proof looks like the one shown in Figure 227.
∗∗
Challenge 549 s Why are there many small holes in the ceilings of many office rooms?

Motion Mountain – The Adventure of Physics
∗∗
Which physical observable determines the wavelength of water waves emitted when a
Challenge 550 ny stone is thrown into a pond?
∗∗
Ref. 2 Yakov Perelman lists the following four problems in his delightful physics problem book.
1. A stone falling into a lake produces circular waves. What is the shape of waves pro-
Challenge 551 s duced by a stone falling into a river, where the water flows?
2. It is possible to build a lens for sound, in the same way as it is possible to build lenses

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 552 s for light. What would such a lens look like?
Challenge 553 s 3. What is the sound heard inside a shell?
4. Light takes about eight minutes to travel from the Sun to the Earth. What con-
Challenge 554 s sequence does this have for the timing of sunrise?
∗∗
Typically, sound of a talking person produces a pressure variation of 20 mPa on the ear.
Challenge 555 ny How is this value determined?
The ear is indeed a sensitive device. As mentioned, most cases of sea mammals, like
whales, swimming onto the shore are due to ear problems: usually some military device
(either sonar signals or explosions) has destroyed their ear so that they became deaf and
lose orientation.
∗∗
Why is the human ear, shown in Figure 228, so complex? The outer part, the pinna or
auricola, concentrates the sound pressure at the tympanic membrane; it produces a gain
of 3 dB. The tympanic membrane, or eardrum, is made in such a way as to always oscil-
late in fundamental mode, thus without any nodes. The tympanic membrane has a (very
wide) resonance at 3 kHz, in the region where the ear is most sensitive. The eardrum
transmits its motion, using the ossicles, into the inner ear. This mechanism thus trans-
forms air waves into water waves in the inner ear, where they are detected. The efficiency
oscillations and waves 325

Motion Mountain – The Adventure of Physics
F I G U R E 228 The human ear
(© Northwestern University).

with which this transformation takes place is almost ideal; using the language of wave
theory, ossicles are, above all, impedance transformers. Why does the ear transform air
waves to water waves? Because water allows a smaller detector than air. Can you explain

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 556 ny why?
∗∗
Infrasound, inaudible sound below 20 Hz, is a modern topic of research. In nature, infra-
sound is emitted by earthquakes, volcanic eruptions, wind, thunder, waterfalls, meteors
and the surf. Glacier motion, seaquakes, avalanches and geomagnetic storms also emit
Ref. 238 infrasound. Human sources include missile launches, traffic, fuel engines and air com-
pressors.
It is known that high intensities of infrasound lead to vomiting or disturbances of the
sense of equilibrium (140 dB or more for 2 minutes), and even to death (170 dB for 10
minutes). The effects of lower intensities on human health are not yet known.
Infrasound can travel several times around the world before dying down, as the ex-
plosion of the Krakatoa volcano showed in 1883. With modern infrasound detectors, sea
surf can be detected hundreds of kilometres away. Sea surf leads to a constant ‘hum’ of
the Earth’s crust at frequencies between 3 and 7 mHz. The Global Infrasound Network
uses infrasound to detect nuclear weapon tests, earthquakes and volcanic eruptions; it
can even count meteors. Only very rarely can meteors be heard with the human ear. (See
can-ndc.nrcan.gc.ca/is_infrasound-en.php.)
∗∗
The method used to deduce the sine waves contained in a signal, illustrated in Figure 205,
326 10 simple motions of extended bodies

is called the Fourier transformation. It is of importance throughout science and tech-
nology. In the 1980s, an interesting generalization became popular, called the wavelet
transformation. In contrast to Fourier transformations, wavelet transformations allow us
to localize signals in time. Wavelet transformations are used to compress digitally stored
images in an efficient way, to diagnose aeroplane turbine problems, and in many other
Ref. 239 applications.
∗∗
If you like engineering challenges, here is one that is still open. How can one make a
Challenge 557 r robust and efficient system that transforms the energy of sea waves into electricity?
∗∗
If you are interested in ocean waves, you might also enjoy the science of oceanography.
For an introduction, see the open-source textbooks at oceanworld.tamu.edu.
∗∗

Motion Mountain – The Adventure of Physics
What is the smallest structure in nature that has standing waves on its surface? The
largest?
∗∗
In our description of extended bodies, we assumed that each spot of a body can be fol-
lowed separately throughout its motion. Is this assumption justified? What would happen
Challenge 558 r if it were not?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
A special type of wave appears in explosions and supersonic flight: shock waves. In a
shock wave, the density or pressure of a gas changes abruptly, over distances of a few
micrometres. Studying shock waves is a research field in itself; shock waves determine
the flight of bullets, the snapping of whips and the effects of detonations.
Around a body moving with supersonic speed, the sound waves form a cone, as shown
in Figure 229. When the cone passes an observer on the ground, the cone leads to a sonic
boom. What is less well-known is that the boom can be amplified. If an aeroplane accel-
erates through the sound barrier, certain observers at the ground will hear two booms
or even a so-called superboom, because cones from various speeds can superpose at cer-
Ref. 240 tain spots on the ground. A plane that performs certain manoeuvres, such as a curve at
high speed, can even produce a superboom at a predefined spot on the ground. In con-
trast to normal sonic booms, superbooms can destroy windows, eardrums and lead to
trauma, especially in children. Unfortunately, they are regularly produced on purpose by
frustrated military pilots in various places of the world.
∗∗
What have swimming swans and ships have in common? The wake behind them. Despite
the similarity, this phenomenon has no relation to the sonic boom. In fact, the angle of
the wake is the same for ducks and ships: the angle is independent of the speed they travel
or of the size of the moving body, provided the water is deep enough.
Page 295 As explained above, water waves in deep water differ from sound waves: their group
oscillations and waves 327

moving shock wave :
moving ‘sonic boom’

sound source
sound
moving through
waves
medium at
supersonic speed

supersonic
sound source

Motion Mountain – The Adventure of Physics
boom
moving
on ground

F I G U R E 229 The shock wave created by a body in supersonic motion leads to a ‘sonic boom’ that
moves through the air; it can be made visible by Schlieren photography or by water condensation

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
(photo © Andrew Davidhazy, Gary Settles, NASA).

velocity is one half the phase velocity. (Can you deduce this from the dispersion relation
𝜔 = √𝑔𝑘 between angular frequency and wave vector, valid for deep water waves?)
Challenge 559 e Water waves will interfere where most of the energy is transported, thus around the
group velocity. For this reason, in the graph shown in Figure 230, the diameter of each
wave circle is always half the distance of their leftmost point O to the apex A. As a result,
the half angle of the wake apex obeys
1
sin 𝛼 = 3
giving a wake angle 2𝛼 = 38.942° . (107)

Figure 230 also allows deducing the curves that make up the wave pattern of the wake,
Ref. 241 using simple geometry.
It is essential to note that the fixed wake angle is valid only in deep water, i.e., only in
water that is much deeper than the wavelength of the involved waves. In other words,
for a given depth, the wake has the fixed shape only up to a maximum source speed. For
high speeds, the wake angle narrows, and the pattern inside the wake changes.
∗∗
Bats fly at night using echolocation. Dolphins also use it. Sonar, used by fishing vessels
328 10 simple motions of extended bodies

Motion Mountain – The Adventure of Physics
T

P
α
α
A C O A
(ship
or
swan)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 230 The wakes behind a ship and behind a swan, and the way to deduce the shape (photos
© Wikimedia, Christopher Thorn).

to look for fish, copies the system of dolphins. Less well known is that humans have the
Ref. 242 same ability. Have you ever tried to echolocate a wall in a completely dark room? You will
be surprised at how easily this is possible. Just make a loud hissing or whistling noise that
Challenge 560 e stops abruptly, and listen to the echo. You will be able to locate walls reliably.
∗∗
Birds sing. If you want to explore how this happens, look at the impressive X-ray film of
a singing bird found at the www.indiana.edu/~songbird/multi/cineradiography_index.
html website.
∗∗
Every soliton is a one-dimensional structure. Do two-dimensional analogues exist? This
issue was open for many years. Finally, in 1988, Boiti, Leon, Martina and Pempinelli
Ref. 243 found that a certain evolution equation, the so-called Davey–Stewartson equation, can
have solutions that are localized in two dimensions. These results were generalized by
oscillations and waves 329

F I G U R E 231 The calculated motion of a dromion across a two-dimensional substrate. (QuickTime ﬁlm

Motion Mountain – The Adventure of Physics
© Jarmo Hietarinta)

Fokas and Santini and further generalized by Hietarinta and Hirota. Such a solution is
today called a dromion. Dromions are bumps that are localized in two dimensions and
can move, without disappearing through diffusion, in non-linear systems. An example is

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
shown in Figure 231. However, so far, no such solution has be observed in experiments;
this is one of the most important experimental challenges left open in non-linear science.
∗∗
Water waves have not lost their interest up to this day. Most of all, two-dimensional solu-
tions of solitonic water waves remain a topic of research. The experiments are simple, the
mathematics is complicated and the issues are fascinating. In two dimensions, crests can
even form hexagonal patterns! The relevant equation for shallow waves, the generaliz-
ation of the Korteweg–de Vries equation to two dimensions, is called the Kadomtsev–
Petviashvili equation. It leads to many unusual water waves, including cnoidal waves,
solitons and dromions, some of which are shown in Figure 232. The issue of whether
rectangular patterns exist is still open, and the exact equations and solutions for deep
water waves are also unknown.
For moving water, waves are even more complex and show obvious phenomena, such
as the Doppler effect, and less obvious ones, such as bores, i.e., the wave formed by the
incoming tide in a river, and the subsequent whelps. Even a phenomenon as common as
Ref. 244 the water wave is still a field of research.
∗∗
How does the tone produced by blowing over a bottle depend on the dimension? For
Ref. 245 bottles that are bulky, the frequency 𝑓, the so-called cavity resonance, is found to depend
330 10 simple motions of extended bodies

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 232 Unusual water waves in shallow water: (top) in an experimental water tank and in a storm
in North Carolina, (bottom) an almost pure cnoidal wave near Panama and two such waves crossing at
the Ile de Ré (photo © Diane Henderson, Anonymous, Wikimedia).

Challenge 561 ny on the volume 𝑉 of the bottle:

𝑐 𝐴 1
𝑓= √ or 𝑓 ∼ (108)
2π 𝑉𝐿 √𝑉

where 𝑐 is the speed of sound, 𝐴 is the area of the opening, and 𝐿 is the length of the
Challenge 562 e neck of the bottle. Does the formula agree with your observations?
In fact, tone production is a complicated process, and specialized books exist on the
Ref. 246 topic. For example, when overblowing, a saxophone produces a second harmonic, an
octave, whereas a clarinet produces a third harmonic, a quint (more precisely, a twelfth).
oscillations and waves 331

Why is this the case? The theory is complex, but the result simple: instruments whose
cross-section increases along the tube, such as horns, trumpets, oboes or saxophones,
overblow to octaves. For air instruments that have a (mostly) cylindrical tube, the ef-
fect of overblowing depends on the tone generation mechanism. Flutes overblow to the
octave, but clarinets to the twelfth.
∗∗
Many acoustical systems do not only produce harmonics, but also subharmonics. There
is a simple way to observe production of subharmonics: sing with your ears below water,
in the bathtub. Depending on the air left in your ears, you can hear subharmonics of your
own voice. The effect is quite special.
∗∗
The origin of the sound of cracking joints, for example the knuckles of the fingers, is a
well-known puzzle. How would you test the conjecture that it is due to cavitation? What
Challenge 563 e would you do to find out definitively?

Motion Mountain – The Adventure of Physics
∗∗
When a kilometre-deep hole is drilled into the Earth, and the sound at the bottom of the
pit is studied, one finds that the deep Earth is filled with low-level, low-frequency sound.
The oscillation periods can be longer than 300 s, i.e., with frequencies as low as 3 mHz.
The majority of the sounds are due to the waves of the ocean. The sounds are modified in
frequency by non-linear effects and can be detected even at the bottom of the sea and in
the middle of continents. These sounds are now regularly used to search for oil and gas.
The origins of specific sound bands are still subject of research; some might be related to

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the atmosphere or even to the motion of magma in the mantle and in volcanoes.
∗∗
All babies cry. Research showed that their sounds are composed of four basic patterns.
It has been shown that the composition of the four patterns depends on the mother’s
language, even for newborn babies.
∗∗
Among the most impressive sound experiences are the singing performances of coun-
tertenors and of the even higher singing male sopranos. If you ever have the chance to
hear one, do not miss the occasion.

Summary on waves and oscillations
In nature, apart from the motion of bodies, we observe also the motion of waves. Wave
groups are often used as signals. Waves have energy, momentum and angular mo-
mentum. Waves exist in solids, liquids and gases, as well as along material interfaces.
Waves can interfere, diffract, refract, disperse, dampen out and, if transverse, can be po-
larized. Solitary waves, i.e., waves with only one crest, are a special case of waves in media
with specific non-linear dispersion.
Oscillations are a special case of waves; usually they are standing waves. Oscillation
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
10 simple motions of extended bodies

and waves only appear in extended systems.
332
C h a p t e r 11

D O E X T E N DE D B ODI E S E X I ST ?
– L I M I T S OF C ON T I N U I T Y

W
e have just discussed the motion of bodies that are extended.
e have found that all extended bodies, be they solid of fluid, show
ave motion. But are extended bodies actually found in nature? Strangely enough,
this question has been one of the most intensely discussed questions in physics. Over
the centuries, it has reappeared again and again, at each improvement of the description

Motion Mountain – The Adventure of Physics
of motion; the answer has alternated between the affirmative and the negative. Many
thinkers have been imprisoned, and many still are being persecuted, for giving answers
that are not politically correct! In fact, the issue already arises in everyday life.

Mountains and fractals
Whenever we climb a mountain, we follow the outline of its shape. We usually describe
this outline as a curved two-dimensional surface. But is this correct? There are alternat-
ive possibilities. A popular one is the idea that mountains might be fractal surfaces. A
fractal was defined by Benoît Mandelbrot as a set that is self-similar under a countable

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
but infinite number of magnification values. We have already encountered fractal lines.
Page 55 An example of an algorithm for building a (random) fractal surface is shown on the right
Ref. 247 side of Figure 233. It produces shapes which look remarkably similar to real mountains.
The results are so realistic that they are used in Hollywood films. If this description were
correct, mountains would be extended, but not continuous.
But mountains could also be fractals of a different sort, as shown in the left side of
Figure 233. Mountain surfaces could have an infinity of small and smaller holes. In fact,
we could also imagine that mountains are described as three-dimensional versions of the
left side of the figure. Mountains would then be some sort of mathematical Swiss cheese.
Can you devise an experiment to decide whether fractals provide the correct description
Challenge 564 s for mountains? To settle the issue, we study chocolate bars, bones, trees and soap bubbles.

C an a cho colate bar last forever?
Any child knows how to make a chocolate bar last forever: eat half the remainder every
day. However, this method only works if matter is scale-invariant. In other words, the
method only works if matter is either fractal, as it then would be scale-invariant for a
discrete set of zoom factors, or continuous, in which case it would be scale-invariant for
any zoom factor. Which case, if either, applies to nature?
Page 52 We have already encountered a fact making continuity a questionable assumption:
continuity would allow us, as Banach and Tarski showed, to multiply food and any other
334 11 extended bodies

n=1
i=4

n=2

n=5

n=∞

Motion Mountain – The Adventure of Physics
F I G U R E 233 Floors (left) and mountains (right) could be fractals; for mountains this approximation is
often used in computer graphics (image © Paul Martz).

matter by clever cutting and reassembling. Continuity would allow children to eat the
same amount of chocolate every day, without ever buying a new bar.

⊳ Matter is thus not continuous.

Now, fractal chocolate is not ruled out in this way; but other experiments settle the ques-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
tion. Indeed, we note that melted materials do not take up much smaller volumes than
solid ones. We also find that even under the highest pressures, materials do not shrink.
Thus we conclude again

⊳ Matter is thus not a fractal.

What then is its structure?
To get an idea of the structure of matter we can take fluid chocolate, or even just some
oil – which is the main ingredient of chocolate anyway – and spread it out over a large
surface. For example, we can spread a drop of oil onto a pond on a day without rain or
wind; it is not difficult to observe which parts of the water are covered by the oil and
which are not: they change in reflection properties and the sky reflected in the water
has a different colour on the water and on the oil. A small droplet of oil cannot cover a
Challenge 565 s surface larger than – can you guess the value? The oil-covered water and the uncovered
water have different colours. Trying to spread the oil film further inevitably rips it apart.
The child’s method of prolonging chocolate thus does not work forever: it comes to a
sudden end. The oil experiment, which can even be conducted at home, shows that there
is a minimum thickness of oil films, with a value of about 2 nm. The experiment shows*

* The oil experiment was popularized by Thomson-Kelvin, a few decades after Loschmidt’s determination
limits of matter continuity 335

Motion Mountain – The Adventure of Physics
F I G U R E 234 The spreading of a
droplet of half a microlitre of oil,
delivered with a micropipette, on a

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
square metre of water covered with
thin lycopodium powder
(© Wolfgang Rueckner).

that there is a smallest size in oil. Oil is made of tiny components. Is this valid for all
matter?

The case of Galileo Galilei
After the middle ages, Galileo (b. 1564 Pisa, d. 1642 Arcetri) was the first to state that
all matter was made of smallest parts, which he called piccolissimi quanti, i.e., smallest
quanta. Today, they are called atoms. However, Galileo paid dearly for this statement.
Indeed, during his life, Galileo was under attack for two reasons: because of his ideas
on the motion of the Earth, and because of his ideas about atoms.* The discovery of the

of the size of molecules. It is often claimed that Benjamin Franklin was the first to conduct the oil experi-
ment; that is wrong. Franklin did not measure the thickness, and did not even consider the question of the
thickness. He did pour oil on water, but missed the most important conclusion that could be drawn from
it. Even geniuses do not discover everything.
* To get a clear view of the matters of dispute in the case of Galileo, especially those of interest to physicists,
the best text is the excellent book by P ietro Redondi, Galileo eretico, Einaudi, 1983, translated into
336 11 extended bodies

importance of both issues is the merit of the great historian Pietro Redondi, a collab-
orator of another great historian, Pierre Costabel. One of Redondi’s research topics is
the history of the dispute between the Jesuits, who at the time defended orthodox theo-
logy, and Galileo and the other scientists. In the 1980s, Redondi discovered a document
of that time, an anonymous denunciation called G3, that allowed him to show that the
condemnation of Galileo to life imprisonment for his views on the Earth’s motion was
organized by his friend the Pope to protect him from a sure condemnation to death over
a different issue: atoms.
Page 337 Galileo defended the view, explained in detail shortly, that since matter is not scale
invariant, it must be made of ‘atoms’ or, as he called them, piccolissimi quanti. This was
and still is a heresy, because atoms of matter contradict the central Catholic idea that in
the Eucharist the sensible qualities of bread and wine exist independently of their sub-
stance. The distinction between substance and sensible qualities, introduced by Thomas
Aquinas, is essential to make sense of transubstantiation, the change of bread and wine
Ref. 248 into human flesh and blood, which is a central tenet of the Catholic faith. Indeed, a true
Catholic is still not allowed to believe in atoms to the present day, because the idea that

Motion Mountain – The Adventure of Physics
matter is made of atoms contradicts transubstantiation. In contrast, protestant faith usu-
ally does not support transubstantiation and thus has no problems with atoms.
In Galileo’s days, church tribunals punished heresy, i.e., personal opinions deviating
from orthodox theology, by the death sentence. But Galileo was not sentenced to death.
Galileo’s life was saved by the Pope by making sure that the issue of transubstantiation
would not be a topic of the trial, and by ensuring that the trial at the Inquisition be
organized by a papal commission led by his nephew, Francesco Barberini. But the Pope
also wanted Galileo to be punished, because he felt that his own ideas had been mocked
in Galileo’s book Il Dialogo and also because, under attack for his foreign policy, he was

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
not able to ignore or suppress the issue.
As a result, in 1633 the seventy-year-old Galileo was condemned to a prison sentence,
‘after invoking the name of Jesus Christ’, for ‘suspicion of heresy’ (and thus not for
heresy), because he did not comply with an earlier promise not to teach that the Earth
Ref. 249 moves. Indeed, the motion of the Earth contradicts what the Christian bible states. Ga-
lileo was convinced that truth was determined by observation, the Inquisition that it was
determined by a book – and by itself. In many letters that Galileo wrote throughout his
life, he expressed his conviction that observational truth could never be a heresy. The
trial showed him the opposite: he was forced to state that he erred in teaching that the
Ref. 250 Earth moves. After a while, the Pope reduced the prison sentence to house arrest.
Galileo’s condemnation on the motion of the Earth was not the end of the story. In
the years after Galileo’s death, also atomism was condemned in several trials against
Galileo’s ideas and his followers. But the effects of these trials were not those planned
by the Inquisition. Only twenty years after the famous trial, around 1650, every astro-
nomer in the world was convinced of the motion of the Earth. And the result of the
trials against atomism was that at the end of the 17th century, practically every scientist
in the world was convinced that atoms exist. The trials accelerated an additional effect:
after Galileo and Descartes, the centre of scientific research and innovation shifted from

English as Galileo Heretic, Princeton University Press, 1987. It is also available in many other languages; an
updated edition that includes the newest discoveries appeared in 2004.
limits of matter continuity 337

Catholic countries, like Italy or France, to protestant countries. In these, such as the Neth-
erlands, England, Germany or the Scandinavian countries, the Inquisition had no power.
This shift is still felt today.
It is a sad story that in 1992, the Catholic church did not revoke Galileo’s condemna-
tion. In that year, Pope John Paul II gave a speech on the Galileo case. Many years before,
he had asked a study commission to re-evaluate the trial, because he wanted to express
his regrets for what had happened and wanted to rehabilitate Galileo. The commission
Ref. 252 worked for twelve years. But the bishop that presented the final report was a crook: he
avoided citing the results of the study commission, falsely stated the position of both
parties on the subject of truth, falsely stated that Galileo’s arguments on the motion of
the Earth were weaker than those of the church, falsely summarized the past positions
of the church on the motion of the Earth, avoided stating that prison sentences are not
good arguments in issues of opinion or of heresy, made sure that rehabilitation was not
even discussed, and of course, avoided any mention of transubstantiation. At the end
of this power struggle, Galileo was thus not rehabilitated, in contrast to what the Pope
wanted and in contrast to what most press releases of the time said; the Pope only stated

Motion Mountain – The Adventure of Physics
that ‘errors were made on both sides’, and the crook behind all this was rewarded with
a promotion.*
But that is not the end of the story. The documents of the trial, which were kept locked
when Redondi made his discovery, were later made accessible to scholars by Pope John
Paul II. In 1999, this led to the discovery of a new document, called EE 291, an internal
expert opinion on the atom issue that was written for the trial in 1632, a few months
Ref. 251 before the start of the procedure. The author of the document comes to the conclusion
that Galileo was indeed a heretic in the matter of atoms. The document thus proves that
the cover-up of the transubstantiation issue during the trial of Galileo must have been

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
systematic and thorough, as Redondi had deduced. Indeed, church officials and the Cath-
olic catechism carefully avoid the subject of atoms even today; you can search the Vatican
website www.vatican.va for any mention of them.
But Galileo did not want to attack transubstantiation; he wanted to advance the idea
of atoms. And he did. Despite being condemned to prison in his trial, Galileo’s last book,
the Discorsi, written as a blind old man under house arrest, includes the arguments that
imply the existence of atoms, or piccolissimi quanti. It is an irony of history that today,
quantum theory, named by Max Born after the term used by Galileo for atoms, has be-
Vol. IV, page 15 come the most precise description of nature yet.
Galileo devoted his life to finding and telling the truth about motion and atoms. Let
us explore how he concluded that all matter is made of atoms.

How high can animals jump?
Fleas can jump to heights a hundred times their size, humans only to heights about their
Ref. 253 own size. In fact, biological studies yield a simple observation: most animals, regardless
of their size, achieve about the same jumping height, namely between 0.8 and 2.2 m,
whether they are humans, cats, grasshoppers, apes, horses or leopards. We have explained
Page 80 this fact earlier on.
* We should not be too indignant: the same situation happens in many commercial companies every day;
most industrial employees can tell similar stories.
338 11 extended bodies

At first sight, the observation of constant jumping height seems to be a simple example
of scale invariance. But let us look more closely. There are some interesting exceptions
at both ends of the mass range. At the small end, mites and other small insects do not
achieve such heights because, like all small objects, they encounter the problem of air
resistance. At the large end, elephants do not jump that high, because doing so would
break their bones. But why do bones break at all?
Why are all humans of about the same size? Why are there no giant adults with a
height of ten metres? Why aren’t there any land animals larger than elephants? Galileo
already gave the answer. The bones of which people and animals are made would not
allow such changes of scale, as the bones of giants would collapse under the weight they
have to sustain. A human scaled up by a factor 10 would weigh 1000 times as much, but
its bones would only be 100 times as wide. But why do bones have a finite strength at all?
There is only one explanation: because the constituents of bones stick to each other with
a finite attraction. In contrast to bones, continuous matter – which exists only in cartoons
– could not break at all, and fractal matter would be infinitely fragile. Galileo concluded:

Motion Mountain – The Adventure of Physics
⊳ Matter breaks under finite loads because it is composed of small basic con-
stituents.

Felling trees
Trees are fascinating structures. Take their size. Why do trees have limited size? Already
in the sixteenth century, Galileo knew that it is not possible to increase tree height
without limits: at some point a tree would not have the strength to support its own
weight. He estimated the maximum height to be around 90 m; the actual record, un-
known to him at the time, seems to be 150 m, for the Australian tree Eucalyptus regnans.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
But why does a limit exist at all? The answer is the same as for bones: wood has a finite
strength because it is not scale invariant; and it is not scale invariant because it is made
of small constituents, namely atoms.*
In fact, the derivation of the precise value of the height limit is more involved. Trees
must not break under strong winds. Wind resistance limits the height-to-thickness ratio
Challenge 566 s ℎ/𝑑 to about 50 for normal-sized trees (for 0.2 m < 𝑑 < 2 m). Can you say why? Thinner
trees are limited in height to less than 10 m by the requirement that they return to the
Ref. 255 vertical after being bent by the wind.
Such studies of natural constraints also answer the question of why trees are made
from wood and not, for example, from steel. You could check for yourself that the max-
imum height of a column of a given mass is determined by the ratio 𝐸/𝜌2 between the
Challenge 567 s elastic module and the square of the mass density. For a long time, wood was actually the
Ref. 256 material for which this ratio was highest. Only recently have material scientists managed
to engineer slightly better ratios: the fibre composites.
Why do materials break at all? All observations yield the same answer and confirm
Galileo’s reasoning: because there is a smallest size in materials. For example, bodies
under stress are torn apart at the position at which their strength is minimal. If a body
were completely homogeneous or continuous, it could not be torn apart; a crack could
Ref. 254 * There is another important limiting factor: the water columns inside trees must not break. Both factors
seem to yield similar limiting heights.
limits of matter continuity 339

not start anywhere. If a body had a fractal, Swiss-cheese structure, cracks would have
places to start, but they would need only an infinitesimal shock to do so.

Lit tle hard balls

“
I prefer knowing the cause of a single thing to

”
being king of Persia.
Democritus

Precise observations show that matter is neither continuous nor a fractal: all matter is
made of smallest basic particles. Galileo, who deduced their existence by thinking about
giants and trees, called them smallest quanta. Today they are called atoms, in honour of
a famous argument of the ancient Greeks. Indeed, 2500 years ago, the Greeks asked the
following question. If motion and matter are conserved, how can change and transform-
ation exist? The philosophical school of Leucippus and Democritus of Abdera* studied
two particular observations in special detail. They noted that salt dissolves in water. They
also noted that fish can swim in water. In the first case, the volume of water does not in-

Motion Mountain – The Adventure of Physics
crease when the salt is dissolved. In the second case, when fish advance, they must push
water aside. Leucippus and Democritus deduced that there is only one possible explan-
ation that satisfies these two observations and also reconciles conservation with trans-
formation: nature is made of void and of small, indivisible and conserved particles.**
In short, since matter is hard, has a shape and is divisible, Leucippus and Democritus
imagined it as being made of atoms. Atoms are particles which are hard, have a shape,
but are indivisible. The Greek thus deduced that every example of motion, change and
transformation is due to rearrangements of these particles; change and conservation are
thus reconciled.
In other words, the Greeks imagined nature as a big Lego set. Lego pieces are first of

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
all hard or impenetrable, i.e., repulsive at very small distances. Atoms thus explain why
solids cannot be compressed much. Lego pieces are also attractive at small distances: they
remain stuck together. Atoms thus explain that solids exist. Finally, lego bricks have no
interaction at large distances. Atoms thus explain the existence of gases. (Actually, what

* Leucippus of Elea (Λευκιππος) (c. 490 to c. 430 bce), Greek philosopher; Elea was a small town south
of Naples. It lies in Italy, but used to belong to the Magna Graecia. Democritus (Δεμοκριτος) of Abdera
(c. 460 to c. 356 or 370 bce), also a Greek philosopher, was arguably the greatest philosopher who ever lived.
Together with his teacher Leucippus, he was the founder of the atomic theory; Democritus was a much-
admired thinker and a contemporary of Socrates. The vain Plato never even mentions him, as Democritus
was a danger to his own fame. Democritus wrote many books which all have been lost; they were not copied
during the Middle Ages because of his scientific and rational worldview, which was felt to be a danger by
religious zealots who had the monopoly on the copying industry. Nowadays, it has become common to
claim – incorrectly – that Democritus had no proof for the existence of atoms. That is a typical example of
disinformation with the aim of making us feel superior to the ancients.
** The story is told by Lucretius, in full Titus Lucretius Carus, in his famous text De rerum natura, around
60 bce. (An English translation can be found on www.perseus.tufts.edu/hopper/text?doc=Lucr.+1.1.) Lu-
cretius relates many other proofs; in Book 1, he shows that there is vacuum in solids – as proven by porosity
and by density differences – and in gases – as proven by wind. He shows that smells are due to particles, and
Challenge 568 ny that so is evaporation. (Can you find more proofs?) He also explains that the particles cannot be seen due to
their small size, but that their effects can be felt and that they allow explaining all observations consistently.
Especially if we imagine particles as little balls, we cannot avoid calling this a typically male idea. (What
Challenge 569 d would be the female approach?)
340 11 extended bodies

the Greeks called ‘atoms’ partly corresponds to what today we call ‘molecules’. The latter
term was introduced in 1811 by Amedeo Avogadro* in order to clarify the distinction. But
we can forget this detail for the moment.)
Since atoms are invisible, it took many years before all scientists were convinced by
the experiments showing their existence. In the nineteenth century, the idea of atoms
was beautifully verified by a large number of experiments, such as the discovery of the
Page 389 ‘laws’ of chemistry and those of gas behaviour. We briefly explore the most interesting
ones.

The sound of silence
Climbing the slopes of any mountain, we arrive in a region of the forest covered with
deep snow. We stop for a minute and look around. It is already dark; all the animals
are asleep; there is no wind and there are no sources of sound. We stand still, without
breathing, and listen to the silence. (You can have the same experience also in a sound
studio such as those used for musical recordings, or in a quiet bedroom at night.) In

Motion Mountain – The Adventure of Physics
situations of complete silence, the ear automatically becomes more sensitive**; we then
have a strange experience. We hear two noises, a lower- and a higher-pitched one, which
are obviously generated inside the ear. Experiments show that the higher note is due to
the activity of the nerve cells in the inner ear. The lower note is due to pulsating blood
streaming through the head. But why do we hear a noise at all?
Many similar experiments confirm that whatever we do, we can never eliminate noise,
i.e., random fluctuations, from measurements. This unavoidable type of random fluctu-
ations is called shot noise in physics. The statistical properties of this type of noise actu-
ally correspond precisely to what would be expected if flows, instead of being motions
of continuous matter, were transportation of a large number of equal, small and discrete

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
entities. Therefore, the precise measurement of noise can be used to prove that air and
liquids are made of molecules, that electric current is made of electrons, and even that
light is made of photons. Shot noise is the sound of silence.

⊳ Shot noise is the sound of atoms.

If matter were continuous or fractal, shot noise would not exist.

How to count what cannot be seen
In everyday life, atoms cannot be counted, mainly because they cannot be seen. Inter-
Ref. 257 estingly, the progress of physics allowed scholars to count atoms nevertheless. As men-
tioned, many of these methods use the measurement of noise.***

* Amedeo Avogadro (b. 1776 Turin, d. 1856 Turin) was an important physicist and chemist. Avogadro’s
number is named for him.
** The human ear can detect, in its most sensitive mode, pressure variations at least as small as 20 μPa and
ear drum motions as small as 11 pm.
*** There are also various methods to count atoms by using electrolysis and determining the electron charge,
by using radioactivity, X-ray scattering or by determining Planck’s constant ℏ. We leave them aside here
because these methods actually count atoms. They are more precise, but also less interesting.
limits of matter continuity 341

In physics, the term noise is not only used for the acoustical effect; it is used for any
process that is random. The most famous kind of noise is Brownian motion, the motion
Page 391 of small particles, such as dust or pollen, floating in liquids. Also small particles fall-
ing in air, such as mercury globules or selenium particles, show fluctuations that can be
observed, for example with the help of air flows.
A mirror glued on a quartz fibre hanging in the air, especially at low pressure, changes
orientation randomly, by small amounts, due to the collision of the air molecules. The
random orientation changes, again a kind of noise, can be followed by reflecting a beam
of light on the mirror and watching the light spot at a large distance.
Also density fluctuations, critical opalescence, and critical miscibility of liquids are
forms of noise. It turns out that every kind of noise can be used to count atoms. The
reason is that all noise in nature is related to the particle nature of matter or radiation.
Indeed, all the mentioned methods have been used to count atoms and molecules, and
to determine their sizes. Since the colour of the sky is a noise effect, we can indeed count
air molecules by looking at the sky!
Ref. 258 When Rayleigh explored scattering and the request of Maxwell, he deduced a rela-

Motion Mountain – The Adventure of Physics
tion between the scattering coefficient 𝛽, the index of refraction of air 𝑛 and the particle
density 𝑁/𝑉:
(𝑛 − 1)2
𝛽 = 32π3 4 (109)
3𝜆 𝑁/𝑉

The scattering coefficient is the inverse of the distance at which light is reduced in in-
tensity by a factor of 1/e. Now, this distance can be estimated by determining the longest
distance at which one can see a mountain in clear weather. Observation shows that this
distance is at least 160 km. If we insert this observation in the equation, we can deduce

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 570 s a value for the number of molecules in air, and thus of Avogadro’s constant!
The result of all precise measurements is that a mol of matter – for any gas, that is the
amount of matter contained in 22.4 l of that gas at standard pressure – always contains
the same number of atoms.

⊳ One mol contains 6.0 ⋅ 1023 particles.

The number is called Avogadro’s number, after the first man who understood that
volumes of gases have equal number of molecules, or Loschmidt’s number, after the first
man who measured it.* All the methods to determine Avogadro’s number also allow us
to deduce that most atoms have a size in the range between 0.1 and 0.3 nm. Molecules
are composed of several or many atoms and are correspondingly larger.
How did Joseph Loschmidt** manage to be the first to determine his and Avogadro’s
number, and be the first to determine reliably the size of the components of matter?
Loschmidt knew that the dynamic viscosity 𝜇 of a gas was given by 𝜇 = 𝜌𝑙𝑣/3, where
𝜌 is the density of the gas, 𝑣 the average speed of the components and 𝑙 their mean free
path. With Avogadro’s prediction (made in 1811 without specifying any value) that a

* The term ‘Loschmidt’s number’ is sometimes also used to designate the number of molecules in one cubic
centimetre of gas.
** Joseph Loschmidt (b. 1821 Putschirn, d. 1895 Vienna) chemist and physicist.
342 11 extended bodies

Motion Mountain – The Adventure of Physics
F I G U R E 235 Soap bubbles show visible effects of molecular size: before bursting, soap bubbles show
small transparent spots; they appear black in this picture due to a black background. These spots are
regions where the bubble has a thickness of only two molecules, with no liquid in between (© LordV).

volume 𝑉 of any gas always contains the same number 𝑁 of components, one also has

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
𝑙 = 𝑉/√2π𝑁𝜎2 , where 𝜎 is the cross-section of the components. (The cross-section is
roughly the area of the shadow of an object.) Loschmidt then assumed that when the gas
is liquefied, the volume of the liquid is the sum of the volumes of the particles. He then
measured all the involved quantities, for mercury, and determined 𝑁. He thus determ-
ined the number of particles in one mole of matter, in one cubic centimetre of matter,
and also the size of these particles.

Experiencing atoms
Matter is not continuous nor fractal. Matter contains smallest components with a char-
acteristic size. Can we see effects of single atoms or molecules in everyday life? Yes, we
can. We just need to watch soap bubbles. Soap bubbles have colours. But just before they
burst, on the upper side of the bubble, the colours are interrupted by small transparent
spots, as shown in Figure 235. Why? Inside a bubble, the liquid flows downwards, so that
over time, the bubble gets thicker at the bottom and thinner at the top. After a while, in
some regions all the liquid is gone, and in these regions, the bubble consists only of two
molecular layers of soap molecules.
In fact, the arrangement of soap or oil molecules on water surfaces can be used to
measure Avogadro’s number, as mentioned. This has been done in various ingenious
Ref. 257 ways, and yields an extremely precise value with very simple means.
A simple experiment showing that solids have smallest components is shown in Fig-
limits of matter continuity 343

three
mono-
atomic
steps

lamp eye

F I G U R E 236 Atoms exist: rotating an F I G U R E 237 Atomic steps in broken gallium arsenide
illuminated, perfectly round single crystal crystals (wafers) can be seen under a light microscope.
aluminium rod leads to brightness
oscillations because of the atoms that make
it up

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 238 An effect of atoms: steps on single crystal surfaces – here silicon carbide grown on a
carbon-terminated substrate (left) and on a silicon terminated substrate (right) observed in a simple
light microscope (© Dietmar Siche).

ure 236. A cylindrical rod of pure, single crystal aluminium shows a surprising behaviour
when it is illuminated from the side: its brightness depends on how the rod is oriented,
even though it is completely round. This angular dependence is due to the atomic ar-
rangement of the aluminium atoms in the rod.
It is not difficult to confirm experimentally the existence of a smallest size in crystals.
It is sufficient to break a single crystal, such as a gallium arsenide wafer, in two. The
breaking surface is either completely flat or shows extremely small steps, as shown in
Challenge 571 e Figure 237. These steps are visible under a normal light microscope. (Why?) Similarly,
Figure 238 shows a defect that appeared in crystal growth. It turns out that all such step
heights are multiples of a smallest height: its value is about 0.2 nm. The existence of a
smallest height, corresponding to the height of an atom, contradicts all possibilities of
scale invariance in matter.
344 11 extended bodies

F I G U R E 239 A single barium ion
levitated in a Paul trap (image size
around 2 mm) at the centre of the
picture, visible also to the naked eye in
the original experiment, performed in
1985 (© Werner Neuhauser).

Motion Mountain – The Adventure of Physics
F I G U R E 240 The atoms on the surface F I G U R E 241 The result of moving helium atoms on

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
of a silicon crystal, mapped with an a metallic surface. Both the moving and the
atomic force microscope (© Universität imaging was performed with an atomic force
Augsburg) microscope (© IBM).

Seeing atoms
Nowadays, with advances in technology, single atoms can be seen, photographed, holo-
Ref. 259, Ref. 260 grammed, counted, touched, moved, lifted, levitated and thrown around. And all these
manipulations confirm that like everyday matter, atoms have mass, size, shape and col-
Ref. 261 our. Single atoms have even been used as lamps and as lasers. Some experimental results
are shown in Figure 239, Figure 240, Figure 240 and Figure 241.
The Greek imagined nature as a Lego set. And indeed, many modern researchers in
several fields have fun playing with atoms in the same way that children play with Lego.
A beautiful demonstration of these possibilities is provided by the many applications of
Ref. 262 the atomic force microscope. If you ever have the opportunity to use one, do not miss
it! An atomic force microscope is a simple table-top device which follows the surface of
an object with an atomically sharp needle;* such needles, usually of tungsten, are easily
Ref. 263 manufactured with a simple etching method. The changes in the height of the needle

* A cheap version costs only a few thousand euro, and will allow you to study the difference between a
silicon wafer – crystalline – a flour wafer – granular-amorphous – and a consecrated wafer.
limits of matter continuity 345

position-sensitive
laser
(segmented)
diode
photo- lens
detector

vertical cantilever
piezo
controller tip

horizontal sample
piezo
controllers

F I G U R E 242 The principle and a realization of an atomic force microscope (photograph © Nanosurf).

Motion Mountain – The Adventure of Physics
along its path over the surface are recorded with the help of a deflected light ray, as shown
in Figure 242. With a little care, the atoms of the object can be felt and made visible on
a computer screen. With special types of such microscopes, the needle can be used to
move atoms one by one to specified places on the surface. It is also possible to scan a
surface, pick up a given atom and throw it towards a mass spectrometer to determine
Ref. 264 what sort of atom it is.
Incidentally, the construction of atomic force microscopes is only a small improve-
ment on what nature is building already by the millions; when we use our ears to listen,
we are actually detecting changes in the eardrum position down to 11 pm. In other words,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
we all have two ‘atomic force microscopes’ built into our heads.
Why is it useful to know that matter is made of atoms? Given only the size of atoms, we
are able to understand and deduce many material properties. Atomic size determines the
mass density, the elastic modulus, the surface tension, the thermal expansion coefficient,
the heat of vaporization, the heat of fusion, the viscosity, the specific heat, the thermal
Challenge 572 s diffusivity and the thermal conductivity. Just try.

Curiosities and fun challenges ab ou t solids and atoms
Glass is a solid. Nevertheless, many textbooks suggest that glass is a liquid. This error has
been propagated for about a hundred years, probably originating from a mistranslation of
a sentence in a German textbook published in 1933 by Gustav Tamman, Der Glaszustand.
Challenge 573 s Can you give at least three reasons why glass is a solid and not a liquid?
∗∗
What is the maximum length of a vertically hanging wire? Could a wire be lowered from
Challenge 574 s a suspended geostationary satellite down to the Earth? This would mean we could realize
a space ‘lift’. How long would the cable have to be? How heavy would it be? How would
you build such a system? What dangers would it face?
∗∗
346 11 extended bodies

F I G U R E 243 Rubik’s cube: the complexity

Motion Mountain – The Adventure of Physics
of simple three-dimensional motion
(© Wikimedia).

2-Euro coin
cigarette

beer mat
glass

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 244 How can you move the coin into the glass without
touching anything?

Every student probably knows Rubik’s cube. Can you or did you deduce how Rubik built
the cube without looking at its interior? Also for cubes with other numbers of segments?
Challenge 575 s Is there a limit to the number of segments? These puzzles are even tougher than the
search for the rearrangement of the cube.
∗∗
Challenge 576 e Arrange six matches in such a way that every match touches all the others.
∗∗
Physics is often good to win bets. See Figure 244 for a way to do so, due to Wolfgang
Stalla.
∗∗
Matter is made of atoms. Over the centuries the stubborn resistance of many people to
this idea lead to the loss of many treasures. For over a thousand years, people thought that
genuine pearls could be distinguished from false ones by hitting them with a hammer:
limits of matter continuity 347

only false pearls would break. However, all pearls break. (Also diamonds break in this
situation.) Due to this belief, over the past centuries, all the most beautiful pearls in the
world have been smashed to pieces.
∗∗
Comic books have difficulties with the concept of atoms. Could Asterix really throw Ro-
mans into the air using his fist? Are Lucky Luke’s precise revolver shots possible? Can
Spiderman’s silk support him in his swings from building to building? Can the Roadrun-
ner stop running in three steps? Can the Sun be made to stop in the sky by command?
Can space-ships hover using fuel? Take any comic-book hero and ask yourself whether
Challenge 577 e matter made of atoms would allow him the feats he seems capable of. You will find that
most cartoons are comic precisely because they assume that matter is not made of atoms,
but continuous! In a sense, atoms make life a serious adventure.
∗∗
Can humans start earthquakes? Yes. In fact, several strong earthquakes have been

Motion Mountain – The Adventure of Physics
triggered by humans. This has happened when water dams have been filled, or when
water has been injected into drilling holes. It has also been suggested that the extrac-
tion of deep underground water also causes earthquakes. If this is confirmed by future
research, a sizeable proportion of all earthquakes could be human-triggered. Here is a
simple question on the topic: What would happen if 1000 million Indians, triggered by a
television programme, were to jump at the same time from the kitchen table to the floor?
Challenge 578 s

∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Many caves have stalactites. They form under two conditions: the water dripping from the
ceiling of a cave must contain calcium carbonate, CaCO3 , and the difference between the
carbon dioxide CO2 concentrations in the water and in the air of the cave must be above
a certain minimum value. If these conditions are fulfilled, calcareous sinter is deposited,
and stalactites can form. How can the tip of a stalactite growing down from the ceiling
Challenge 579 s be distinguished from the tip of a stalagmite rising from the floor? Does the difference
exist also for icicles?
∗∗
Fractals do not exist. Which structures in nature approximate them most closely? One
candidate is the lung, as shown in Figure 245. Its bronchi divide over and over, between
26 and 28 times. Each end then arrives at one of the 300 million alveoli, the 0.25 mm
cavities in which oxygen is absorbed into the blood and carbon dioxide is expelled into
the air.
∗∗
How much more weight would your bathroom scales show if you stood on them in a
Challenge 580 s vacuum?
∗∗
One of the most complex extended bodies is the human body. In modern simulations
348 11 extended bodies

F I G U R E 245 The human lung and its alveoli. (© Patrick J. Lynch, C. Carl Jaffe)

of the behaviour of humans in car accidents, the most advanced models include ribs,

Motion Mountain – The Adventure of Physics
vertebrae, all other bones and the various organs. For each part, its specific deformation
properties are taken into account. With such models and simulations, the protection of
passengers and drivers in cars can be optimized.
∗∗
The human body is a remarkable structure. It is stiff and flexible, as the situation de-
mands. Additionally, most stiff parts, the bones, are not attached to other stiff parts. Since
a few years, artists and architects have started exploring such structures. An example of
such a structure, a tower, is shown in Figure 246. It turns out that similar structures –

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
sometimes called tensegrity structures – are good models for the human spine, for ex-
ample. Just search the internet for more examples.
∗∗
Geometric solids are fun by themselves. Every convex polyhedron with 𝑓 faces, 𝑣 vertices
and 𝑒 edges obeys
𝑓+𝑣−𝑒=2. (110)

Challenge 581 ny This is Euler’s polyhedron formula. Find a proof.
∗∗
The deepest hole ever drilled into the Earth is 12 km deep. In 2003, somebody proposed
to enlarge such a hole and then to pour millions of tons of liquid iron into it. He claimed
that the iron would sink towards the centre of the Earth. If a measurement device com-
munication were dropped into the iron, it could send its observations to the surface using
Challenge 582 s sound waves. Can you give some reasons why this would not work?
∗∗
You may not know it, but you surely have already played with a helicoseir. This is the
technical name for a chain or rope, about 1 m in length, which hangs from your hand
and which you then put into rotational motion. Depending on your dexterity, you can
limits of matter continuity 349

Motion Mountain – The Adventure of Physics
F I G U R E 246 A stiff structure in which no rigid piece is attached to any
other one (© Kenneth Snelson).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
generate several stable shapes, with different numbers of stationary nodes. How many
nodes can you achieve? Robots can also do it, as shown on www.youtube.com/watch?
v=EnJdn3XdxEE.
∗∗
The economic power of a nation has long been associated with its capacity to produce
high-quality steel. Indeed, the Industrial Revolution started with the mass production of
steel. Every scientist should know the basic facts about steel. Steel is a combination of
iron and carbon to which other elements, mostly metals, may be added as well. One can
distinguish three main types of steel, depending on the crystalline structure. Ferritic steels
have a body-centred cubic structure, as shown in Figure 247, austenitic steels have a face-
centred cubic structure, and martensitic steels have a body-centred tetragonal structure.
Table 42 gives further details.
∗∗
A simple phenomenon which requires a complex explanation is the cracking of a whip.
Ref. 265 Since the experimental work of Peter Krehl it has been known that the whip cracks when
350 11 extended bodies

TA B L E 42 Steel types, properties and uses.

Ferritic steel Au s t e ni t i c s t e e l Martensitic steel

Definition
‘usual’ steel ‘soft’ steel hardened steel, brittle
body centred cubic (bcc) face centred cubic (fcc) body centred tetragonal (bct)
iron and carbon iron, chromium, nickel, carbon steel and alloys
manganese, carbon
Examples
construction steel most stainless (18/8 Cr/Ni) knife edges
steels
car sheet steel kitchenware drill surfaces
ship steel food industry spring steel, crankshafts
12 % Cr stainless ferrite Cr/V steels for nuclear
reactors
Properties

Motion Mountain – The Adventure of Physics
phases described by the phases described by the phases described by the
iron-carbon phase diagram Schaeffler diagram iron-carbon diagram and the
TTT (time–temperature
transformation) diagram
in equilibrium at RT some alloys in equilibrium at not in equilibrium at RT, but
RT stable
mechanical properties and mechanical properties and mechanical properties and
grain size depend on heat grain size depend on grain size strongly depend on
treatment thermo-mechanical heat treatment

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
pre-treatment
hardened by reducing grain hardened by cold working hard anyway – made by laser
size, by forging, by increasing only irradiation, induction heating,
carbon content or by nitration etc.
grains of ferrite and paerlite, grains of austenite grains of martensite
with cementite (Fe3 C)
ferromagnetic not magnetic or weakly ferromagnetic
magnetic

Challenge 583 ny the tip reaches a velocity of twice the speed of sound. Can you imagine why?
∗∗
A bicycle chain is an extended object with no stiffness. However, if it is made to ro-
tate rapidly, it acquires dynamical stiffness and can roll down an inclined plane or along
the floor. This surprising effect can be watched at the www.iwf.de/iwf/medien/infothek?
Signatur=C+14825 website.
∗∗
Mechanical devices are not covered in this text. There is a lot of progress in the area even
Ref. 266 at present. For example, people have built robots that are able to ride a unicycle. But even
limits of matter continuity 351

Motion Mountain – The Adventure of Physics
F I G U R E 247 Ferritic steels are bcc (body centred cubic), as shown by the famous Atomium in Brussels, a
section of an iron crystal magniﬁed to a height of over 100 m (photo and building are © Asbl Atomium
Vzw – SABAM Belgium 2007).

Ref. 267 the physics of human unicycling is not simple. Try it; it is an excellent exercise to stay
young.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
There are many arguments against the existence of atoms as hard balls. Thomson-Kelvin
Ref. 268 put it in writing and spoke of “the monstrous assumption of infinitely strong and infin-
itely rigid pieces of matter”. Even though Thomson was right in his comment, atoms do
Challenge 584 s exist. Why?
∗∗
Sand has many surprising ways to move, and new discoveries are still made regularly. In
Ref. 269 2001, Sigurdur Thoroddsen and Amy Shen discovered that a steel ball falling on a bed of
sand produces, after the ball has sunk in, a granular jet that jumps out upwards from the
sand. Figure 248 shows a sequence of photographs of the effect. The discovery has led to
a stream of subsequent research.
∗∗
Engineering is not a part of this text. Nevertheless, it is an interesting topic. A few ex-
amples of what engineers do are shown in Figure 249 and Figure 250.

Summary on atoms
Matter is not scale-invariant: in particular, it is neither smooth (continuous) nor fractal.
There are no arbitrarily small parts in matter. Everyday matter is made of countable com-
ponents: everyday matter is made of atoms. This has been confirmed for all solids, liquids
352 11 extended bodies

Motion Mountain – The Adventure of Physics
F I G U R E 248 An example of a
surprising motion of sand: granular jets
(© Amy Shen).

F I G U R E 249 Modern engineering highlights: a lithography machine for the production of integrated
circuits and a paper machine (© ASML, Voith).

and gases. Pictures from atomic force microscopes show that the size and arrangement
of atoms produce the shape and the extension of objects, confirming the Lego model of
matter due to the ancient Greek. Different types of atoms, as well as their various com-
binations, produce different types of substances.
Studying matter in even more detail – as will be done later on – yields the now
well-known idea that matter, at higher and higher magnifications, is made of molecules,
limits of matter continuity 353

Motion Mountain – The Adventure of Physics
F I G U R E 250 Sometimes unusual moving objects cross German roads (© RWE).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
atoms, nuclei, protons and neutrons, and finally, quarks. Atoms also contain electrons. A
final type of matter, neutrino, is observed coming from the Sun and from certain types
of radioactive materials. Even though the fundamental bricks have become somewhat
smaller in the twentieth century, this will not happen in the future. The basic idea of
the ancient Greek remains: matter is made of smallest entities, nowadays called element-
Vol. V, page 261 ary particles. In the parts of our adventure on quantum theory we will explore the con-
sequences in detail. We will discover later on that the discreteness of matter is itself a
consequence of the existence of a smallest change in nature.
Due to the existence of atoms, the description of everyday motion of extended objects
can be reduced to the description of the motion of their atoms. Atomic motion will be
a major theme in the following pages. Two of its consequences are especially important:
pressure and heat. We study them now.
C h a p t e r 12

F LU I D S A N D T H E I R MOT ION

F
luids can be liquids or gases, including plasmas. And the motion of
Ref. 270 luids can be exceedingly intricate, as Figure 251 shows. In fact,
luid motion is common and important – think about breathing, blood circu-
lation or the weather. Exploring it is worthwhile.

Motion Mountain – The Adventure of Physics
What can move in nature? – Flows of all kinds
Before we continue with the exploration of fluids, we have a closer look at all the various
types of motion in the world around us. An overview of the types of motion in nature is
given in Table 43. It lists motion in the various domains of natural sciences that belong to
everyday life: motion of fluids, of matter, of matter types, of heat, of light and of charge.
All these types of motion are the domains of continuum physics. For each domain,
the table gives the quantity whose motion is central to that domain. The moving quantity
– be it mass, volume, entropy or charge – is described by an extensive observable, i.e., an
observable that can accumulate. The motion of an extensive observable is called a flow.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Every type of motion in nature is a flow. The extensive quantity is the ‘stuff’ that flows.
Within continuum physics, there are two domains we have not yet studied: the motion
of charge and light, called electrodynamics, and the motion of heat, called thermodynam-
ics. Once we have explored these domains, we will have completed the first step of our
description of motion: continuum physics. In continuum physics, motion and moving
entities are described with continuous quantities that can take any value, including ar-
bitrarily small or arbitrarily large ones.
But nature is not continuous. We have already seen that matter cannot be indefinitely
divided into ever-smaller entities. In fact, we will discover that there are precise experi-
ments that provide limits to the observed values for every domain of continuum physics.
We will discover that there are both lower and upper limits to mass, to speed, to angular
momentum, to force, to entropy and to charge. The consequences of these discoveries
lead to the next legs of our description of motion: relativity and quantum theory. Re-
lativity is based on upper limits, quantum theory on lower limits. By the way, can you
Challenge 585 e argue that all extensive quantities have a smallest value? The last leg of our description
of motion will explore the unification of quantum theory and general relativity.
Every domain of physics, regardless of which one of the above legs it belongs to, de-
Ref. 299 scribes change in terms an extensive quantity characteristic of the domain and energy.
An observable quantity is called extensive if it increases with system size.
12 fluids and their motion 355

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 251 Examples of laminar ﬂuid motion: a vertical water jet striking a horizontal impactor, two
jets of a glycerol–water mixture colliding at an oblique angle, a water jet impinging on a reservoir in
stationary motion, a glass of wine showing tears (all © John Bush, MIT) and a dripping water tap
(© Andrew Davidhazy).
356 12 fluids and their motion

TA B L E 43 Flows in nature: all extensive quantities, i.e., observable quantities that accumulate and thus
move and ﬂow.

Domain Extensive Current Intens- Energy R e s i s ta nce
qua ntit y ive flow to
qua ntit y transp ort
(energy (flow (driving (power) (intensity
carrier) r at e ) strength) of entropy
genera-
tion)
Rivers mass 𝑚 mass flow 𝑚/𝑡 height 𝑃 = 𝑔ℎ 𝑚/𝑡 𝑅m = 𝑔ℎ𝑡/𝑚
(difference) [m2 /s kg]
𝑔ℎ
Gases volume 𝑉 volume flow 𝑉/𝑡 pressure 𝑝 𝑃 = 𝑝𝑉/𝑡 𝑅V = 𝑝𝑡/𝑉
[kg/s m5 ]
Mechanics momentum 𝑝 force 𝐹 = d𝑝/d𝑡 velocity 𝑣 𝑃 = 𝑣𝐹 𝑅p = Δ𝑉/𝐹 =

Motion Mountain – The Adventure of Physics
𝑡/𝑚 [s/kg]
angular torque angular 𝑃 = 𝜔𝑀 𝑅L = 𝑡/𝑚𝑟2
momentum 𝐿 𝑀 = d𝐿/d𝑡 velocity 𝜔 [s/kg m2 ]
Chemistry amount of substance flow chemical 𝑃 = 𝜇 𝐼𝑛 𝑅𝑛 = 𝜇𝑡/𝑛
substance 𝑛 𝐼𝑛 = d𝑛/d𝑡 potential 𝜇 [Js/mol2 ]
Thermo- entropy 𝑆 entropy flow temperature 𝑃 = 𝑇 𝐼𝑆 𝑅𝑆 = 𝑇𝑡/𝑆
dynamics 𝐼𝑆 = d𝑆/d𝑡 𝑇 [K2 /W]
Electricity charge 𝑞 electrical current electrical 𝑃 = 𝑈𝐼 𝑅 = 𝑈/𝐼

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
𝐼 = d𝑞/d𝑡 potential 𝑈 [Ω]
Magnetism no accumulable magnetic sources are found in nature
Light like all massless radiation, it can flow but cannot accumulate
Nuclear extensive quantities exist, but do not appear in everyday life
physics
Gravitation empty space can move and flow, but the motion is not observed in everyday life

⊳ An extensive quantity describes what can flow in nature.

Examples are mass, volume, momentum, charge and entropy. Table 43 provides a full
overview. In every flow there is also an physical observable that makes things move:

⊳ An intensive quantity describes the driving strength of flow; it quantifies why
it flows.

Examples are height difference, pressure, speed, potential, or temperature.
For every physical system, the state is described by a set of extensive and intensive ob-
servables. Any complete description of motion and any complete observation of a system
12 fluids and their motion 357

thus always requires both types of observables.
The extensive and intensive quantities – what flows and why it flows – for the case of
fluids are volume and pressure. They are central to the description and the understanding
of fluid flows. In the case of heat, the extensive and intensive quantities are entropy – the
quantity that flows – and temperature – the quantity that makes heat flow. We will explore
them shortly. We also note:

⊳ The product of the extensive and intensive quantities is always energy per
time, i.e., always the power of the corresponding flow.

The analogies of Table 43 can be carried even further. In all domains,

⊳ The capacity of a system is defined by the extensive quantity divided by the
intensive quantity.

The capacity measures how easy it is to let ‘stuff’ flow into a system. For electric charge,

Motion Mountain – The Adventure of Physics
the capacity is what is usually called electric capacity. If the number is very large, it is very
easy to add charge. For momentum, the capacity is called mass. Mass measures how easily
momentum can be added to a system. Can you determine the quantities that measure
Challenge 586 e capacity in the other cases?
In all fields it is possible to store energy by using the intensive quantity – such as 𝐸 =
𝐶𝑈2 /2 in a capacitor or 𝐸 = 𝑚𝑣2 /2 in a moving body – or by using the extensive quantity
– such as 𝐸 = 𝐿𝐼2 /2 in a coil or 𝐸 = 𝐹2 /2𝑘 in a spring. Combining and alternating the
two storage methods, we get oscillations.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
⊳ Every flow can produce oscillations.

Challenge 587 ny Can you explain oscillations for the other cases of flow?

The state of a fluid
To describe motion means, first of all, to describe the state of the moving system. For
most fluids, the state is described by specifying, at every point in space, the mass dens-
ity, the velocity, the temperature and the pressure. We thus have two new observables:
Page 383 temperature, which we will explore in the next chapter, and pressure.
Here we concentrate on the other new observable:

⊳ The pressure 𝑝 at a point in a fluid is the force per area that a body of negli-
gible size feels at that point: 𝑝 = 𝐹/𝐴.

Equivalently,

⊳ Pressure is momentum change per area.

The unit of pressure is the pascal: 1 Pa is defined as 1 N/m2 . A selection of pressure values
found in nature is given in Table 44. Pressure is measured with the help of barometers
358 12 fluids and their motion

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 252 Different pressure sensors: an antique barograph, an industrial water pressure sensor, a
pressure sensitive resistor, the lateral line of a ﬁsh, and modern MEMS pressure sensors. (© Anonymous,
ifm, Conrad, Piet Spaans, Proton Mikrotechnik)
12 fluids and their motion 359

TA B L E 44 Some measured pressure values.

O b s e r va t i o n Pressure

Record negative pressure (tension) measured in −140 MPa
water, after careful purification Ref. 271 = −1400 bar
Negative pressure measured in tree sap up to
(xylem) Ref. 272, Ref. 255 −10 MPa
= −100 bar
Negative pressure in gases does not exist
Negative pressure in solids is called
tension
Record vacuum pressure achieved in laboratory 10 pPa
(10−13 torr)
Pressure variation at hearing threshold 20 μPa
Pressure variation at hearing pain 100 Pa
Atmospheric pressure in La Paz, Bolivia 51 kPa

Motion Mountain – The Adventure of Physics
Atmospheric pressure in cruising passenger aircraft 75 kPa
Time-averaged pressure in pleural cavity in human 0.5 kPa
thorax 5 mbar below
atmospheric
pressure
Standard sea-level atmospheric pressure 101.325 kPa
or
1013.25 mbar
or 760 torr

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Healthy human arterial blood pressure at height of 17 kPa,11 kPa
the heart: systolic, diastolic above
atmospheric
pressure
Record pressure produced in laboratory, using a c. 200 GPa
diamond anvil
Pressure at the centre of the Earth c. 370(20) GPa
Pressure at the centre of the Sun c. 24 PPa
Pressure at the centre of a neutron star c. 4 ⋅ 1033 Pa
Planck pressure (maximum pressure possible in 4.6 ⋅ 10113 Pa
nature)

or similar instruments, such as those shown in Figure 252. Also the human body is full
Vol. V, page 39 of pressure sensors; the Merkel cells in the fingertips are examples. We will explore them
later on.
Inside a fluid, pressure has no preferred direction. At the boundary of a fluid, pres-
sure acts on the container wall. Pressure is not a simple property. Can you explain the
observations of Figure 254? If the hydrostatic paradox – an effect of the so-called com-
municating vases – would not be valid, it would be easy to make perpetua mobilia. Can
Challenge 589 e you think about an example? Another puzzle about pressure is given in Figure 255.
360 12 fluids and their motion

F I G U R E 253 Daniel Bernoulli (1700–1782)

Motion Mountain – The Adventure of Physics
F I G U R E 254 The hydrostatic and the hydrodynamic paradox (© IFE).

Challenge 588 s F I G U R E 255 A puzzle: Which method of emptying a container is fastest? Does the method on the
right-hand side work at all?

The air around us has a considerable pressure, of the order of 100 kPa. As a result, it
is not easy to make a vacuum; indeed, everyday forces are often too weak to overcome
air pressure. This is known for several centuries, as Figure 256 shows. Your favourite
physics laboratory should possess a vacuum pump and a pair of (smaller) Magdeburg
12 fluids and their motion 361

F I G U R E 256 The pressure of
air leads to surprisingly large
forces, especially for large
objects that enclose a
vacuum. This was regularly

Motion Mountain – The Adventure of Physics
demonstrated in the years
from 1654 onwards by Otto
von Guericke with the help
of his so-called Magdeburg
hemispheres and, above all,
the various vacuum pumps
that he invented
(© Deutsche Post, Otto-von-
Guericke-Gesellschaft,
Deutsche Fotothek).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
hemispheres; enjoy performing the experiment yourself.

L aminar and turbulent flow
Like all motion, fluid motion obeys energy conservation. For fluids in which no energy
is transformed into heat, the conservation of energy is particularly simple. Motion that
does not generate heat is motion without vortices; such fluid motion is called laminar.
In the special case that, in addition, the speed of the fluid does not depend on time at
all positions, the fluid motion is called stationary. Non-laminar flow is called turbulent.
Figure 251 and Figure 257 show examples.
For motion that is both laminar and stationary, energy conservation can be expressed
with the help of speed 𝑣 and pressure 𝑝:

1
2
𝜌𝑣2 + 𝑝 + 𝜌𝑔ℎ = const (111)

where ℎ is the height above ground, 𝜌 is the mass density and 𝑔 = 9.8 m/s2 is the gravit-
ational acceleration. This is called Bernoulli’s equation.* In this equation, the last term is
only important if the fluid rises against the ground. The first term is the kinetic energy
* Daniel Bernoulli (b. 1700 Bâle, d. 1782 Bâle), important mathematician and physicist. Also his father Jo-
362 12 fluids and their motion

F I G U R E 257 Left: non-stationary and stationary laminar ﬂows; right: an example of turbulent ﬂow
(© Martin Thum, Steve Butler).

Motion Mountain – The Adventure of Physics
(per volume) of the fluid, and the other two terms are potential energies (per volume).
Indeed, the second term is the potential energy (per volume) resulting from the com-
Challenge 590 e pression of the fluid. This is due to a further way to define pressure:

⊳ Pressure is potential energy per volume.

Energy conservation implies that the lower the pressure is, the larger the speed of a fluid
becomes. We can use this relation to measure the speed of a stationary water flow in a

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
tube. We just have to narrow the tube somewhat at one location along the tube, and then
measure the pressure difference before and at the tube restriction. The speed 𝑣 far from
Challenge 591 s the constriction is then given as 𝑣 = 𝑘√𝑝1 − 𝑝2 . (What is the constant 𝑘?) A device using
this method is called a Venturi gauge.
Now think about flowing water. If the geometry is kept fixed and the water speed
is increased – or the relative speed of a body in water is increased – at a certain speed
value, we observe a transition: the water loses its clarity, the flow is not stationary and
not laminar any more. We can observe the transition whenever we open a water tap: at
a certain speed, the flow changes from laminar to turbulent. From this point onward,
Bernoulli’s equation (111) is not valid any more.
A precise description of turbulence has not yet been achieved. This might be the
toughest of all open problems in physics. When the young Werner Heisenberg was asked
to continue research on turbulence, he refused – rightly so – saying it was too difficult; he

hann and his uncle Jakob were famous mathematicians, as were his brothers and some of his nephews.
Daniel Bernoulli published many mathematical and physical results. In physics, he studied the separation
of compound motion into translation and rotation. In 1738 he published the Hydrodynamique, in which
he deduced all results from a single principle, namely the conservation of energy. The so-called Bernoulli
equation states how the pressure of a fluid decreases when its speed increases. He studied the tides and many
complex mechanical problems, and explained the Boyle–Mariotte gas ‘law’. For his publications he won the
prestigious prize of the French Academy of Sciences – a forerunner of the Nobel Prize – ten times.
12 fluids and their motion 363

F I G U R E 258 The moth sailing class: a 30 kg boat that sails above the water using hydrofoils, i.e.,
underwater wings (© Bladerider International).

Motion Mountain – The Adventure of Physics
turned to something easier and he discovered and developed quantum theory instead.
Turbulence is such a vast topic, with many of its concepts still not settled, that despite the
number and importance of its applications, only now, at the beginning of the twenty-first
Ref. 273 century, are its secrets beginning to be unravelled.
It is thought that the equations of motion describing fluids in full generality, the so-
called Navier–Stokes equations, are sufficient to understand turbulence.* But the math-
ematics behind these equations is mind-boggling. There is even a prize of one million
dollars offered by the Clay Mathematics Institute for the completion of certain steps on
the way to solving the equations.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Important systems which show laminar flow, vortices and turbulence at the same time
are wings and sails. (See Figure 258.) All wings and sails work best in laminar mode. The
essence of a wing is that it imparts air a downward velocity with as little turbulence as
possible. (The aim to minimize turbulence is the reason that wings are curved. If the en-
gine is very powerful, a flat wing at an angle also keeps an aeroplane in the air. However,
the fuel consumption increases dramatically. On the other hand, strong turbulence is of
advantage for landing safely.) Around a wing of a flying bird or aeroplane, the downward
velocity of the trailing air leads to a centrifugal force acting on the air that passes above
Ref. 274 the wing. This leads to a lower pressure, and thus to lift. (Wings thus do not rely on the
Bernoulli equation, where lower pressure along the flow leads to higher air speed, as un-
fortunately, many books used to say. Above a wing, the higher speed is related to lower
pressure across the flow.)
The different speeds of the air above and below the wing lead to vortices at the end
of every wing. These vortices are especially important for the take-off of any insect, bird
Vol. V, page 278 and aeroplane. More aspects of wings are explored later on.

* They are named after Claude Navier (b. 1785 Dijon, d. 1836 Paris), important engineer and bridge builder,
and Georges Gabriel Stokes (b. 1819 Skreen, d. 1903 Cambridge), important physicist and mathematician.
364 12 fluids and their motion

upper atmosphere

ozone layer
stratosphere
troposphere

Motion Mountain – The Adventure of Physics
F I G U R E 259 Several layers of the atmosphere are visble in this sunset photograph taken from the
International Space Station, ﬂying at several hundred km of altitude (courtesy NASA).

The atmosphere

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The atmosphere, a thin veil around our planet that is shown in Figure 259, keeps us alive.
The atmosphere is a fluid that surrounds the Earth and consists of 5 ⋅ 1018 kg of gas. The
density decreases with height: 50 % of the mass is below 5.6 km of height, 75 % within
11 km, 90 % within 16 km and 99,999 97 % within 100 km.*
At sea level, the atmospheric density is, on average, 1.29 kg/m3 – about 1/800th of
that of water – and the pressure is 101.3 kPa; both values decrease with altitude. The
composition of the atmosphere at sea level is given on page 516. Also the composition
varies with altitude; furthermore, the composition depends on the weather and on the
pollution level.
The structure of the atmosphere is given in Table 45. The atmosphere ceases to behave
as a gas above the thermopause, somewhere between 500 and 1000 km; above that alti-
tude, there are no atomic collisions any more. In fact, we could argue that the atmosphere
ceases to behave as an everyday gas above 150 km, when no audible sound is transmitted
any more, not even at 20 Hz, due to the low atomic density.

* The last height is called the Kármán line; it is the conventional height at which a flying system cannot use
lift to fly any more, so that it is often used as boundary between aeronautics and astronautics.
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
365

F I G U R E 260 The main layers of the atmosphere (© Sebman81).
12 fluids and their motion
366 12 fluids and their motion

TA B L E 45 The layers of the atmosphere.

L ay e r A lt i t u d e D e ta i l s

Exosphere > 500 to mainly composed of hydrogen and helium, includes
about the magnetosphere, temperature above 1000°C,
10 000 km contains many artificial satellites and sometimes
aurora phenomena, includes, at its top, the luminous
geocorona
Boundary: thermopause between 500 above: no ‘gas’ properties, no atomic collisions;
or exobase and 1000 km below: gas properties, friction for satellites; altitude
varies with solar activity
Thermosphere from 85 km composed of oxygen, helium, hydrogen and ions,
to temperature of up to 2500°C, pressure 1 to 10 μPa;
thermopause infrasound speed around 1000 m/s; no transmission
of sound above 20 Hz at altitudes above 150 km;
contains the International Space Station and many
satellites; featured the Sputnik and the Space Shuttle

Motion Mountain – The Adventure of Physics
Heterosphere all above separate concept that includes all layers that show
turbopause diffusive mixing, i.e., most of the thermosphere and
the exosphere
Boundary: 100 km boundary between diffusive mixing (above) and
turbopause or turbulent mixing (below)
homopause
Homosphere everything separate concept that includes the lowest part of the
below thermosphere and all layers below it
turbopause

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Boundary: mesopause 85 km temperature between −100°C and −85°C, lowest
temperature ‘on’ Earth; temperature depends on
season; contains ions, includes a sodium layer that is
used to make guide stars for telescopes
Mesosphere from temperature decreases with altitude, mostly
stratopause to hydrogen, contains noctilucent clouds, sprites, elves,
mesopause ions; burns most meteors, shows atmospheric tides
and a circulation from summer to winter pole
Ionosphere or 60 km to a separate concept that includes all layers that
magnetosphere 1000 km contain ions, thus the exosphere, the thermosphere
and a large part of the mesosphere
Boundary: stratopause 50 to 55 km maximum temperature between stratosphere and
(or mesopeak) mesosphere; pressure around 100 Pa, temperature
−15°C to −3°C
Stratosphere up to the stratified, no weather phenomena, temperature
stratopause increases with altitude, dry, shows quasi-biennial
oscillations, contains the ozone layer in its lowest
20 km, as well as aeroplanes and some balloons
12 fluids and their motion 367

TA B L E 45 (Continued) The layers of the atmosphere.

L ay e r A lt i t u d e D e ta i l s

Boundary: tropopause 6 to 9 km at temperature −50°C, temperature gradient vanishes,
the poles, 17 no water any more
to 20 km at
the equator
Troposphere up to the contains water and shows weather phenomena;
tropopause contains life, mountains and aeroplanes; makes stars
flicker; temperature generally decreases with
altitude; speed of sound is around 340 m/s
Boundary: planetary 0.2 to 2 km part of the troposphere that is influenced by friction
boundary layer or with the Earth’s surface; thickness depends on
peplosphere landscape and time of day

The physics of blo od and breath

Motion Mountain – The Adventure of Physics
Fluid motion is of vital importance. There are at least four fluid circulation systems in-
side the human body. First, blood flows through the blood system by the heart. Second,
air is circulated inside the lungs by the diaphragm and other chest muscles. Third,
lymph flows through the lymphatic vessels, moved passively by body muscles. Fourth,
the cerebrospinal fluid circulates around the brain and the spine, moved by motions of
the head. For this reason, medical doctors like the simple statement: every illness is ulti-
mately due to bad circulation.
Why do living beings have circulation systems? Circulation is necessary because diffu-
Challenge 592 e sion is too slow. Can you detail the argument? We now explore the two main circulation

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
systems in the human body.
Blood keeps us alive: it transports most chemicals required for our metabolism to
and from the various parts of our body. The flow of blood is almost always laminar;
turbulence only exists in the venae cavae, near the heart. The heart pumps around
80 ml of blood per heartbeat, about 5 l/min. At rest, a heartbeat consumes about 1.2 J.
The consumption is sizeable, because the dynamic viscosity of blood ranges between
3.5 ⋅ 10−3 Pa s (3.5 times higher than water) and 10−2 Pa s, depending on the diameter of
the blood vessel; it is highest in the tiny capillaries. The speed of the blood is highest in
the aorta, where it flows with 0.5 m/s, and lowest in the capillaries, where it is as low as
0.3 mm/s. As a result, a substance injected in the arm arrives in the feet between 20 and
60 s after the injection.
In fact, all animals have similar blood circulation speeds, usually between 0.2 m/s and
Challenge 593 ny 0.4 m/s. Why?
To achieve blood circulation, the heart produces a (systolic) pressure of about 16 kPa,
corresponding to a height of about 1.6 m of blood. This value is needed by the heart to
pump blood through the brain. When the heart relaxes, the elasticity of the arteries keeps
the (diastolic) pressure at around 10 kPa. These values are measured at the height of the
heart.* The values vary greatly with the position and body orientation at which they are

* The blood pressure values measured on the two upper arms also differ; for right-handed people, the pres-
sure in the right arm is higher.
368 12 fluids and their motion

measured: the systolic pressure at the feet of a standing adult reaches 30 kPa, whereas it
is 16 kPa in the feet of a lying person. For a standing human, the pressure in the veins
in the foot is 18 kPa, larger than the systolic pressure in the heart. The high pressure
values in the feet and legs are one of the reasons that lead to varicose veins. Nature uses
many tricks to avoid problems with blood circulation in the legs. Humans leg veins have
valves to prevent the blood from flowing downwards; giraffes have extremely thin legs
with strong and tight skin in the legs for the same reason. The same happens for other
large animals.
At the end of the capillaries, the pressure is only around 2 kPa. The lowest blood pres-
sure is found in veins that lead back from the head to the heart, where the pressure can
even be slightly negative. Because of blood pressure, when a patient receives an (intra-
venous) infusion, the bag must have a minimum height above the infusion point where
the needle enters the body; values of about 0.8 to 1 m cause no trouble. (Is the height
difference also needed for person-to-person transfusions of blood?) Since arteries have
higher blood pressure, for the more rare arterial infusions, hospitals usually use arterial
pumps, to avoid the need for unpractical heights of 2 m or more.

Motion Mountain – The Adventure of Physics
Ref. 275 Recent research has demonstrated what was suspected for a long time: in the capil-
laries, the red blood cells change shape and motion. The shape change depends on the
capillary diameter and the flow speed. In large vessels, red blood cells usually tumble in
the bloodstream. In smaller blood vessels, they roll and in still smaller ones they deform
in various ways. These changes explain how blood flows more easily, i.e., with lower vis-
cosity, in thinner vessels, It is even conjectured that disturbing these shape changes might
be related to specific symptoms and illnesses.
The physics of breathing is equally interesting. A human cannot breathe at any depth
under water, even if he has a tube going to the surface, as shown in Figure 261. At a few

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
metres of depth, trying to do so is inevitably fatal! Even at a depth of 50 cm only, the
human body can only breathe in this way for a few minutes and can get badly hurt for
Challenge 594 s life. Why?
Inside the lungs, the gas exchange with the blood occurs in around 300 millions of
little spheres, the alveoli, with a diameter between 0.2 and 0.6 mm. They are shown in
Page 348 Figure 245. To avoid that the large ones grow and the small ones collapse – as in the
Page 389 experiment of Figure 276– the alveoli are covered with a phospholipid surfactant that
reduces their surface tension. In newborns, the small radius of the alveoli and the low
level of surfactant is the reason that the first breaths, and sometimes also the subsequent
ones, require a large effort.
We need around 2 % of our energy for breathing alone. The speed of air in the throat
is 3 m/s for normal breathing; when coughing, it can be as high as 50 m/s. The flow of
air in the bronchi is turbulent; the noise can be heard in a quiet environment. In normal
breathing, the breathing muscles, in the thorax and in the belly, exchange 0.5 l of air; in
a deep breath, the volume can reach 4 l.
Breathing is especially tricky in unusual situations. After scuba diving* at larger
depths than a few meters for more than a few minutes, it is important to rise slowly, to

* Originally, ‘scuba’ is the abbreviation of ‘self-contained underwater breathing apparatus’. The central
device in it, the ‘aqua lung’, was invented by Emile Gagnan and Jacques Cousteau; it keeps the air pressure
always at the same level as the water pressure.
12 fluids and their motion 369

air

tube

danger!

water

Motion Mountain – The Adventure of Physics
F I G U R E 261 Attention, danger! Trying to do this will
destroy your lung irreversibly and possibly kill you.

Getting water from A to B

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
A (1) the Roman solution: an aqueduct
B

A
B

(2) the cost-effective solution: a tube F I G U R E 262 Wasting
money because of lack
of knowledge about
ﬂuids.

Challenge 595 e avoid a potentially fatal embolism. Why? The same can happen to participants in high-
altitude flights with balloons or aeroplanes, to high-altitude parachutists and to cosmo-
nauts.
370 12 fluids and their motion

Curiosities and fun challenges ab ou t fluids
What happens if people do not know the rules of nature? The answer is the same since
2000 years ago: taxpayers’ money is wasted or health is in danger. One of the oldest ex-
amples, the aqueducts from Roman time, is shown in Figure 262. Aqueducts only exist
because Romans did not know how fluids move. The figure tells why there are no aque-
ducts any more.
We note that using a 1 or 2 m water hose in the way shown in Figure 262 or in Fig-
Challenge 596 s ure 255 to transport gasoline can be dangerous. Why?
∗∗
Take an empty milk carton, and make a hole on one side, 1 cm above the bottom. Then
make two holes above it, each 5 cm above the previous one. If you fill the carton with
water and put it on a table, which of the three streams will reach further away? And if
Challenge 597 e you put the carton on the edge of the table, so that the streams fall down on the floor?
∗∗

Motion Mountain – The Adventure of Physics
Your bathtub is full of water. You have an unmarked 3-litre container and an unmarked
Challenge 598 e 5-litre container. How can you get 4 litres of water from the bathtub?
∗∗
What is the easiest way to create a supersonic jet of air? Simply drop a billiard ball into a
Ref. 276 bucket full of water. It took a long time to discover this simple method. Enjoy researching
the topic.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Fluids are important for motion. Spiders have muscles to flex their legs, but no muscles
to extend them. How do they extend their legs?
In 1944, Ellis discovered that spiders extend their legs by hydraulic means: they in-
crease the pressure of a fluid inside their legs; this pressure stretches the leg like the water
Ref. 277 pressure stiffens a garden hose. If you prefer, spider legs thus work a bit like the arm of an
excavator. That is why spiders have bent legs when they are dead. The fluid mechanism
works well: it is also used by jumping spiders.
∗∗
Where did the water in the oceans – whose amount is illustrated in Figure 263 – come
from? Interestingly enough, this question is not fully settled! In the early age of the Earth,
the high temperatures made all water evaporate and escape into space. So where did
today’s water come from? (For example, could the hydrogen come from the radioactivity
of the Earth’s core?) The most plausible proposal is that the water comes from comets.
Comets are made, to a large degree, of ice. Comets hitting the Earth in the distant past
seem to have formed the oceans. In 2011, it was shown for the first time, by the Herschel
Ref. 278 infrared space telescope of the European Space Agency, that comets from the Kuiper belt
– in contrast to comets from the inner Solar System – have ice of the same oxygen isotope
composition as the Earth’s oceans. The comet origin of oceans seems settled.
12 fluids and their motion 371

F I G U R E 263 All the water on Earth
would form a sphere with a radius
of about 700 km, as illustrated in
this computer-generated graphic.
(© Jack Cook, Adam Nieman, Woods
Hole Oceanographic Institution,
Howard Perlman, USGS)

Motion Mountain – The Adventure of Physics
∗∗
The physics of underwater diving, in particular the physics of apnoea diving, is full of
Ref. 279 wonders and of effects that are not yet understood. For example, every apnoea champion
knows that it is quite hard to hold your breath for five or six minutes while sitting in a
chair. But if the same is done in a swimming pool, the feat becomes readily achievable
for modern apnoea champions. It is still not fully clear why this is the case.
There are many apnoea diving disciplines. In 2009, the no-limit apnoea diving re-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
cord is at the incredible depth of 214 m, achieved by Herbert Nitsch. The record static
apnoea time is over eleven minutes, and, with hyperventilation with pure oxygen, over
22 minutes. The dynamic apnoea record, without fins, is 213 m.
When an apnoea diver reaches a depth of 100 m, the water pressure corresponds to a
weight of over 11 kg on each square centimetre of his skin. To avoid the problems of ear
pressure compensation at great depths, a diver has to flood the mouth and the trachea
with water. His lungs have shrunk to one eleventh of their original size, to the size of
apples. The water pressure shifts almost all blood from the legs and arms into the thorax
and the brain. At 150 m, there is no light, and no sound – only the heartbeat. And the
heartbeat is slow: there is only a beat every seven or eight seconds. He becomes relaxed
and euphoric at the same time. None of these fascinating observations is fully under-
stood.
Sperm whales, Physeter macrocephalus, can stay below water more than half an hour,
and dive to a depth of more than 3000 m. Weddell seals, Leptonychotes weddellii, can stay
below water for an hour and a half. The mechanisms are unclear, but seem to involve
haemoglobine and neuroglobine. The research into the involved mechanisms is inter-
esting because it is observed that diving capability strengthens the brain. For example,
bowhead whales, Balaena mysticetus, do not suffer strokes nor brain degeneration, even
though they reach over 200 years in age.
∗∗
372 12 fluids and their motion

Apnoea records show the beneficial effects of oxygen on human health. An oxygen bottle
is therefore a common item in professional first aid medical equipment.
∗∗
What is the speed record for motion underwater? Probably only few people know: it
is a military secret. In fact, the answer needs to be split into two. The fastest published
speed for a projectile underwater, almost fully enclosed in a gas bubble, is 1550 m/s, faster
than the speed of sound in water, achieved over a distance of a few metres in a military
laboratory in the 1990s. The fastest system with an engine seems to be a torpedo, also
moving mainly in a gas bubble, that reaches over 120 m/s, thus faster than any formula
1 racing car. The exact speed achieved is higher and secret, as the method of enclosing
objects underwater in gas bubbles, called supercavitation, is a research topic of military
engineers all over the world.
The fastest fish, the sailfish Istiophorus platypterus, reaches 22 m/s, but speeds up to
30 m/s are suspected. Underwater, the fastest manned objects are military submarines,
whose speeds are secret, but believed to be around 21 m/s. (All military naval engineers

Motion Mountain – The Adventure of Physics
in this world, with the enormous budgets they have, are not able to make submarines
that are faster than fish. The reason that aeroplanes are faster than birds is evident: aero-
planes were not developed by military engineers, but by civilian engineers.) The fast-
est human-powered submarines reach around 4 m/s. We can estimate that if human-
powered submarine developers had the same development budget as military engineers,
their machines would probably be faster than nuclear submarines.
There are no record lists for swimming under water. Underwater swimming is known
to be faster than above-water breast stroke, backstroke or dolphin stroke: that is the
reason that swimming underwater over long distances is forbidden in competitions in

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
these styles. However, it is not known whether crawl-style records are faster or slower
than records for the fastest swimming style below water. Which one is faster in your own
Challenge 599 e case?
∗∗
How much water is necessary to moisten the air in a room in winter? At 0°C, the vapour
pressure of water is 6 mbar, 20°C it is 23 mbar. As a result, heating air in the winter gives
Challenge 600 e at most a humidity of 25 %. To increase the humidity by 50 %, about 1 litre of water per
100 m3 is needed.
∗∗
Surface tension can be dangerous. A man coming out of a swimming pool is wet. He
carries about half a kilogram of water on his skin. In contrast, a wet insect, such as a
house fly, carries many times its own weight. It is unable to fly and usually dies. Therefore,
most insects stay away from water as much as they can – or at least use a long proboscis.
∗∗
The human heart pumps blood at a rate of about 0.1 l/s. A typical capillary has the dia-
meter of a red blood cell, around 7 μm, and in it, the blood moves at a speed of half a
Challenge 601 s millimetre per second. How many capillaries are there in a human?
12 fluids and their motion 373

∗∗
You are in a boat on a pond with a stone, a bucket of water and a piece of wood. What
happens to the water level of the pond after you throw the stone in it? After you throw
Challenge 602 s the water into the pond? After you throw the piece of wood?
∗∗
Challenge 603 s A ship leaves a river and enters the sea. What happens?
∗∗
Put a rubber air balloon over the end of a bottle and let it hang inside the bottle. How
Challenge 604 e much can you blow up the balloon inside the bottle?
∗∗
Put a rubber helium balloon in your car. You accelerate and drive around bends. In which
Challenge 605 s direction does the balloon move?

Motion Mountain – The Adventure of Physics
∗∗
Put a small paper ball into the neck of a horizontal bottle and try to blow it into the
Challenge 606 e bottle. The paper will fly towards you. Why?
∗∗
It is possible to blow an egg from one egg cup to a second one just behind it. Can you
Challenge 607 e perform this trick?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In the seventeenth century, engineers who needed to pump water faced a challenge. To
pump water from mine shafts to the surface, no water pump managed more than 10 m
of height difference. For twice that height, one always needed two pumps in series, con-
Challenge 608 s nected by an intermediate reservoir. Why? How then do trees manage to pump water
upwards for larger heights?
∗∗
When hydrogen and oxygen are combined to form water, the amount of hydrogen
needed is exactly twice the amount of oxygen, if no gas is to be left over after the re-
Challenge 609 s action. How does this observation confirm the existence of atoms?
∗∗
Challenge 610 s How are alcohol-filled chocolate pralines made? Note that the alcohol is not injected into
them afterwards, because there would be no way to keep the result tight enough.
∗∗
How often can a stone jump when it is thrown over the surface of water? The present
world record was achieved in 2002: 40 jumps. More information is known about the pre-
Ref. 280 vious world record, achieved in 1992: a palm-sized, triangular and flat stone was thrown
with a speed of 12 m/s (others say 20 m/s) and a rotation speed of about 14 revolutions
per second along a river, covering about 100 m with 38 jumps. (The sequence was filmed
374 12 fluids and their motion

stone

water
F I G U R E 264 What is your
personal stone-skipping
record?

water
jet

Motion Mountain – The Adventure of Physics
F I G U R E 265 Heron’s fountain in operation.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
with a video recorder from a bridge.)
What would be necessary to increase the number of jumps? Can you build a machine
Challenge 611 r that is a better thrower than yourself?
∗∗
The most abundant component of air is nitrogen (about 78 %). The second component
Challenge 612 s is oxygen (about 21 %). What is the third one?
∗∗
Which everyday system has a pressure lower than that of the atmosphere and usually
Challenge 613 s kills a person if the pressure is raised to the usual atmospheric value?
∗∗
Water can flow uphill: Heron’s fountain shows this most clearly. Heron of Alexandria
(c. 10 to c. 70) described it 2000 years ago; it is easily built at home, using some plastic
Challenge 614 s bottles and a little tubing. How does it work? How is it started?
∗∗
12 fluids and their motion 375

A light bulb is placed, underwater, in a stable steel cylinder with a diameter of 16 cm.
An original Fiat Cinquecento car (500 kg) is placed on a piston pushing onto the water
Challenge 615 s surface. Will the bulb resist?
∗∗
Challenge 616 s What is the most dense gas? The most dense vapour?
∗∗
Every year, the Institute of Maritime Systems of the University of Rostock organizes a
contest. The challenge is to build a paper boat with the highest carrying capacity. The
paper boat must weigh at most 10 g and fulfil a few additional conditions; the carrying
capacity is measured by pouring small lead shot onto it, until the boat sinks. The 2008
Challenge 617 e record stands at 5.1 kg. Can you achieve this value? (For more information, see the www.
paperboat.de website.)
∗∗

Motion Mountain – The Adventure of Physics
Challenge 618 s Is it possible to use the wind to move against the wind, head-on?
∗∗
Measuring wind speed is an important task. Two methods allow measuring the wind
speed at an altitude of about 100 m above the ground: sodar, i.e., sound detection and
ranging, and lidar, i.e., light detection and ranging. Two typical devices are shown in
Figure 266. Sodar works also for clear air, whereas lidar needs aerosols.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
A modern version of an old question – originally posed by the physicist Daniel Colladon
(b. 1802 Geneva, d. 1893 Geneva) – is the following version by Yakov Perelman. A ship
of mass 𝑚 in a river is pulled by horses walking along the river bank attached by ropes.
If the river is of superfluid helium, meaning that there is no friction between ship and
river, what energy is necessary to pull the ship upstream along the river until a height ℎ
Challenge 619 s has been gained?
∗∗
An urban legend pretends that at the bottom of large waterfalls there is not enough air
Challenge 620 e to breathe. Why is this wrong?
∗∗
The Swiss physicist and inventor Auguste Piccard (b. 1884 Basel, d. 1962 Lausanne) was
a famous explorer. Among others, he explored the stratosphere: he reached the record
height of 16 km in his aerostat, a hydrogen gas balloon. Inside the airtight cabin hanging
under his balloon, he had normal air pressure. However, he needed to introduce several
ropes attached at the balloon into the cabin, in order to be able to pull and release them,
as they controlled his balloon. How did he get the ropes into the cabin while at the same
Challenge 621 s time preventing air from leaving?
∗∗
376 12 fluids and their motion

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 266 Two wind
measuring systems: a sodar
system and a lidar system
(© AQSystems, Leosphere).

A human in air falls with a limiting speed of about 50 m/s (the precise value depends
on clothing). How long does it take to fall from a plane at 3000 m down to a height of
Challenge 622 e 200 m?
∗∗
To get an idea of the size of Avogadro’s and Loschmidt’s number, two questions are usu-
ally asked. First, on average, how many molecules or atoms that you breathe in with every
breath have previously been exhaled by Caesar? Second, on average, how many atoms of
Challenge 623 s Jesus do you eat every day? Even though the Earth is large, the resulting numbers are still
telling.
∗∗
A few drops of tea usually flow along the underside of the spout of a teapot (or fall
onto the table). This phenomenon has even been simulated using supercomputer simula-
12 fluids and their motion 377

140°C 210°C

Motion Mountain – The Adventure of Physics
F I G U R E 267 A water droplet on a pan: an example of the Leidenfrost effect (© Kenji Lopez-Alt).

Ref. 281 tions of the motion of liquids, by Kistler and Scriven, using the Navier–Stokes equations.
Teapots are still shedding drops, though.
∗∗
The best giant soap bubbles can be made by mixing 1.5 l of water, 200 ml of corn syrup

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
and 450 ml of washing-up liquid. Mix everything together and then let it rest for four
hours. You can then make the largest bubbles by dipping a metal ring of up to 100 mm
Challenge 624 s diameter into the mixture. But why do soap bubbles burst?
∗∗
A drop of water that falls into a pan containing moderately hot oil evaporates immedi-
ately. However, if the oil is really hot, i.e., above 210°C, the water droplet dances on the oil
surface for a considerable time. Cooks test the temperature of oil in this way. Why does
Challenge 625 s this so-called Leidenfrost effect take place? The effect is named after the theologian and
physician Johann Gottlob Leidenfrost (b. 1715 Rosperwenda, d. 1794 Duisburg). For an
instructive and impressive demonstration of the Leidenfrost effect with water droplets,
see the video featured at www.thisiscolossal.com/2014/03/water-maze/. The video also
shows water droplets running uphill and running through a maze.
The Leidenfrost effect also allows one to plunge the bare hand into molten lead or
liquid nitrogen, to keep liquid nitrogen in one’s mouth, to check whether a pressing iron
is hot, or to walk over hot coal – if one follows several safety rules, as explained by Jearl
Ref. 282 Walker. (Do not try this yourself! Many things can go wrong.) The main condition is
that the hand, the mouth or the feet must be wet. Walker lost two teeth in a demonstra-
tion and badly burned his feet in a walk when the condition was not met. You can see
some videos of the effect for a hand in liquid nitrogen on www.popsci.com/diy/article/
2010-08/cool-hand-theo and for a finger in molten lead on www.popsci.com/science/
378 12 fluids and their motion

F I G U R E 268 Which funnel empties more rapidly?

article/2012-02/our-columnist-tests-his-trust-science-dipping-his-finger-molten-lead.

Motion Mountain – The Adventure of Physics
∗∗
Challenge 626 s Why don’t air molecules fall towards the bottom of the container and stay there?
∗∗
Challenge 627 s Which of the two water funnels in Figure 268 is emptied more rapidly? Apply energy
Ref. 283 conservation to the fluid’s motion (the Bernoulli equation) to find the answer.
∗∗
As we have seen, fast flow generates an underpressure. How do fish prevent their eyes

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 628 s from popping when they swim rapidly?
∗∗
Golf balls have dimples for the same reasons that tennis balls are hairy and that shark
and dolphin skin is not flat: deviations from flatness reduce the flow resistance because
Challenge 629 ny many small eddies produce less friction than a few large ones. Why?
∗∗
The recognized record height reached by a helicopter is 12 442 m above sea level, though
12 954 m has also been claimed. (The first height was reached in 1972, the second in 2002,
both by French pilots in French helicopters.) Why, then, do people still continue to use
their legs in order to reach the top of Mount Sagarmatha, the highest mountain in the
Challenge 630 s world?
∗∗
A loosely knotted sewing thread lies on the surface of a bowl filled with water. Putting
a bit of washing-up liquid into the area surrounded by the thread makes it immediately
Challenge 631 e become circular. Why?
∗∗
Challenge 632 s How can you put a handkerchief under water using a glass, while keeping it dry?
12 fluids and their motion 379

∗∗
Are you able to blow a ping-pong ball out of a funnel? What happens if you blow through
a funnel towards a burning candle?
∗∗
The fall of a leaf, with its complex path, is still a topic of investigation. We are far from
being able to predict the time a leaf will take to reach the ground; the motion of the air
around a leaf is not easy to describe. One of the simplest phenomena of hydrodynamics
remains one of its most difficult problems.
∗∗
Fluids exhibit many interesting effects. Soap bubbles in air are made of a thin spherical
film of liquid with air on both sides. In 1932, anti-bubbles, thin spherical films of air
Ref. 284 with liquid on both sides, were first observed. In 2004, the Belgian physicist Stéphane
Dorbolo and his team showed that it is possible to produce them in simple experiments,
and in particular, in Belgian beer.

Motion Mountain – The Adventure of Physics
∗∗
Have you ever dropped a Mentos candy into a Diet Coca Cola bottle? You will get an
Challenge 633 e interesting effect. (Do it at your own risk...) Is it possible to build a rocket in this way?
∗∗
Challenge 634 e A needle can swim on water, if you put it there carefully. Just try, using a fork. Why does
it float?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
3
The Rhine emits about 2 300 m /s of water into the North Sea, the Amazon River about
Challenge 635 e 120 000 m3 /s into the Atlantic. How much is this less than 𝑐3 /4𝐺?
∗∗
Fluids exhibit many complex motions. To see an overview, have a look at the beautiful
Challenge 636 e collection on the website serve.me.nus.edu.sg/limtt. Among fluid motion, vortex rings, as
emitted by smokers or volcanoes, have often triggered the imagination. (See Figure 269.)
One of the most famous examples of fluid motion is the leapfrogging of vortex rings,
shown in Figure 270. Lim Tee Tai explains that more than two leapfrogs are extremely
hard to achieve, because the slightest vortex ring misalignment leads to the collapse of
Ref. 285 the system.
∗∗
A surprising effect can be observed when pouring shampoo on a plate: sometimes a thin
stream is ejected from the region where the shampoo hits the plate. This so-called Kaye
effect is best enjoyed in the beautiful movie produced by the University of Twente found
on the youtube.com/watch?v=GX4_3cV_3Mw website.
∗∗
Challenge 637 e Most mammals take around 30 seconds to urinate. Can you find out why?
380 12 fluids and their motion

F I G U R E 269 A smoke ring, around 100 m in size, ejected

Motion Mountain – The Adventure of Physics
from Mt. Etna’s Bocca Nova in 2000 (© Daniela Szczepanski at
www.vulkanarchiv.de and www.vulkane.net).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 270 Two leapfrogging vortex rings. (QuickTime ﬁlm © Lim Tee Tai)

∗∗
Aeroplanes toilets are dangerous places. In the 1990s, a fat person sat on the toilet seat
and pushed the ‘flush’ button while sitting. (Never try this yourself.) The underpressure
exerted by the toilet was so strong that it pulled out the intestine and the person had to
be brought into hospital. (Everything ended well, by the way.)
∗∗
If one surrounds water droplets with the correct type of dust, the droplets can roll along
inclined planes. They can roll with a speed of up to 1 m/s, whereas on the same surface,
water would flow a hundred times more slowly. When the droplets get too fast, they
become flat discs; at even higher speeds, they get a doughnut shape. Such droplets can
Ref. 286 even jump and swim.
12 fluids and their motion 381

F I G U R E 271 Water droplets covered with pollen rolling over inclined planes at 35 degrees (inclination
not shown) have unexpected shapes. The grey lines have a length of 1 cm; the spherical droplets had
initial radii of 1.3 mm and 2.5 mm, respectively, and the photographs were taken at 9 ms and 23 ms time
intervals (© David Quéré).

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 272 A computer visualization of the pressure regions surrounding a racing bicycle and a
motorcycle following it (© Technical University Eindhoven).

∗∗
It is well known that it is easier to ride a bicycle behind a truck, a car or a motorcycle,
because wind drag is lower in such situations. Indeed, the cycling speed record, achieved
by Fred Rompelberg in 1990, is 74.7 m/s, or 268.8 km/h, and was achieved with a car
Ref. 298 driving in front of the bicycle. It is more surprising that a motorcycle driving behind a
bicycle also reduces the drag, up to nine per cent. In 2016, Bert Blocken and his team
confirmed this effect in the wind tunnel and with numeric simulations. The effect is thus
sufficient to determine the winner of racing prologues or time trials. In fact, three mo-
torcycles riding in a row behind the bicycle can reduce the drag for the bicycle riding in
front by up to 14 per cent. The effect occurs because the following motorcycles cause a
reduction of the low-pressure region – shown in red in Figure 272 – behind the rider;
382 12 fluids and their motion

this reduces the aerodynamic drag for the bicycle rider.
∗∗
All fluids can sustain vortices. Forexample, in air there are cyclones, tornados and sea
hoses. But vortices also exist in the sea. They have a size between 100 m and 10 km; their
short life last from a few hours up to a day. Sea vortices mix the water layers in the sea
and they transport heat, nutrients and seaweed. Research on their properties and effects
is just at its beginning.

Summary on fluids
The motion of a fluid is the motion of its constituent particles. The motion of fluids al-
lows for swimming, flying, breathing, blood circulation, vortices and turbulence. Fluid
motion can be laminar or turbulent. Laminar flow that lacks any internal friction is de-
scribed by Bernoulli’s equation, i.e., by energy conservation. Laminar flow with internal
friction is beyond the scope of this text; so is turbulent flow. The exact description of tur-

Motion Mountain – The Adventure of Physics
bulent fluid motion is the most complicated problem of physics and not yet fully solved.

ON H E AT A N D MOT ION R EV E R S A L
I N VA R IA NC E

S
pilled milk never returns into its container by itself. Any hot object, left alone,
tarts to cool down with time; it never heats up. These and many other observations
how that numerous processes in nature are irreversible. On the other hand, for every
a stone that flies along a path, it is possible to throw a stone that follows the path in the
reverse direction. The motion of stones is reversible. Further observations show that irre-

Motion Mountain – The Adventure of Physics
versibility is only found in systems composed of a many particles, and that all irreversible
systems involve heat.
Our everyday experience, including human warmth, ovens and stoves, shows that heat
flows. Heat moves. But the lack of reversibility also shows that heat moves in a special
way. Since heat appears in many-particle systems, we are led to explore the next global
approach for the description of motion: statistical physics. Statistical physics, which in-
cludes thermodynamics, the study of heat and temperature, explains the origin of many
material properties, and also the observed irreversibility of heat flow.
Does irreversibility mean that motion, at a fundamental level, is not invariant under
reversal, as Nobel Prize winner Ilya Prigogine, one of the fathers of self-organization,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
thought? In this chapter, we show that despite his other achievements, he was wrong.**
To deduce this result, we first need to know the basic facts about temperature and heat;
Ref. 287 then we discuss irreversibility and motion reversal.

Temperature
Macroscopic bodies, i.e., bodies made of many atoms, have temperature. Only bodies
made of few atoms do not have a temperature. Ovens have high temperature, and refri-
gerators low temperature. Temperature changes have important effects: matter changes
from solid to liquid to gaseous to plasma state. With a change in temperature, matter also
changes size, colour, magnetic properties, stiffness and many other properties.
Temperature is an aspect of the state of a body. In other words, two otherwise identical
bodies can be characterized and distinguished by their temperature. This is well-known
to criminal organizations around the world that rig lotteries. When a blind-folded child
is asked to draw a numbered ball from a set of such balls, such as in Figure 274, it is often
told beforehand to draw only hot or cold balls. The blindfolding also helps to hide the

** Many even less serious thinkers often ask the question in the following term: is motion time-invariant?
The cheap press goes even further, and asks whether motion has an ‘arrow’ or whether time has a preferred
Page 48 ‘direction of flow’. We have already shown above that this is nonsense. We steer clear of such phrases in the
following.
384 13 on heat and motion reversal invariance

F I G U R E 273 Braking generates heat on the ﬂoor and in the tire (© Klaus-Peter Möllmann and Michael
Vollmer).

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 274 A rigged lottery shows that temperature is
an aspect of the state of a body (© ISTA).

Ref. 289 tears due to the pain.
The temperature of a macroscopic body is an aspect of its state. In particular, the tem-
perature is an intensive quantity or variable. In short, temperature describes the intensity
of heat flow. An overview of measured temperature values is given in Table 46.
We observe that any two bodies in contact tend towards the same temperature: tem-
perature is contagious. In other words, temperature describes a situation of equilibrium.
Temperature thus behaves like pressure and any other intensive variable: it is the same for
all parts of a system. We call heating the increase of temperature, and cooling its decrease.
How is temperature measured? The eighteenth century produced the clearest answer:
temperature is best defined and measured by the expansion of gases. For the simplest, so-
called ideal gases, the product of pressure 𝑝 and volume 𝑉 is proportional to temperature:

𝑝𝑉 ∼ 𝑇 . (112)

The proportionality constant is fixed by the amount of gas used. (More about it shortly.)
The ideal gas relation allows us to determine temperature by measuring pressure and
13 on heat and motion reversal invariance 385

volume. This is the way (absolute) temperature has been defined and measured for about
Ref. 288 a century. To define the unit of temperature, we only have to fix the amount of gas used.
Page 453 It is customary to fix the amount of gas at 1 mol; for example, for oxygen this is 32 g. The
proportionality constant for 1 mol, called the ideal gas constant 𝑅, is defined to be

𝑅 = 8.314 462 618 J/(mol K) . (113)

This numerical value has been chosen in order to yield the best approximation to the
independently defined Celsius temperature scale. Fixing the ideal gas constant in this
way defines 1 K, or one Kelvin, as the unit of temperature.
In general, if we need to determine the temperature of an object, we thus need to take
a mole of an ideal gas, put it in contact with the object, wait a while, and then measure the
pressure and the volume of the gas. The ideal gas relation (112) then gives the temperature
of the object.
For every substance, a characteristic material property is the triple point. The triple
point is the only temperature at which the solid, liquid, and gaseous phase coexists. For

Motion Mountain – The Adventure of Physics
water, it lies at 0.01°C (and at a partial vapour pressure of 611.657 Pa). Using the triple
point of water, we can define the Kelvin in simple terms: a temperature increase of one
Kelvin is the temperature increase that makes an ideal gas at the triple point of water
Challenge 638 e increase in volume, keeping the pressure fixed, by a fraction of 1/273.16 or 0.366 09 %.
Most importantly, the ideal gas relation shows that there is a lowest temperature in
nature, namely that temperature at which an ideal gas would have a vanishing volume.
For an ideal gas, this would happen at 𝑇 = 0 K. In reality, other effects, like the volume of
the atoms themselves, prevent the volume of a real gas from ever reaching zero exactly.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
⊳ The unattainability of zero temperature is called the third principle of ther-
modynamics.

In fact, the temperature values achieved by a civilization can be used as a measure of
its technological achievements. We can define in this way the Bronze Age (1.1 kK, 3500
b ce), the Iron Age (1.8 kK, 1000 b ce), the Electric Age (3 kK from c. 1880) and the Atomic
Age (several MK, from 1944) in this way. Taking into account also the quest for lower
Ref. 290 temperatures, we can define the Quantum Age (4 K, starting 1908). All these thoughts
lead to a simple question: what exactly are heating and cooling? What happens in these
processes?

TA B L E 46 Some temperature values.

O b s e r va t i o n Te m p e r at u r e

Lowest,but unattainable, temperature 0 K = −273.15°C
In the context of lasers, it can make (almost) sense to talk about negative temperature.
Temperature a perfect vacuum would have at Earth’s surface 40 zK
Vol. V, Page 146
Sodium gas in certain laboratory experiments – coldest mat- 0.45 nK
ter system achieved by man and possibly in the universe
Temperature of neutrino background in the universe c. 2 K
386 13 on heat and motion reversal invariance

TA B L E 46 (Continued) Some temperature values.

O b s e r va t i o n Te m p e r at u r e

Temperature of (photon) cosmic background radiation in the 2.7 K
universe
Liquid helium 4.2 K
Oxygen triple point 54.3584 K
Liquid nitrogen 77 K
Coldest weather ever measured (Antarctic) 185 K = −88°C
Freezing point of water at standard pressure 273.15 K = 0.00°C
Triple point of water 273.16 K = 0.01°C
Average temperature of the Earth’s surface 287.2 K
Smallest uncomfortable skin temperature 316 K (10 K above normal)
Interior of human body 310.0 ± 0.5 K = 36.8 ± 0.5°C
Temperature of most land mammals 310 ± 3 K = 36.8 ± 2°C

Motion Mountain – The Adventure of Physics
Hottest weather ever measured 343.8 K = 70.7°C
Boiling point of water at standard pressure 373.13 K or 99.975°C
Temperature of hottest living things: thermophile bacteria 395 K = 122°C
Large wood fire, liquid bronze c. 1100 K
Freezing point of gold 1337.33 K
Liquid, pure iron 1810 K
Bunsen burner flame up to 1870 K
Light bulb filament 2.9 kK
Melting point of hafnium carbide 4.16 kK

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Earth’s centre 5(1) kK
Sun’s surface 5.8 kK
Air in lightning bolt 30 kK
Hottest star’s surface (centre of NGC 2240) 250 kK
Space between Earth and Moon (no typo) up to 1 MK
Centre of white dwarf 5 to 20 MK
Sun’s centre 20 MK
Centre of the accretion disc in X-ray binary stars 10 to 100 MK
Inside the JET fusion tokamak 100 MK
Centre of hottest stars 1 GK
Maximum temperature of systems without electron–positron ca. 6 GK
pair generation
Universe when it was 1 s old 100 GK
Hagedorn temperature 1.9 TK
Heavy ion collisions – highest man-made value up to 3.6 TK
Planck temperature – nature’s upper temperature limit 1032 K
13 on heat and motion reversal invariance 387

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 275 Thermometers: a Galilean thermometer (left), the row of infrared sensors in the jaw of the
emerald tree boa Corallus caninus, an infrared thermometer to measure body temperature in the ear, a
nautical thermometer using a bimetal, a mercury thermometer, and a thermocouple that is attached to
a voltmeter for read-out (© Wikimedia, Ron Marcus, Braun GmbH, Universum, Wikimedia,
Thermodevices).

Thermal energy
Around us, friction slows down moving bodies, and, while doing so, heats them up. The
‘creation’ of heat by friction can be observed in many experiments. An example is shown
Ref. 291 in Figure 273. Such experiments show that heat can be generated from friction, just by
continuous rubbing, without any limit. This endless ‘creation’ of heat implies that heat
is neither a material fluid nor a substance extracted from the body – which in this case
would be consumed after a certain time – but something else. Indeed, today we know that
heat, even though it behaves in some ways like a fluid, is due to the disordered motion
of particles. The conclusion from all these explorations is:
388 13 on heat and motion reversal invariance

⊳ Friction is the transformation of mechanical (i.e., ordered) energy into (dis-
ordered) thermal energy, i.e., into the disordered motion of the particles
making up a material.

Heating and cooling is thus the flow of disordered energy. In order to increase the tem-
perature of 1 kg of water by 1 K using friction, 4.2 kJ of mechanical energy must be sup-
plied. The first to measure this quantity with precision was, in 1842, the physician Ju-
lius Robert Mayer (b. 1814 Heilbronn, d. 1878 Heilbronn). He described his experiments
as proofs of the conservation of energy; indeed, he was the first person to state energy
conservation! It is something of an embarrassment to modern physics that a medical
doctor was the first to show the conservation of energy, and furthermore, that he was
ridiculed by most physicists of his time. Worse, conservation of energy was accepted by
scientists only when it was publicized many years later by two authorities: Hermann von
Helmholtz – himself also a physician turned physicist – and William Thomson, who also
cited similar, but later experiments by James Joule.* All of them acknowledged Mayer’s
priority. Marketing by William Thomson eventually led to the naming of the unit of en-

Motion Mountain – The Adventure of Physics
ergy after Joule. In summary, two medical doctors proved to all experts on motion:

⊳ In a closed system, the sum of mechanical energy and thermal energy is
constant. This is called the first principle of thermodynamics.

Equivalently, it is impossible to produce mechanical energy without paying for it with
some other form of energy. This is an important statement because among others it
means that humanity will stop living one day. Indeed, we live mostly on energy from
the Sun; since the Sun is of finite size, its energy content will eventually be consumed.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 639 s Can you estimate when this will happen?
Page 110 The first principle of thermodynamics, the conservation of energy, implies:

⊳ There is no perpetuum mobile ‘of the first kind’.

In particular, no machine can run without energy input. For this very reason, we need
food to eat: the energy in the food keeps us alive. If we stop eating, we die. The conserva-
tion of energy also makes most so-called ‘wonders’ impossible: in nature, energy cannot
be created, but is conserved.
Thermal energy is a form of energy. Thermal energy can be stored, accumulated,
transferred, transformed into mechanical energy, electrical energy or light. In short,
thermal energy can be transformed into motion, into work, and thus into money.
The first principle of thermodynamics also allows us to formulate what a car engine
achieves. Car engines are devices that transform hot matter – the hot exploding fuel
inside the cylinders – into motion of the car wheels. Car engines, like steam engines, are
thus examples of heat engines.

* Hermann von Helmholtz (b. 1821 Potsdam, d. 1894 Berlin), important scientist. William Thomson-Kelvin
(b. 1824 Belfast, d. 1907 Netherhall), important physicist. James Prescott Joule (b. 1818 Salford, d. 1889 Sale),
physicist. Joule is pronounced so that it rhymes with ‘cool’, as his descendants like to stress. (The pronun-
ciation of the name ‘Joule’ varies from family to family.)
13 on heat and motion reversal invariance 389

F I G U R E 276 Which balloon wins when the tap is
opened? Note: this is also how aneurysms grow in
arteries.

The study of heat and temperature is called thermostatics if the systems concerned
are at equilibrium, and thermodynamics if they are not. In the latter case, we distinguish
situations near equilibrium, when equilibrium concepts such as temperature can still be
Page 415 used, from situations far from equilibrium, where such concepts often cannot be applied.
Does it make sense to distinguish between thermal energy and heat? It does. Many
older texts use the term ‘heat’ to mean the same as thermal energy. However, this is
confusing to students and experts:

Motion Mountain – The Adventure of Physics
⊳ In this text, ‘heat’ is used, in accordance with modern approaches, as the
everyday term for entropy.

Both thermal energy and heat flow from one body to another, and both accumulate. Both
have no measurable mass.* Both the amount of thermal energy and the amount of heat
inside a body increase with increasing temperature. The precise relation will be given
shortly. But heat has many other interesting properties and stories to tell. Of these, two
are particularly important: first, heat is due to particles; and second, heat is at the heart

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
of the difference between past and future. These two stories are intertwined.

Why d o ballo ons take up space? – The end of continuit y
Heat properties are material-dependent. Studying thermal properties therefore should
enable us to understand something about the constituents of matter. Now, the simplest
materials of all are gases.** Gases need space: any amount of gas has pressure and volume.
It did not take a long time to show that gases could not be continuous. One of the first
scientists to think about gases as made up of atoms or molecules was Daniel Bernoulli.
Bernoulli reasoned that if gases are made up of small particles, with mass and mo-
mentum, he should be able to make quantitative predictions about the behaviour of
gases, and check them with experiments. If the particles fly around in a gas, then the
pressure of a gas in a container is produced by the steady flow of particles hitting the
wall. Bernoulli understood that if he reduced the volume to one-half, the particles in
the gas would need only to travel half as long to hit a wall: thus the pressure of the gas

* This might change in future, when mass measurements improve in precision, thus allowing the detection
Vol. II, page 71 of relativistic effects. In this case, temperature increase may be detected through its related mass increase.
However, such changes are noticeable only with twelve or more digits of precision in mass measurements.
** By the way, the word gas is a modern construct. It was coined by the alchemist and physician Jo-
han Baptista van Helmont (b. 1579 Brussels, d. 1644 Vilvoorde), to sound similar to ‘chaos’. It is one of
the few words which have been invented by one person and then adopted all over the world.
390 13 on heat and motion reversal invariance

F I G U R E 277 What happened here? (© Johan de Jong)

Motion Mountain – The Adventure of Physics
would double. He also understood that if the temperature of a gas is increased while its
volume is kept constant, the speed of the particles would increase. Combining these res-
ults, Bernoulli concluded that if the particles are assumed to behave as tiny, hard and
perfectly elastic balls, the pressure 𝑝, the volume 𝑉 and the temperature 𝑇 must be re-
Challenge 640 s lated by
𝑝𝑉 = 𝑛𝑅𝑇 = 𝑘𝑁𝑇 . (114)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In this so-called ideal gas relation, 𝑅 is the ideal gas constant; it has the value 𝑅 =
8.314 462 618 J/(mol K) and 𝑛 is the number of moles. In the alternative writing of
the gas relation, 𝑁 is the number of particles contained in the gas and 𝑘 is the
Boltzmann constant, one of the fundamental constants of nature, is given by 𝑘 =
Page 392 1.380 648 52(79) ⋅ 10−23 J/K. (More about this constant is told below.) A gas made of
particles with such a textbook behaviour is called an ideal gas. Relation (114), often also
called the ideal gas ‘law’, was known before Bernoulli; the relation has been confirmed
by experiments at room and higher temperatures, for all known gases.
Bernoulli derived the ideal gas relation, with a specific prediction for the proportion-
ality constant 𝑅, from the single assumption that gases are made of small particles with
mass. This derivation provides a clear argument for the existence of atoms and for their
behaviour as normal, though small objects. And indeed, we have already seen above how
Page 340 𝑁 can be determined experimentally.
The ideal gas model helps us to answer questions such as the one illustrated in Fig-
ure 276. Two identical rubber balloons, one filled up to a larger size than the other, are
Challenge 641 s connected via a pipe and a valve. The valve is opened. Which one deflates?
The ideal gas relation states that hotter gases, at given pressure, need more volume.
Challenge 642 e The relation thus explains why winds and storms exist, why hot air balloons rise – even
those of Figure 277 – why car engines work, why the ozone layer is destroyed by certain
gases, or why during the extremely hot summer of 2001 in the south of Turkey, oxygen
masks were necessary to walk outside during the day.
13 on heat and motion reversal invariance 391

The ideal gas relation also explains why on the 21st of August 1986, over a thousand
people and three-thousand livestock were found dead in their homes in Cameroon. They
were living below a volcano whose crater contains a lake, Lake Nyos. It turns out that the
volcano continuously emits carbon dioxide, or CO2 , into the lake. The carbon dioxide
is usually dissolved in the water. But in August 1986, an unknown event triggered the
release of a bubble of around one million tons of CO2 , about a cubic kilometre, into the
atmosphere. Because carbon dioxide (2.0 kg/m3 ) is denser than air (1.2 kg/m3 ), the gas
flowed down into the valleys and villages below the volcano. The gas has no colour and
smell, and it leads to asphyxiation. It is unclear whether the outgassing system installed
in the lake after the event is sufficiently powerful to avoid a recurrence of the event.
Using the ideal gas relation you are now able to explain why balloons increase in size
as they rise high up in the atmosphere, even though the air is colder there. The largest
balloon built so far had a diameter, at high altitude, of 170 m, but only a fraction of that
Challenge 643 ny value at take-off. How much?
Now you can also take up the following challenge: how can you measure the weight
Challenge 644 s of a car or a bicycle with a ruler only?

Motion Mountain – The Adventure of Physics
The picture of gases as being made of hard constituents without any long-distance
interactions breaks down at very low temperatures. However, the ideal gas relation (114)
Ref. 292 can be improved to overcome these limitations, by taking into account the deviations due
to interactions between atoms or molecules. This approach is now standard practice and
Ref. 293 allows us to measure temperatures even at extremely low values. The effects observed
below 80 K, such as the solidification of air, frictionless transport of electrical current, or
Vol. V, page 108 frictionless flow of liquids, form a fascinating world of their own, the beautiful domain
Vol. V, page 103 of low-temperature physics. The field will be explored later on.
Not long after Bernoulli, chemists found strong arguments confirming the existence

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
of atoms. They discovered that chemical reactions occur under ‘fixed proportions’: only
specific ratios of amounts of chemicals react. Many researchers, including John Dalton,
deduced that this property occurs because in chemistry, all reactions occur atom by atom.
For example, two hydrogen atoms and one oxygen atoms form one water molecule in this
way – even though these terms did not exist at the time. The relation is expressed by the
chemical formula H2 O. These arguments are strong, but did not convince everybody.
Finally, the existence of atoms was confirmed by observing the effects of their motion
even more directly.

Brownian motion
If fluids are made of particles moving randomly, this random motion should have ob-
servable effects. An example of the observed motion is shown in Figure 280. The particles
seem to follow a random zig-zag movement. The first description is by Lucretius, in the
year 60 b ce, in his poem De rerum natura. In it, Lucretius tells about a common obser-
vation: in air that is illuminated by the Sun, dust particles seem to dance.
In 1785, Jan Ingenhousz saw that coal dust particles never come to rest. Indeed, un-
der a microscope it is easy to observe that coal dust or other small particles in or on a
liquid never come to rest. Ingenhousz discovered what is called Brownian motion today.
40 years after him, the botanist Robert Brown was the first Englishman to repeat the
observation, this time for small particles floating in vacuoles inside pollen. Further ex-
392 13 on heat and motion reversal invariance

Motion Mountain – The Adventure of Physics
Vol. III, page 183 F I G U R E 278 An image of pollen grains – ﬁeld size about 0.3 mm – made with an electron microscope
(Dartmouth College Electron Microscope Facility).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
periments by many other researchers showed that the observation of a random motion is
independent of the type of particle and of the type of liquid. In other words, Ingenhousz
had discovered a fundamental form of noise in nature.
Around 1860, the random motion of particles in liquids was attributed by various
researchers to the molecules of the liquid that were colliding with the particles. In 1905
Ref. 294 and 1906, Marian von Smoluchowski and, independently, Albert Einstein argued that this
attribution could be tested experimentally, even though at that time nobody was able to
observe molecules directly. The test makes use of the specific properties of thermal noise.
It had already been clear for a long time that if molecules, i.e., indivisible matter
particles, really existed, then thermal energy had to be disordered motion of these con-
stituents and temperature had to be the average energy per degree of freedom of the con-
Page 393 stituents. Bernoulli’s model of Figure 279 implies that for monatomic gases the kinetic
Challenge 645 ny energy 𝑇kin per particle is given by

𝑇kin = 32 𝑘𝑇 (115)

where 𝑇 is temperature. The so-called Boltzmann constant 𝑘 = 1.4 ⋅ 10−23 J/K is the
standard conversion factor between temperature and energy.* At a room temperature

* The Boltzmann constant 𝑘 was discovered and named by Max Planck, in the same work in which he also
13 on heat and motion reversal invariance 393

F I G U R E 279 The basic idea of
statistical mechanics about
gases: gases are systems of
moving particles, and pressure
is due to their collisions with
the container.

probability density evolution

Motion Mountain – The Adventure of Physics
F I G U R E 280 Example paths for particles in Brownian motion and their displacement distribution.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
of 293 K, the kinetic energy of a particle is thus 6 zJ.
If you use relation (115) to calculate the speed of air molecules at room temperature,
Challenge 646 e you get values of several hundred metres per second, about the speed of sound! Given
this large speed, why does smoke from a candle take so long to diffuse through a room
that has no air currents? Rudolph Clausius (b. 1822 Köslin, d. 1888 Bonn) answered this
question in the mid-nineteenth century: smoke diffusion is slowed by the collisions with
air molecules, in the same way as pollen particles collide with molecules in liquids. Since
flows are usually more effective than diffusion, the materials that show no flows at all
are those where the importance of diffusion is most evident: solids. Metal hardening and
semiconductor production are examples.
The description of Brownian motion can be tested by following the displacement of
pollen particles under the microscope. At first sight, we might guess that the average

discovered what is now called Planck’s constant ℏ, the quantum of action. For more details on Max Planck,
Vol. III, page 149 see later on.
Planck named the Boltzmann constant after the important physicist Ludwig Boltzmann (b. 1844 Vienna,
d. 1906 Duino), who is most famous for his work on thermodynamics. Boltzmann explained all thermo-
dynamic phenomena and observables, above all entropy itself, as results of the behaviour of molecules. It
seems that Boltzmann committed suicide partly because of the animosities of his fellow physicists towards
his ideas and himself. Nowadays, his work is standard textbook material.
394 13 on heat and motion reversal invariance

distance the pollen particle has moved after 𝑛 collisions should be zero, because the mo-
lecule velocities are random. However, this is wrong, as experiment shows.
An increasing average square displacement, written ⟨𝑑2 ⟩, is observed for the pollen
particle. It cannot be predicted in which direction the particle will move, but it does
move. If the distance the particle moves after one collision is 𝑙, the average square dis-
Challenge 647 ny placement after 𝑛 collisions is given, as you should be able to show yourself, by

⟨𝑑2 ⟩ = 𝑛𝑙2 . (116)

For molecules with an average velocity 𝑣 over time 𝑡 this gives

⟨𝑑2 ⟩ = 𝑛𝑙2 = 𝑣𝑙𝑡 . (117)

In other words, the average square displacement increases proportionally with time. Of
course, this is only valid because the liquid is made of separate molecules. Repeatedly
measuring the position of a particle should give the distribution shown in Figure 280 for

Motion Mountain – The Adventure of Physics
the probability that the particle is found at a given distance from the starting point. This
Ref. 295 is called the (Gaussian) normal distribution. In 1908, Jean Perrin* performed extensive
experiments in order to test this prediction. He found that equation (117) correspon-
ded completely with observations, thus convincing everybody that Brownian motion is
indeed due to collisions with the molecules of the surrounding liquid, as had been ex-
pected.** Perrin received the 1926 Nobel Prize in Physics for these experiments.
Einstein also showed that the same experiment could be used to determine the num-
ber of molecules in a litre of water (or equivalently, the Boltzmann constant 𝑘). Can you
Challenge 648 d work out how he did this?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Why stones can be neither smo oth nor fractal, nor made of
lit tle hard balls
The exploration of temperature yields another interesting result. Researchers first stud-
ied gases, and measured how much energy was needed to heat them by 1 K. The result
is simple: all gases share only a few values, when the number of molecules 𝑁 is taken
into account. Monatomic gases (in a container with constant volume and at sufficient
temperature) require 3𝑁𝑘/2, diatomic gases (and those with a linear molecule) 5𝑁𝑘/2,
Page 392 and almost all other gases 3𝑁𝑘, where 𝑘 = 1.4 ⋅ 10−23 J/K is the Boltzmann constant.
The explanation of this result was soon forthcoming: each thermodynamic degree of
freedom*** contributes the energy 𝑘𝑇/2 to the total energy, where 𝑇 is the temperature.

* Jean Perrin (b. 1870 Lille, d. 1942 New York), important physicist, devoted most of his career to the ex-
perimental proof of the atomic hypothesis and the determination of Avogadro’s number; in pursuit of this
aim he perfected the use of emulsions, Brownian motion and oil films. His Nobel Prize speech (nobelprize.
org/physics/laureates/1926/perrin-lecture.html) tells the interesting story of his research. He wrote the in-
fluential book Les atomes and founded the Centre National de la Recherche Scientifique. He was the first to
speculate, in 1901, that an atom might be similar to a small solar system.
Ref. 296 ** In a delightful piece of research, Pierre Gaspard and his team showed in 1998 that Brownian motion is
Page 424 also chaotic, in the strict physical sense given later on.
*** A thermodynamic degree of freedom is, for each particle in a system, the number of dimensions in which
13 on heat and motion reversal invariance 395

match
head

F I G U R E 281 The ﬁre pump.

Motion Mountain – The Adventure of Physics
So the number of degrees of freedom in physical bodies is finite. Bodies are not continu-
ous, nor are they fractals: if they were, their specific thermal energy would be infinite.
Matter is indeed made of small basic entities.
All degrees of freedom contribute to the specific thermal energy. At least, this is what
classical physics predicts. Solids, like stones, have 6 thermodynamic degrees of freedom
and should show a specific thermal energy of 3𝑁𝑘. At high temperatures, this is indeed
observed. But measurements of solids at room temperature yield lower values, and the
lower the temperature, the lower the values become. Even gases show values lower than
those just mentioned, when the temperature is sufficiently low. In other words, molecules
and atoms behave differently at low energies: atoms are not immutable little hard balls.

Entropy

“
– It’s irreversible.

”
– Like my raincoat!
Mel Brooks, Spaceballs, 1987

Every domain of physics describes change in terms of three quantities: energy, as well
Ref. 299 as an intensive and an extensive quantity characteristic of the domain. In the domain
of thermal physics, the intensive quantity is temperature. What is the corresponding ex-
tensive quantity?
The obvious guess would be ‘heat’. Unfortunately, the quantity that physicists usually
call ‘heat’ is not the same as what we call ‘heat’ in our everyday speech. For this historical
reason, we need to introduce a new term. The extensive quantity corresponding to what
we call ‘heat’ in everyday speech is called entropy in physics.*

it can move plus the number of dimensions in which it is kept in a potential. Atoms in a solid have six,
particles in monatomic gases have only three; particles in diatomic gases or rigid linear molecules have five.
Ref. 297 The number of degrees of freedom of larger molecules depends on their shape.
* The term ‘entropy’ was invented by the physicist Rudolph Clausius (b. 1822 Köslin, d. 1888 Bonn) in 1865.
He formed it from the Greek ἐν ‘in’ and τρόπος ‘direction’, to make it sound similar to ‘energy’. The term
396 13 on heat and motion reversal invariance

Entropy describes the amount of everyday heat. Entropy is measured in joule per
kelvin or J/K; some example values (per amount of matter) are listed in Table 47 and
Table 48. Entropy describes everyday heat in the same way as momentum describes
everyday motion. Correspondingly, temperature describes the intensity of everyday heat
or entropy, in the same way that speed describes the intensity of motion.
When two objects of different speeds collide, a flow of momentum takes place between
them. Similarly, when two objects differing in temperature are brought into contact,
an entropy flow takes place between them. We now define the concept of entropy –
‘everyday heat’ – more precisely and explore its properties in some more detail.
Entropy measures the degree to which energy is mixed up inside a system, that is, the
degree to which energy is spread or shared among the components of a system. When
all components of a system – usually the molecules or atoms – move in the same way,
in concert, the entropy of the system is low. When the components of the system move
completely independently, randomly, the entropy is large. In short, entropy measures the
amount of disordered energy content per temperature in a system. That is the reason that
it is measured in J/K.

Motion Mountain – The Adventure of Physics
Entropy is an extensive quantity, like charge and momentum. It is measured by trans-
ferring it to a measurement apparatus. The simplest measurement apparatus is a mixture
of water and ice. When an amount 𝑆 of entropy is transferred to the mixture, the amount
of melted ice is a measure of the transferred entropy.
More precisely, the entropy Δ𝑆 flowing into a system is measured by measuring the
energy 𝐸 flowing into the system, and recording the temperature 𝑇 that occurs during
the process:
𝑇end
𝑑𝐸
Δ𝑆 = ∫ . (118)
𝑇start 𝑇

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Often, this can be approximated as Δ𝑆 = 𝑃 Δ𝑡/𝑇, where 𝑃 is the power of the heating
device, Δ𝑡 is the heating time, and 𝑇 is the average temperature.
Since entropy measures an amount, an extensive quantity, and not an intensity, en-
tropy adds up when identical systems are composed into one. When two one-litre bottles
of water at the same temperature are poured together, the entropy of the water adds up.
Again, this corresponds to the behaviour of momentum: it also adds up when systems
are composed.
Like any other extensive quantity, entropy can be accumulated in a body, and entropy
can flow into or out of bodies. When we transform water into steam by heating it, we
say that we add entropy to the water. We also add entropy when we transform ice into
liquid water. After either transformation, the added entropy is contained in the warmer
phase. Indeed, we can measure the entropy we add by measuring how much ice melts or
how much water evaporates. In short, entropy is the exact term for what we call ‘heat’ in
everyday speech.
Whenever we dissolve a block of salt in water, the entropy of the total system must
increase, because the disorder increases. We now explore this process.

entropy has always had the meaning given here.
In contrast, what physicists traditionally called ‘heat’ is a form of energy and not an extensive quantity
in general.
13 on heat and motion reversal invariance 397

TA B L E 47 Some measured speciﬁc entropy values.

Process/System E n t r o p y va l u e

Carbon, solid, in diamond form 2.43 J/K mol
Carbon, solid, in graphite form 5.69 J/K mol
Melting of ice 1.21 kJ/K kg = 21.99 J/K mol
Iron, solid, under standard conditions 27.2 J/K mol
Magnesium, solid, under standard condi- 32.7 J/K mol
tions
Water, liquid, under standard conditions 70.1(2) J/K mol
Boiling of 1 kg of liquid water at 101.3 kPa 6.03 kJ/K= 110 J/K mol
Helium gas under standard conditions 126.15 J/K mol
Hydrogen gas under standard conditions 130.58 J/K mol
Carbon gas under standard conditions 158 J/K mol
Water vapour under standard conditions 188.83 J/K mol

Motion Mountain – The Adventure of Physics
Oxygen O2 under standard conditions 205.1 J/K mol
C2 H6 gas under standard conditions 230 J/K mol
C3 H8 gas under standard conditions 270 J/K mol
C4 H10 gas under standard conditions 310 J/K mol
C5 H12 gas under standard conditions 348.9 J/K mol
TiCl4 gas under standard conditions 354.8 J/K mol

TA B L E 48 Some typical entropy values per particle at

M at e r i a l Entropy per
pa rt i c l e
Monatomic solids 0.3 𝑘 to 10 𝑘
Diamond 0.29 𝑘
Graphite 0.68 𝑘
Lead 7.79 𝑘
Monatomic gases 15-25 𝑘
Helium 15.2 𝑘
Radon 21.2 𝑘
Diatomic gases 15 𝑘 to 30 𝑘
Polyatomic solids 10 𝑘 to 60 𝑘
Polyatomic liquids 10 𝑘 to 80 𝑘
Polyatomic gases 20 𝑘 to 60 𝑘
Icosane 112 𝑘
398 13 on heat and motion reversal invariance

Entropy from particles
Once it had become clear that heat and temperature are due to the motion of microscopic
particles, people asked what entropy was microscopically. The answer can be formulated
in various ways. The two most extreme answers are:

⊳ Entropy measures the (logarithm of the) number 𝑊 of possible microscopic
states. A given macroscopic state can have many microscopic realizations.
The logarithm of this number, multiplied by the Boltzmann constant 𝑘, gives
the entropy.*
⊳ Entropy is the expected number of yes-or-no questions, multiplied by 𝑘 ln 2,
the answers of which would tell us everything about the system, i.e., about
its microscopic state.

In short, the higher the entropy, the more microstates are possible. Through either of
these definitions, entropy measures the quantity of randomness in a system. In other

Motion Mountain – The Adventure of Physics
words, entropy measures the transformability of energy: higher entropy means lower
transformability. Alternatively, entropy measures the freedom in the choice of microstate
that a system has. High entropy means high freedom of choice for the microstate. For ex-
ample, when a molecule of glucose (a type of sugar) is produced by photosynthesis, about
40 bits of entropy are released. This means that after the glucose is formed, 40 additional
yes-or-no questions must be answered in order to determine the full microscopic state
of the system. Physicists often use a macroscopic unit; most systems of interest are large,
and thus an entropy of 1023 bits is written as 1 J/K. (This is only approximate. Can you
Challenge 649 ny find the precise value?)
To sum up, entropy is thus a specific measure for the characterization of the disorder

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 300 of thermal systems. Three points are worth making here. First of all, entropy is not the
measure of disorder, but one measure of disorder. It is therefore not correct to use entropy
as a synonym for the concept of disorder, as is often done in popular literature. Entropy
is only defined for systems that have a temperature, in other words, only for systems that
are in or near equilibrium. (For systems far from equilibrium, no measure of disorder
has been found yet; probably none is possible.) In fact, the use of the term entropy has
degenerated so much that sometimes one has to call it thermodynamic entropy for clarity.
Secondly, entropy is related to information only if information is defined also as
−𝑘 ln 𝑊. To make this point clear, take a book with a mass of one kilogram. At room
temperature, its entropy content is about 4 kJ/K. The printed information inside a book,
say 500 pages of 40 lines with each containing 80 characters out of 64 possibilities, cor-
responds to an entropy of 4 ⋅ 10−17 J/K. In short, what is usually called ‘information’
in everyday life is a negligible fraction of what a physicist calls information. Entropy
is defined using the physical concept of information.
Finally, entropy is not a measure for what in normal life is called the complexity of a
Ref. 301 situation. In fact, nobody has yet found a quantity describing this everyday notion. The
Challenge 650 ny task is surprisingly difficult. Have a try!

* When Max Planck went to Austria to search for the anonymous tomb of Boltzmann in order to get him
buried in a proper grave, he inscribed the formula 𝑆 = 𝑘 ln 𝑊 on the tombstone. (Which physicist would
finance the tomb of another, nowadays?)
13 on heat and motion reversal invariance 399

In summary, if you hear the term entropy used with a different meaning than the
expression 𝑆 = 𝑘 ln 𝑊, beware. Somebody is trying to get you, probably with some ideo-
logy.

The characteristic entropy of nature – the quantum of
information
Before we complete our discussion of thermal physics we must point out in another way
the importance of the Boltzmann constant 𝑘. We have seen that this constant appears
whenever the granularity of matter plays a role; it expresses the fact that matter is made
of small basic entities. The most striking way to put this statement is the following:

⊳ There is a characteristic entropy change in nature for single particles: Δ𝑆 ≈ 𝑘.

This result is almost 100 years old; it was stated most clearly (with a different numerical
Ref. 302 factor) by Leo Szilard. The same point was made by Léon Brillouin (again with a different
numerical factor). The statement can also be taken as the definition of the Boltzmann

Motion Mountain – The Adventure of Physics
Ref. 303
constant 𝑘.
The existence of a characteristic entropy change in nature is a powerful statement. It
eliminates the possibility of the continuity of matter and also that of its fractality. A char-
acteristic entropy implies that matter is made of a finite number of small components.
The characteristic entropy expresses the fact that matter is made of particles.* The char-
acteristic entropy also shows that Galilean physics cannot be correct: Galilean physics
assumes that arbitrarily small quantities do exist. The entropy limit is the first of several
limits to motion that we will encounter in our adventure. After we have found all limits,
we can start the final leg that leads to the unified description of motion.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The existence of a characteristic quantity also implies a limit on the precision of meas-
urements. Measurements cannot have infinite precision. This limitation is usually stated
in the form of an indeterminacy relation. Indeed, the existence of a characteristic entropy
can be rephrased as an indeterminacy relation between the temperature 𝑇 and the inner
energy 𝑈 of a system:

1 𝑘
Δ Δ𝑈 ⩾ . (119)
𝑇 2

Ref. 305 This relation** was given by Niels Bohr; it was discussed by Werner Heisenberg, who
Vol. VI, page 31 called it one of the basic indeterminacy relations of nature. The Boltzmann constant (di-
Ref. 306 vided by 2) thus fixes the smallest possible entropy value in nature. For this reason, Gilles
Ref. 303 Cohen-Tannoudji calls it the quantum of information and Herbert Zimmermann calls it
the quantum of entropy.
The relation (119) points towards a more general pattern. For every minimum value for
an observable, there is a corresponding indeterminacy relation. We will come across this

* The chracteristic entropy change implies that matter is made of tiny spheres; the minimum action, which
we will encounter in quantum theory, implies that these spheres are actually small clouds.
Ref. 304 ** It seems that the historical value for the right-hand side, 𝑘, has to be corrected to 𝑘/2, for the same reason
that the quantum of action ℏ appears with a factor 1/2 in Heisenberg’s indeterminacy relations.
400 13 on heat and motion reversal invariance

several times in the rest of our adventure, most importantly in the case of the quantum
Vol. IV, page 15 of action and Heisenberg’s indeterminacy relation.
The existence of a characteristic entropy has numerous consequences. First of all, it
sheds light on the third principle of thermodynamics. A characteristic entropy implies
that absolute zero temperature is not achievable. Secondly, a characteristic entropy ex-
plains why entropy values are finite instead of infinite. Thirdly, it fixes the absolute value
of entropy for every system; in continuum physics, entropy, like energy, is only defined
up to an additive constant. The quantum of entropy settles all these issues.
The existence of a characteristic value for an observable implies that an indetermin-
acy relation appears for any two quantities whose product yields that observable. For
example, entropy production rate and time are such a pair. Indeed, an indeterminacy
relation connects the entropy production rate 𝑃 = d𝑆/d𝑡 and the time 𝑡:

𝑘
Δ𝑃 Δ𝑡 ⩾ . (120)
2

Motion Mountain – The Adventure of Physics
From this and the previous relation (119) it is possible to deduce all of statistical phys-
Ref. 306, Ref. 304 ics, i.e., the precise theory of thermostatics and thermodynamics. We will not explore
this further here. (Can you show that the third principle follows from the existence of a
Challenge 651 ny characteristic entropy?) We will limit ourselves to one of the cornerstones of thermody-
namics: the second principle.

Is every thing made of particles?

“
A physicist is the atom’s way of knowing about

”
atoms.

Historically, the study of statistical mechanics has been of fundamental importance for
physics. It provided the first demonstration that physical objects are made of interact-
ing particles. The story of this topic is in fact a long chain of arguments showing that
all the properties we ascribe to objects – including size, stiffness, colour, mass density,
magnetism, thermal or electrical conductivity – result from the interactions of the many
particles they consist of.

⊳ All objects are made of interacting particles.

This discovery has often been called the main result of modern science.
Page 356 How was this discovery made? Table 43 listed the main extensive quantities used in
physics. Extensive quantities are able to flow. It turns out that all flows in nature are
composed of elementary processes, as shown in Table 49. We have already seen that flows
of mass, volume, charge, entropy and substance are composed. Later, quantum theory
will show the same for flows of angular momentum and of the nuclear quantum numbers.

⊳ All flows are made of particles.

The success of this idea has led many people to generalize it to the statement: ‘Everything
13 on heat and motion reversal invariance 401

TA B L E 49 Some minimum ﬂow values found in nature.

O b s e r va t i o n Minimum flow

Matter flow one molecule or one atom or one particle
Volume flow one molecule or one atom or one particle
Angular momentum flow Planck’s quantum of action
Chemical amount of substance one molecule, one atom or one particle
Entropy flow the minimum entropy
Charge flow one elementary charge
Light flow one single photon, Planck’s quantum of action

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 282 A 111 crystal
surface of a gold single crystal,
every bright dot being an atom,
with a surface dislocation
(© CNRS).

we observe is made of parts.’ This approach has been applied with success to chem-
Ref. 308 istry with molecules, materials science and geology with crystals, electricity with elec-
trons, atoms with elementary particles, space with points, time with instants, light with
photons, biology with cells, genetics with genes, neurology with neurons, mathematics
with sets and relations, logic with elementary propositions, and even to linguistics with
morphemes and phonemes. All these sciences have flourished on the idea that everything
is made of related parts. The basic idea seems so self-evident that we find it difficult even
Challenge 652 e to formulate an alternative; try!
However, in the case of the whole of nature, the idea that nature is a sum of related
Vol. VI, page 106 parts is incorrect and only approximate. It turns out to be a prejudice, and a prejudice so
entrenched that it retarded further developments in physics in the latter decades of the
twentieth century. In particular, it does not apply to elementary particles or to space:
402 13 on heat and motion reversal invariance

⊳ Elementary particles and space are not made of parts.

Finding the correct description for the whole of nature, for elementary particles and for
space is the biggest challenge of our adventure. It requires a complete change in thinking
habits. There is a lot of fun ahead.

“
Jede Aussage über Komplexe läßt sich in eine
Aussage über deren Bestandteile und in
diejenigen Sätze zerlegen, welche die Komplexe

”
vollständig beschreiben.*
Ludwig Wittgenstein, Tractatus, 2.0201

The second principle of thermodynamics
In contrast to several other important extensive quantities, entropy is not conserved. On
the one hand, in closed systems, entropy accumulates and never decreases; the sharing
or mixing of energy among the components of a system cannot be undone. On the other

Motion Mountain – The Adventure of Physics
hand, the sharing or mixing can increase spontaneously over time. Entropy is thus only
‘half conserved’. What we call thermal equilibrium is simply the result of the highest
possible mixing. Entropy allows us to define the concept of equilibrium more precisely
as the state of maximum entropy, or maximum energy sharing among the components of
a system. In short, the entropy of a closed system increases until it reaches the maximum
possible value, the equilibrium value.
The non-conservation of entropy has far-reaching consequences. When a piece of
rock is detached from a mountain, it falls, tumbles into the valley, heating up a bit, and
eventually stops. The opposite process, whereby a rock cools and tumbles upwards, is
never observed. Why? We could argue that the opposite motion does not contradict any

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 653 s rule or pattern about motion that we have deduced so far.
Rocks never fall upwards because mountains, valleys and rocks are made of many
particles. Motions of many-particle systems, especially in the domain of thermodynam-
ics, are called processes. Central to thermodynamics is the distinction between reversible
processes, such as the flight of a thrown stone, and irreversible processes, such as the
afore-mentioned tumbling rock. Irreversible processes are all those processes in which
friction and its generalizations play a role. Irreversible processes are those processes that
increase the sharing or mixing of energy. They are important: if there were no friction,
Ref. 309 shirt buttons and shoelaces would not stay fastened, we could not walk or run, coffee
machines would not make coffee, and maybe most importantly of all, we would have no
Vol. IV, page 159 memory.
Irreversible processes, in the sense in which the term is used in thermodynamics,
transform macroscopic motion into the disorganized motion of all the small microscopic
components involved: they increase the sharing and mixing of energy. Irreversible pro-
cesses are therefore not strictly irreversible – but their reversal is extremely improbable.
We can say that entropy measures the ‘amount of irreversibility’: it measures the degree
of mixing or decay that a collective motion has undergone.

* ‘Every statement about complexes can be resolved into a statement about their constituents and into the
propositions that describe the complexes completely.’
13 on heat and motion reversal invariance 403

Entropy is not conserved. Indeed, entropy – ‘heat’ – can appear out of nowhere, spon-
taneously, because energy sharing or mixing can happen by itself. For example, when
two different liquids of the same temperature are mixed – such as water and sulphuric
acid – the final temperature of the mix can differ. Similarly, when electrical current flows
through material at room temperature, the system can heat up or cool down, depending
on the material.
All experiments on heat agree with the so-called second principle of thermodynamics,
which states:

⊳ The entropy in a closed system tends towards its maximum.

Motion Mountain – The Adventure of Physics
In sloppy terms, ‘entropy ain’t what it used to be.’ In this statement, a closed system is a
system that does not exchange energy or matter with its environment. Can you think of
Challenge 654 s an example?
In a closed system, entropy never decreases. Even everyday life shows us that in a
closed system, such as a room, the disorder increases with time, until it reaches some
maximum. To reduce disorder, we need effort, i.e., work and energy. In other words, in
order to reduce the disorder in a system, we need to connect the system to an energy
source in some clever way. For this reason, refrigerators need electrical current or some
other energy source.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 310 In 1866, Ludwig Boltzmann showed that the second principle of thermodynamics res-
Challenge 655 ny ults from the principle of least action. Can you imagine and sketch the general ideas?
Because entropy never decreases in closed systems, white colour does not last.
Whenever disorder increases, the colour white becomes ‘dirty’, usually grey or brown.
Perhaps for this reason white objects, such as white clothes, white houses and white un-
derwear, are valued in our society. White objects defy decay.
The second principle implies that heat cannot be transformed to work completely. In
other words, every heat engine needs cooling: that is the reason for the holes in the front
of cars. The first principle of thermodynamics then states that the mechanical power of
a heat engine is the difference between the inflow of thermal energy at high temperature
and the outflow of thermal energy at low temperature. If the cooling is insufficient –
for example, because the weather is too hot or the car speed too low – the power of the
engine is reduced. Every driver knows this from experience.
In summary, the concept of entropy, corresponding to what is called ‘heat’ in every-
day life – but not to what is traditionally called ‘heat’ in physics! – describes the ran-
domness of the internal motion in matter. Entropy is not conserved: in a closed system,
entropy never decreases, but it can increase until it reaches a maximum value. The non-
conservation of entropy is due to the many components inside everyday systems. The
large number of components lead to the non-conservation of entropy and therefore ex-
plain, among many other things, that many processes in nature never occur backwards,
even though they could do so in principle.
404 13 on heat and motion reversal invariance

Why can ’ t we remember the fu ture?

“
It’s a poor sort of memory which only works

”
backwards.
Lewis Carroll, Alice in Wonderland

Page 40 When we first discussed time, we ignored the difference between past and future. But
obviously, a difference exists, as we do not have the ability to remember the future. This
is not a limitation of our brain alone. All the devices we have invented, such as tape
recorders, photographic cameras, newspapers and books, only tell us about the past. Is
there a way to build a video recorder with a ‘future’ button? Such a device would have to
Challenge 656 e solve a deep problem: how would it distinguish between the near and the far future? It
does not take much thought to see that any way to do this would conflict with the second
principle of thermodynamics. That is unfortunate, as we would need precisely the same
Challenge 657 ny device to show that there is faster-than-light motion. Can you find the connection?
In summary, the future cannot be remembered because entropy in closed systems
tends towards a maximum. Put even more simply, memory exists because the brain is

Motion Mountain – The Adventure of Physics
made of many particles, and so the brain is limited to the past. However, for the most
simple types of motion, when only a few particles are involved, the difference between the
past and the future disappears. For few-particle systems, there is no difference between
times gone by and times approaching. We could say that the future differs from the
past only in our brain, or equivalently, only because of friction. Therefore the difference
between the past and the future is not mentioned frequently in this walk, even though
it is an essential part of our human experience. But the fun of the present adventure is
precisely to overcome our limitations.

Flow of entropy

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
We know from daily experience that transport of an extensive quantity always involves
friction. Friction implies the generation of entropy. In particular, the flow of entropy itself
produces additional entropy. For example, when a house is heated, entropy is produced
in the wall. Heating means keeping a temperature difference Δ𝑇 between the interior and
the exterior of the house. The heat flow 𝐽 traversing a square metre of a wall is given by

𝐽 = 𝜅Δ𝑇 = 𝜅(𝑇i − 𝑇e ) (121)

where 𝜅 is a constant characterizing the ability of the wall to conduct heat. While con-
ducting heat, the wall also produces entropy. The entropy production 𝜎 is proportional
to the difference between the interior and the exterior entropy flows. In other words, one
has
𝐽 𝐽 (𝑇 − 𝑇e )2
𝜎= − =𝜅 i . (122)
𝑇e 𝑇i 𝑇i 𝑇e

Note that we have assumed in this calculation that everything is near equilibrium in
each slice parallel to the wall, a reasonable assumption in everyday life. A typical case of
a good wall has 𝜅 = 1 W/m2 K in the temperature range between 273 K and 293 K. With
13 on heat and motion reversal invariance 405

this value, one gets an entropy production of

𝜎 = 5 ⋅ 10−3 W/m2 K . (123)

Can you compare the amount of entropy that is produced in the flow with the amount
Challenge 658 ny that is transported? In comparison, a good goose-feather duvet has 𝜅 = 1.5 W/m2 K,
which in shops is also called 15 tog.*
The insulation power of materials is usually measured by the constant 𝜆 = 𝜅𝑑 which
is independent of the thickness 𝑑 of the insulating layer. Values in nature range from
about 2000 W/K m for diamond, which is the best conductor of all, down to between
0.1 W/K m and 0.2 W/K m for wood, between 0.015 W/K m and 0.05 W/K m for wools,
cork and foams, and the small value of 5 ⋅ 10−3 W/K m for krypton gas.
Entropy can be transported in three ways: through heat conduction, as just mentioned,
via convection, used for heating houses, and through radiation, which is possible also
through empty space. For example, the Earth radiates about 1.2 W/m2 K into space, in
total thus about 0.51 PW/K. The entropy is (almost) the same that the Earth receives

Motion Mountain – The Adventure of Physics
from the Sun. If more entropy had to be radiated away than received, the temperature
of the surface of the Earth would have to increase. This is called the greenhouse effect or
global warming. Let’s hope that it remains small in the near future.

Do isolated systems exist?
In all our discussions so far, we have assumed that we can distinguish the system under
investigation from its environment. But do such isolated or closed systems, i.e., systems
not interacting with their environment, actually exist? Probably our own human condi-
tion was the original model for the concept: we do experience having the possibility to

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
act independently of our environment. An isolated system may be simply defined as a
system not exchanging any energy or matter with its environment. For many centuries,
scientists saw no reason to question this definition.
The concept of an isolated system had to be refined somewhat with the advent of
quantum mechanics. Nevertheless, the concept of an isolated system provides useful and
precise descriptions of nature also in the quantum domain, if some care is used. Only in
the final part of our walk will the situation change drastically. There, the investigation
of whether the universe is an isolated system will lead to surprising results. (What do
Challenge 659 s you think? A strange hint: your answer is almost surely wrong.) We’ll take the first steps
towards the answer shortly.

Curiosities and fun challenges ab ou t reversibilit y and heat
Running backwards is an interesting sport. The 2006 world records for running back-
wards can be found on www.recordholders.org/en/list/backwards-running.html. You
will be astonished how much these records are faster than your best personal forward-
Challenge 660 e running time.

* The unit tog is not as bad as the official unit (not a joke) BthU ⋅ h/sqft/cm/°F used in some remote
provinces of our galaxy.
406 13 on heat and motion reversal invariance

∗∗
Ref. 313 In 1912, Emile Borel noted that if a gram of matter on Sirius was displaced by one cen-
timetre, it would change the gravitational field on Earth by a tiny amount only. But this
tiny change would be sufficient to make it impossible to calculate the path of molecules
in a gas after a fraction of a second.
∗∗
If heat really is the disordered motion of atoms, a big problem appears. When two atoms
collide head-on, in the instant of smallest distance, neither atom has velocity. Where does
the kinetic energy go? Obviously, it is transformed into potential energy. But that implies
that atoms can be deformed, that they have internal structure, that they have parts, and
thus that they can in principle be split. In short, if heat is disordered atomic motion,
atoms are not indivisible! In the nineteenth century this argument was put forward in
order to show that heat cannot be atomic motion, but must be some sort of fluid. But
since we know that heat really is kinetic energy, atoms must be divisible, even though
their name means ‘indivisible’. We do not need an expensive experiment to show this!

Motion Mountain – The Adventure of Physics
Vol. IV, page 79 We will discover more about them later on in our exploration.
∗∗
Compression of air increases its temperature. This is shown directly by the fire pump, a
variation of a bicycle pump, shown in Figure 281. (For a working example, see the web
page www.de-monstrare.nl). A match head at the bottom of an air pump made of trans-
parent material is easily ignited by the compression of the air above it. The temperature
of the air after compression is so high that the match head ignites spontaneously.
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
In the summer, temperature of the air can easily be measured with a clock. Indeed, the
rate of chirping of most crickets depends on temperature. For example, for a cricket
Ref. 311 species most common in the United States, by counting the number of chirps during
8 seconds and adding 4 yields the air temperature in degrees Celsius.
∗∗
How long does it take to cook an egg? This issue has been researched in extensive de-
tail; of course, the time depends on what type of cooked egg you want, how large it is,
and whether it comes from the fridge or not. There is even a formula for calculating the
Ref. 312 cooking time! Egg white starts hardening at 62°C, the yolk starts hardening at 65°C. The
best-tasting hard eggs are formed at 69°C, half-hard eggs at 65°C, and soft eggs at 63°C.
If you cook eggs at 100°C (for a long time) , the white gets the consistency of rubber and
the yolk gets a green surface that smells badly, because the high temperature leads to the
formation of the smelly H2 S, which then bonds to iron and forms the green FeS. Note
that when temperature is controlled, the time plays no role; ‘cooking’ an egg at 65°C for
10 minutes or 10 hours gives the same result.
∗∗
It is possible to cook an egg in such a way that the white is hard but the yolk remains
Challenge 661 s liquid. Can you achieve the opposite? Research has even shown how you can cook an
13 on heat and motion reversal invariance 407

1 2 3 4

F I G U R E 283 Can you boil water in this paper cup?

Challenge 662 e egg so that the yolk remains at the centre. Can you imagine the method?
∗∗
Not only gases, but also most other materials expand when the temperature rises. As
a result, the electrical wires supported by pylons hang much lower in summer than in

Motion Mountain – The Adventure of Physics
Challenge 663 s winter. True?
∗∗
Ref. 314 The following is a famous problem asked by Fermi. Given that a human corpse cools
down four hours after death, what is the minimum number of calories needed per day
Challenge 664 ny in our food?
∗∗
The energy contained in thermal motion is not negligible. For example, a 1 g bullet trav-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
elling at the speed of sound has a kinetic energy of only 0.04 kJ= 0.01 kcal. What is its
Challenge 665 e thermal energy content?
∗∗
Challenge 666 s How does a typical, 1500 m3 hot-air balloon work?
∗∗
If you do not like this text, here is a proposal. You can use the paper to make a cup, as
shown in Figure 283, and boil water in it over an open flame. However, to succeed, you
Challenge 667 s have to be a little careful. Can you find out in what way?
∗∗
Mixing 1 kg of water at 0°C and 1 kg of water at 100°C gives 2 kg of water at 50°C. What
Challenge 668 s is the result of mixing 1 kg of ice at 0°C and 1 kg of water at 100°C?
∗∗
Temperature has many effects. In the past years, the World Health Organization found
that drinking liquids that are hotter than 65°C – including coffee, chocolate or tea –
causes cancer in the oesophagus, independently of the type of liquid.
∗∗
408 13 on heat and motion reversal invariance

invisible pulsed
laser beam
emitting sound

laser
cable
to amplifier

F I G U R E 284 The invisible loudspeaker.

Motion Mountain – The Adventure of Physics
Ref. 315 The highest recorded air temperature in which a man has survived is 127°C. This was
tested in 1775 in London, by the secretary of the Royal Society, Charles Blagden, together
with a few friends, who remained in a room at that temperature for 45 minutes. Inter-
estingly, the raw steak which he had taken in with him was cooked (‘well done’) when
he and his friends left the room. What condition had to be strictly met in order to avoid
Challenge 669 s cooking the people in the same way as the steak?
∗∗
Is the Boltzmann constant 𝑘 really the smallest possible value for entropy in nature? How

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
then can the entropy per particle of Krypton be as low as 0.3𝑘 per particle? The answer
to this paradox is that a single free particle has more entropy, in fact more than 𝑘, than a
bound one. The limit entropy 𝑘 is thus valid for a physical system, such as the crystal as
a whole, but not separately for the entropy value for each bound particle that is part of a
system.
∗∗
The influential astronomer Anders Celsius (b. 1701 Uppsala, d. 1744 Uppsala) originally
set the freezing point of water at 100 degrees and the boiling point at 0 degrees. Shortly
afterwards, the scale was reversed to the one in use now. However, this is not the whole
Ref. 316 story. With the official definition of the kelvin and the degree Celsius, at the standard
pressure of 101 325 Pa, water boils at 99.974°C. Can you explain why it is not 100°C any
Challenge 670 s more?
∗∗
−30
Challenge 671 s Can you fill a bottle precisely with 1 ± 10 kg of water?
∗∗
One gram of fat, either butter or human fat, contains 38 kJ of chemical energy (or, in an-
cient units more familiar to nutritionists, 9 kcal). That is the same value as that of petrol.
Challenge 672 s Why are people and butter less dangerous than petrol?
13 on heat and motion reversal invariance 409

∗∗
In 1992, the Dutch physicist Martin van der Mark invented a loudspeaker which works by
heating air with a laser beam. He demonstrated that with the right wavelength and with
a suitable modulation of the intensity, a laser beam in air can generate sound. The effect
at the basis of this device, called the photoacoustic effect, appears in many materials. The
best laser wavelength for air is in the infrared domain, on one of the few absorption lines
of water vapour. In other words, a properly modulated infrared laser beam that shines
through the air generates sound. Such light can be emitted from a small matchbox-sized
semiconductor laser hidden in the ceiling and shining downwards. The sound is emit-
ted in all directions perpendicular to the beam. Since infrared laser light is not (usually)
visible, Martin van der Mark thus invented an invisible loudspeaker! Unfortunately, the
efficiency of present versions is still low, so that the power of the speaker is not yet suffi-
cient for practical applications. Progress in laser technology should change this, so that in
the future we should be able to hear sound that is emitted from the centre of an otherwise
empty room.

Motion Mountain – The Adventure of Physics
∗∗
A famous exam question: How can you measure the height of a building with a baro-
Challenge 673 s meter, a rope and a ruler? Find at least six different ways.
∗∗
What is the approximate probability that out of one million throws of a coin you get
Challenge 674 s exactly 500 000 heads and as many tails? You may want to use Stirling’s formula 𝑛! ≈
√2π𝑛 (𝑛/𝑒)𝑛 to calculate the result.*
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 675 s Does it make sense to talk about the entropy of the universe?
∗∗
Challenge 676 ny Can a helium balloon lift the tank which filled it?
∗∗
All friction processes, such as osmosis, diffusion, evaporation, or decay, are slow. They
take a characteristic time. It turns out that any (macroscopic) process with a time-scale
is irreversible. This is no real surprise: we know intuitively that undoing things always
takes more time than doing them. That is again the second principle of thermodynamics.
∗∗
It turns out that storing information is possible with negligible entropy generation. How-
Ref. 317 ever, erasing information requires entropy. This is the main reason why computers, as
well as brains, require energy sources and cooling systems, even if their mechanisms
would otherwise need no energy at all.

* There are many improvements to Stirling’s formula. A simple one is Gosper’s formula 𝑛! ≈
√(2𝑛 + 1/3)π (𝑛/𝑒)𝑛 . Another is √2π𝑛 (𝑛/e)𝑛 e1/(12𝑛+1) < 𝑛! < √2π𝑛 (𝑛/e)𝑛 e1/(12𝑛) .
410 13 on heat and motion reversal invariance

∗∗
When mixing hot rum and cold water, how does the increase in entropy due to the mix-
Challenge 677 ny ing compare with the entropy increase due to the temperature difference?
∗∗
Why aren’t there any small humans, say 10 mm in size, as in many fairy tales? In fact,
Challenge 678 s there are no warm-blooded animals of that size at all. Why not?
∗∗
Shining a light onto a body and repeatedly switching it on and off produces sound. This
is called the photoacoustic effect, and is due to the thermal expansion of the material.
By changing the frequency of the light, and measuring the intensity of the noise, one
reveals a characteristic photoacoustic spectrum for the material. This method allows us
to detect gas concentrations in air of one part in 109 . It is used, among other methods, to
study the gases emitted by plants. Plants emit methane, alcohol and acetaldehyde in small
quantities; the photoacoustic effect can detect these gases and help us to understand the

Motion Mountain – The Adventure of Physics
processes behind their emission.
∗∗
What is the rough probability that all oxygen molecules in the air would move away from
Challenge 679 ny a given city for a few minutes, killing all inhabitants?
∗∗
If you pour a litre of water into the sea, stir thoroughly through all the oceans and then
Challenge 680 ny take out a litre of the mixture, how many of the original atoms will you find?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Challenge 681 s How long would you go on breathing in the room you are in if it were airtight?
∗∗
Heat loss is a larger problem for smaller animals, because the surface to volume ratio
increases when size decreases. As a result, small animals are found in hot climate, large
animals are found in cold climates. This is true for bears, birds, rabbits, insects and many
other animal families. For the same reason, small living beings need high amounts of
food per day, when calculated in body weight, whereas large animals need far less food.
∗∗
What happens if you put some ash onto a piece of sugar and set fire to the whole?
Challenge 682 s (Warning: this is dangerous and not for kids.)
∗∗
Entropy calculations are often surprising. For a system of 𝑁 particles with two states
each, there are 𝑊all = 2𝑁 states. For its most probable configuration, with exactly half the
particles in one state, and the other half in the other state, we have 𝑊max = 𝑁!/((𝑁/2)!)2.
Now, for a macroscopic system of particles, we might typically have 𝑁 = 1024 . That gives
𝑊all ≫ 𝑊max ; indeed, the former is 1012 times larger than the latter. On the other hand,
13 on heat and motion reversal invariance 411

Challenge 683 ny we find that ln 𝑊all and ln 𝑊max agree for the first 20 digits! Even though the configura-
tion with exactly half the particles in each state is much more rare than the general case,
Challenge 684 ny where the ratio is allowed to vary, the entropy turns out to be the same. Why?
∗∗
If heat is due to motion of atoms, our built-in senses of heat and cold are simply detectors
Challenge 685 ny of motion. How could they work?
By the way, the senses of smell and taste can also be seen as motion detectors, as they
Challenge 686 e signal the presence of molecules flying around in air or in liquids. Do you agree?
∗∗
The Moon has an atmosphere, although an extremely thin one, consisting of sodium
(Na) and potassium (K). This atmosphere has been detected up to nine Moon radii from
its surface. The atmosphere of the Moon is generated at the surface by the ultraviolet
Challenge 687 s radiation from the Sun. Can you estimate the Moon’s atmospheric density?

Motion Mountain – The Adventure of Physics
∗∗
Does it make sense to add a line in Table 43 for the quantity of physical action? A column?
Challenge 688 ny Why?
∗∗
Diffusion provides a length scale. For example, insects take in oxygen through their skin.
As a result, the interiors of their bodies cannot be much more distant from the surface
than about a centimetre. Can you list some other length scales in nature implied by dif-
Challenge 689 s fusion processes?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Rising warm air is the reason why many insects are found in tall clouds in the evening.
Many insects, especially that seek out blood in animals, are attracted to warm and humid
air.
∗∗
Thermometers based on mercury can reach 750°C. How is this possible, given that mer-
Challenge 690 s cury boils at 357°C?
∗∗
Challenge 691 s What does a burning candle look like in weightless conditions?
∗∗
It is possible to build a power station by building a large chimney, so that air heated by
the Sun flows upwards in it, driving a turbine as it does so. It is also possible to make a
power station by building a long vertical tube, and letting a gas such as ammonia rise into
it which is then liquefied at the top by the low temperatures in the upper atmosphere; as
it falls back down a second tube as a liquid – just like rain – it drives a turbine. Why are
Challenge 692 s such schemes, which are almost completely non-polluting, not used yet?
412 13 on heat and motion reversal invariance

cold air

hot air hot air
(exhaust valve) exhaust

room temperature compressed
(compressed) air air cold air

F I G U R E 285 The design of the Wirbelrohr or Ranque–Hilsch vortex tube, and a commercial version,
about 40 cm in size, used to cool manufacturing processes (© Coolquip).

∗∗
One of the most surprising devices ever invented is the Wirbelrohr or Ranque–Hilsch
vortex tube. By blowing compressed air at room temperature into it at its midpoint, two

Motion Mountain – The Adventure of Physics
flows of air are formed at its ends. One is extremely cold, easily as low as −50°C, and one
extremely hot, up to 200°C. No moving parts and no heating devices are found inside.
Challenge 693 s How does it work?
∗∗
Thermoacoustic engines, pumps and refrigerators provide many strange and fascinating
applications of heat. For example, it is possible to use loud sound in closed metal cham-
bers to move heat from a cold place to a hot one. Such devices have few moving parts
Ref. 318 and are being studied in the hope of finding practical applications in the future.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Challenge 694 s Does a closed few-particle system contradict the second principle of thermodynamics?
∗∗
What happens to entropy when gravitation is taken into account? We carefully left grav-
itation out of our discussion. In fact, gravitation leads to many new problems – just try to
think about the issue. For example, Jacob Bekenstein has discovered that matter reaches
Challenge 695 s its highest possible entropy when it forms a black hole. Can you confirm this?
∗∗
The numerical values – but not the units! – of the Boltzmann constant 𝑘 =
1.38 ⋅ 10−23 J/K and the combination ℎ/𝑐𝑒 – where ℎ is Planck’s constant, 𝑐 the speed of
light and 𝑒 the electron charge – agree in their exponent and in their first three digits.
Challenge 696 s How can you dismiss this as mere coincidence?
∗∗
Mixing is not always easy to perform. The experiment of Figure 286 gives completely
Challenge 697 s different results with water and glycerine. Can you guess them?
∗∗
13 on heat and motion reversal invariance 413

ink droplets

ink stripe

F I G U R E 286 What happens to the ink stripe if the
inner cylinder is turned a few times in one direction,
and then turned back by the same amount?

Motion Mountain – The Adventure of Physics
Challenge 698 s How can you get rid of chewing gum in clothes?
∗∗
With the knowledge that air consists of molecules, Maxwell calculated that ‘‘each particle
Challenge 699 ny makes 8 077 200 000 collisions per second’’. How did he do this?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
A perpetuum mobile ‘of the second kind’ is a machine that converts heat into motion
without the use of a second, cooler bath. Entropy implies that such a device does not
Challenge 700 e exist. Can you show this?
∗∗
There are less-well known arguments proving the existence of atoms. In fact, two every-
Challenge 701 ny day observations prove the existence of atoms: reproduction and memory. Why?
∗∗
In the context of lasers and of spin systems, it is fun to talk about negative temperature.
Challenge 702 s Why is this not really sensible?

Summary on heat and time-invariance
Microscopic motion due to gravity and electric interactions, thus all microscopic motion
in everyday life, is reversible: such motion can occur backwards in time. In other words,
motion due to gravity and electromagnetism is symmetric under motion reversal or, as is
often incorrectly stated, under ‘time reversal’.
Nevertheless, everyday motion is irreversible, because there are no completely closed
systems in everyday life. Lack of closure leads to fluctuations; fluctuations lead to friction.
414 13 on heat and motion reversal invariance

Equivalently, irreversibility results from the extremely low probability of motion reversal
in many-particle systems. Macroscopic irreversibility does not contradict microscopic
reversibility.
For these reasons, in everyday life, entropy in closed systems never decreases. This
leads to a famous issue: how can biological evolution be reconciled with entropy in-
crease? Let us have a look.

SE L F-ORG A N I Z AT ION A N D C HAO S
– T H E SI M PL IC I T Y OF C OM PL E X I T Y

“
To speak of non-linear physics is like calling

”
zoology the study of non-elephant animals.
Ref. 319 Stanislaw Ulam

I
Page 242 n our list of global descriptions of motion, the study of self-organization
s the high point. Self-organization is the appearance of order. In physics, order

Motion Mountain – The Adventure of Physics
Ref. 320 s a term that includes shapes, such as the complex symmetry of snowflakes; patterns,
such as the stripes of zebras and the ripples on sand; and cycles, such as the creation
of sound when singing. When we look around us, we note that every example of what
Challenge 703 s we call beauty is a combination of shapes, patterns and cycles. (Do you agree?) Self-
organization can thus be called the study of the origin of beauty. Table 50 shows how
frequently the appearance of order shapes our environment.

TA B L E 50 Some rhythms, patterns and shapes observed in nature.

O b s e r va t i o n Driving Restoring T y p. S c a l e

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
‘force ’ ‘force’
Fingerprint chemical reactions diffusion 0.1 mm
Clock ticking falling weight friction 1s
Chalk squeaking due to motion friction 600 Hz
stick-slip instability
Musical note generation in bow motion friction 600 Hz
violin
Musical note generation in air flow turbulence 400 Hz
flute
Train oscillations motion friction 0.3 Hz
transversally to the track
Flow structures in water flow turbulence 10 cm
waterfalls and fountains
Jerky detachment of scotch pulling speed sticking friction 0.1 Hz
tape
Radius oscillations in extrusion speed friction 10 cm
spaghetti and polymer fibre
production
Patterns on buckled metal deformation stiffness depend on thickness
plates and foils
416 14 self-organization and chaos

TA B L E 50 (Continued) Some rhythms, patterns and shapes observed in nature.

O b s e r va t i o n Driving Restoring T y p. S c a l e
‘force ’ ‘force’
Flapping of flags in steady air flow stiffness 20 cm
wind
Dripping of water tap water flow surface tension 1 Hz
Bubble stream from a beer dissolved gas surface tension 0.1 Hz, 1 mm
glass irregularity pressure
Raleigh–Bénard instability temperature diffusion 0.1 Hz, 1 mm
gradient
Couette–Taylor flow speed gradient friction 0.1 Hz, 1 mm
Bénard–Marangoni flow, surface tension viscosity 0.1 Hz, 1 mm
sea wave generation
Karman wakes, Emmon momentum viscosity from mm to km
spots, Osborne Reynolds

Motion Mountain – The Adventure of Physics
flow
Regular bangs in a car flow pressure resonances 0.3 Hz
exhaustion pipe
Regular cloud flow diffusion 0.5 km
arrangements
El Niño flow diffusion 5 to 7 years
Wine arcs on glass walls surface tension binary mixture 0.1 Hz, 1 mm
Ferrofluids surfaces in magnetic energy gravity 3 mm
magnetic fields

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Patterns in liquid crystals electric energy stress 1 mm, 3 s
Flickering of aging electron flow diffusion 1 Hz
fluorescence lights
Surface instabilities of electron flow diffusion 1 cm
welding
Tokamak plasma electron flow diffusion 10 s
instabilities
Snowflake formation and concentration surface diffusion 10 μm
other dendritic growth gradient
processes
Solidification interface entropy flow surface tension 1 mm
patterns, e.g. in CBr4
Periodic layers in metal concentration diffusion 10 μm
corrosion gradients
Hardening of steel by cold strain dislocation motion 5 μm
working
Labyrinth structures in particle flow dislocation motion 5 μm
proton irradiated metals
Patterns in laser irradiated laser irradiation diffusion 50 μm
Cd-Se alloys
the simplicity of complexity 417

TA B L E 50 (Continued) Some rhythms, patterns and shapes observed in nature.

O b s e r va t i o n Driving Restoring T y p. S c a l e
‘force ’ ‘force’
Dislocation patterns and strain dislocation motion 10 μm 100 s
density oscillations in
fatigued Cu single crystals
Laser light emission, its pumping energy light losses 10 ps to 1 ms
cycles and chaotic regimes
Rotating patterns from light energy diffusion 1 mm
shining laser light on the
surface of certain
electrolytes
Belousov-Zhabotinski concentration diffusion 1 mm, 10 s
reaction patterns and gradients
cycles

Motion Mountain – The Adventure of Physics
Flickering of a burning heat and thermal and substance 0.1 s
candle concentration diffusion
gradients
Regular sequence of hot heat and thermal and substance 1 cm
and cold flames in concentration diffusion
carbohydrate combustion gradients
Feedback whistle from amplifiers electric losses 1 kHz
microphone to loudspeaker
Any electronic oscillator in power supply resistive losses 1 kHz to 30 GHz
radio sets, television sets,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
computers, mobile phones,
etc.
Periodic geyser eruptions underground evaporation 10 min
heating
Periodic earthquakes at tectonic motion ruptures 1 Ms
certain faults
Hexagonal patterns in heating heat diffusion 1m
basalt rocks
Hexagonal patterns on dry regular temperature water diffusion 0.5 m
soil changes
Periodic intensity changes nuclear fusion energy emission 3 Ms
of the Cepheids and other
stars
Convection cells on the nuclear fusion energy emission 1000 km
surface of the Sun
Formation and oscillations charge separation resistive losses 100 ka
of the magnetic field of the due to convection
Earth and other celestial and friction
bodies
Wrinkling/crumpling strain stiffness 1 mm
transition
418 14 self-organization and chaos

TA B L E 50 (Continued) Some rhythms, patterns and shapes observed in nature.

O b s e r va t i o n Driving Restoring T y p. S c a l e
‘force ’ ‘force’
Patterns of animal furs chemical diffusion 1 cm
concentration
Growth of fingers and chemical diffusion 1 cm
limbs concentration
Symmetry breaking in probably molecular diffusion 1m
embryogenesis, such as the chirality plus
heart on the left chemical
concentration
Cell differentiation and chemical diffusion 10 μm to 30 m
appearance of organs concentration
during growth
Prey–predator oscillations reproduction hunger 3 to 17 a

Motion Mountain – The Adventure of Physics
Thinking neuron firing heat dissipation 1 ms, 100 μm

Appearance of order
The appearance of order is a general observation across nature. Fluids in particular ex-
hibit many phenomena where order appears and disappears. Examples include the more
or less regular flickering of a burning candle, the flapping of a flag in the wind, the regular
stream of bubbles emerging from small irregularities in the surface of a beer or cham-
Page 355 pagne glass, and the regular or irregular dripping of a water tap. Figure 251 shows some
additional examples, and so do the figures in this chapter. Other examples include the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
appearance of clouds and of regular cloud arrangements in the sky. It can be fascinating
to ponder, during an otherwise boring flight, the mechanisms behind the formation of
Challenge 704 e the cloud shapes and patterns you see from the aeroplane. A typical cloud has a mass
density of 0.3 to 5 g/m3 , so that a large cloud can contain several hundred tons of water.
Other cases of self-organization are mechanical, such as the formation of mountain
ranges when continents move, the creation of earthquakes, or the formation of laughing
folds at the corners of human eyes.
All growth processes are self-organization phenomena. The appearance of order is
found in the cell differentiation in an embryo inside a woman’s body; the formation of
colour patterns on tigers, tropical fish and butterflies; the symmetrical arrangements of
Page 46 flower petals; the formation of biological rhythms; and so on.
Have you ever pondered the incredible way in which teeth grow? A practically in-
organic material forms shapes in the upper and the lower rows fitting exactly into each
Vol. III, page 242 other. How this process is controlled is still a topic of research. Also the formation, before
and after birth, of neural networks in the brain is another process of self-organization.
Even the physical processes at the basis of thinking, involving changing electrical signals,
is to be described in terms of self-organization.
Biological evolution is a special case of growth. Take the evolution of animal shapes.
It turns out that snake tongues are forked because that is the most efficient shape for
Ref. 321 following chemical trails left by prey and other snakes of the same species. (Snakes smell
the simplicity of complexity 419

a funnel

y
digital video
camera
x

b
d
c

Motion Mountain – The Adventure of Physics
d

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 287 Examples of self-organization for sand: spontaneous appearance of a temporal cycle (a
and b), spontaneous appearance of a periodic pattern (b and c), spontaneous appearance of a
spatiotemporal pattern, namely solitary waves (right) (© Ernesto Altshuler et al.).

with the help of their tongue.) How many tips would the tongues of flying reptiles need,
Challenge 705 e such as flying dragons?
Ref. 322 The fixed number of fingers in human hands are also consequence of self-
Vol. III, page 298 organization. The number of petals of flowers may or may not be due to self-organization.
Studies into the conditions required for the appearance or disappearance of order have
shown that their description requires only a few common concepts, independently of the
details of the physical system. This is best seen looking at a few simple examples.
420 14 self-organization and chaos

TA B L E 51 Patterns and a cycle on horizontal sand and on sand-like surfaces in the sea and on land.

Pa t t e r n / c y c l e P e r i o d Amplitude Origin

Under water
Ripples 5 cm 5 mm water waves
Megaripples 1m 0.1 m tides
Sand waves 100 to 800 m 5m tides
Sand banks 2 to 10 km 2 to 20 m tides
In air
Ripples 0.1 m 0.05 m wind
Singing sand 65 to 110 Hz up to 105 dB wind on sand dunes, avalanches
making the dune vibrate
Road 0.3 to 0.9 m 0.05 m wheels
corrugations

Motion Mountain – The Adventure of Physics
Ski moguls 5 to 6 m up to 1 m skiers
Elsewhere
On Mars a few km few tens of m wind

Self-organization in sand
All the richness of self-organization reveals itself in the study of plain sand. Why do sand
dunes have ripples, as does the sand floor at the bottom of the sea? How do avalanches
occur on steep heaps of sand? How does sand behave in hourglasses, in mixers, or in
vibrating containers? The results are often surprising.
An overview of self-organization phenomena in sand is given in Table 51. For example,
as recently as 2006, the Cuban research group of Ernesto Altshuler and his colleagues
Ref. 323 discovered solitary waves on sand flows (shown in Figure 287). They had already dis-
covered the revolving river effect on sand piles, shown in the same figure, in 2002. Even
the simplicity of complexity 421

n = 21

n = 23

time

Motion Mountain – The Adventure of Physics
F I G U R E 289 Oscillons formed by shaken F I G U R E 290 Magic numbers: 21 spheres, when swirled
bronze balls; horizontal size is about 2 cm in a dish, behave differently from non-magic numbers,
(© Paul Umbanhowar) like 23, of spheres (redrawn from photographs
© Karsten Kötter).

more surprisingly, these effects occur only for Cuban sand, and a few rare other types of
sand. The reasons are still unclear.
Ref. 324 Similarly, in 1996 Paul Umbanhowar and his colleagues found that when a flat con-
tainer holding tiny bronze balls (around 0.165 mm in diameter) is shaken up and down

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in vacuum at certain frequencies, the surface of this bronze ‘sand’ forms stable heaps.
They are shown in Figure 289. These heaps, so-called oscillons, also bob up and down.
The oscillons can move and interact with one another.
Oscillons in bronze sand are a simple example of a general effect in nature:

⊳ Discrete systems with non-linear interactions can exhibit localized
Ref. 325 excitations.

This fascinating topic is just beginning to be researched. It might well be that one day it
will yield results relevant to our understanding of the growth of organisms.
Sand shows many other pattern-forming processes.
— A mixture of sand and sugar, when poured onto a heap, forms regular layered struc-
tures that in cross-section look like zebra stripes.
— Horizontally rotating cylinders with binary mixtures inside them separate the mix-
ture out over time.
— Take a container with two compartments separated by a 1 cm wall. Fill both halves
with sand and rapidly shake the whole container with a machine. Over time, all the
sand will spontaneously accumulate in one half of the container.
— In sand, people have studied the various types of sand dunes that ‘sing’ when the
Ref. 326 wind blows over them.
422 14 self-organization and chaos

Ref. 327 — Also the corrugations formed by traffic on roads without tarmac, the washboard

Motion Mountain – The Adventure of Physics
roads shown in Figure 288, are an example of self-organization. These corrugation
Challenge 706 s patterns often move, over time, against the traffic direction. Can you explain why?
Page 319 The moving ski moguls mentioned above also belong here.

In fact, the behaviour of sand and dust is proving to be such a beautiful and fascinating
topic that the prospect of each human returning to dust does not look so grim after all.

Self-organization of spheres
A stunningly simple and beautiful example of self-organization is the effect discovered in

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 328 1999 by Karsten Kötter and his group. They found that the behaviour of a set of spheres
swirled in a dish depends on the number of spheres used. Usually, all the spheres get
continuously mixed up. But for certain ‘magic’ numbers, such as 21, stable ring patterns
emerge, for which the outside spheres remain outside and the inside ones remain inside.
The rings, best seen by colouring the spheres, are shown in Figure 290.

C onditions for the appearance of order
The many studies of self-organizing systems have changed our understanding of nature
in a number of ways. First of all, they have shown that patterns and shapes are similar to
cycles: all are due to motion. Without motion, and thus without history, there is no order,
Ref. 329 neither patterns nor shapes nor rhythms. Every pattern has a history; every pattern is a
result of motion. As an example, Figure 291 shows how a snowflake grows.
Secondly, patterns, shapes and rhythms are due to the organized motion of large num-
bers of small constituents. Systems which self-organize are always composite: they are co-
operative structures.
Thirdly, all these systems obey evolution equations which are non-linear in the mac-
roscopic configuration variables. Linear systems do not self-organize.
Fourthly, the appearance and disappearance of order depends on the strength of a
driving force or driving process, the so-called order parameter.
Finally, all order and all structures appear when two general types of motion compete
the simplicity of complexity 423

with each other, namely a ‘driving’, energy-adding process, and a ‘dissipating’, braking
mechanism. Thermodynamics thus plays a role in all self-organization. Self-organizing
systems are always dissipative systems, and are always far from equilibrium. When the
driving and the dissipation are of the same order of magnitude, and when the key beha-
viour of the system is not a linear function of the driving action, order may appear.*

The mathematics of order appearance
Every pattern, every shape and every rhythm or cycle can be described by some observ-
able 𝐴 that describes the amplitude of the pattern, shape or rhythm. For example, the
amplitude 𝐴 can be a length for sand patterns, or a chemical concentration for biological
systems, or an air pressure for sound appearance.
Order appears when the amplitude 𝐴 differs from zero. To understand the appearance
of order, we have to understand the evolution of the amplitude 𝐴. The study of order has
shown that this amplitude always follows similar evolution equations, independently of
the physical mechanism of the system. This surprising result unifies the whole field of

Motion Mountain – The Adventure of Physics
self-organization.
All self-organizing systems at the onset of order appearance can be described by equa-
tions for the pattern amplitude 𝐴 of the general form

∂𝐴(𝑡, 𝑥)
= 𝜆𝐴 − 𝜇|𝐴|2 𝐴 + 𝜅 Δ𝐴 + higher orders . (124)
∂𝑡
Here, the observable 𝐴 – which can be a real or a complex number, in order to describe
phase effects – is the observable that appears when order appears, such as the oscillation
amplitude or the pattern amplitude. The first term 𝜆𝐴 is the driving term, in which 𝜆

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
is a parameter describing the strength of the driving. The next term is a typical non-
linearity in 𝐴, with 𝜇 a parameter that describes its strength, and the third term 𝜅 Δ𝐴 =
𝜅(∂2 𝐴/∂𝑥2 + ∂2 𝐴/∂𝑦2 + ∂2 𝐴/∂𝑧2 ) is a typical diffusive and thus dissipative term.
We can distinguish two main situations. In cases where the dissipative term plays no
role (𝜅 = 0), we find that when the driving parameter 𝜆 increases above zero, a temporal
Challenge 707 ny oscillation appears, i.e., a stable limit cycle with non-vanishing amplitude. In cases where
the diffusive term does play a role, equation (124) describes how an amplitude for a spatial
oscillation appears when the driving parameter 𝜆 becomes positive, as the solution 𝐴 = 0
Challenge 708 ny then becomes spatially unstable.
In both cases, the onset of order is called a bifurcation, because at this critical value
of the driving parameter 𝜆 the situation with amplitude zero, i.e., the homogeneous (or
unordered) state, becomes unstable, and the ordered state becomes stable. In non-linear
systems, order is stable. This is the main conceptual result of the field. Equation (124) and
its numerous variations allow us to describe many phenomena, ranging from spirals,

* To describe the ‘mystery’ of human life, terms like ‘fire’, ‘river’ or ‘tree’ are often used as analogies. These
are all examples of self-organized systems: they have many degrees of freedom, have competing driving
and braking forces, depend critically on their initial conditions, show chaos and irregular behaviour, and
sometimes show cycles and regular behaviour. Humans and human life resemble them in all these respects;
thus there is a solid basis for their use as metaphors. We could even go further and speculate that pure beauty
is pure self-organization. The lack of beauty indeed often results from a disturbed equilibrium between
external braking and external driving.
424 14 self-organization and chaos

oscillation, quasiperiodic
fixed point limit cycle motion chaotic motion

configuration variables configuration variables

F I G U R E 292 Examples of different types of motion in conﬁguration space.

waves, hexagonal patterns, and topological defects, to some forms of turbulence. For

Motion Mountain – The Adventure of Physics
Ref. 330
every physical system under study, the main task is to distil the observable 𝐴 and the
parameters 𝜆, 𝜇 and 𝜅 from the underlying physical processes.
In summary, the appearance of order can be described to generally valid equations.
Self-organization is, fundamentally, a simple process. In short: beauty is simple.
Self-organization is a vast field which is yielding new results almost by the week. To
discover new topics of study, it is often sufficient to keep your eye open; most effects are
Challenge 709 e comprehensible without advanced mathematics. Enjoy the hunting!

Chaos

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Most systems that show self-organization also show another type of motion. When the
driving parameter of a self-organizing system is increased to higher and higher values,
order becomes more and more irregular, and in the end one usually finds chaos.
For physicists, c ha o Tc motion is the most irregular type of motion.* Chaos can be
𝑖

defined independently of self-organization, namely as that motion of systems for which
small changes in initial conditions evolve into large changes of the motion (exponentially
with time). This is illustrated in Figure 293. More precisely,

⊳ (Physical) chaos is irregular motion characterized by a positive Lyapounov
exponent in the presence of a strictly valid evolution.

A simple chaotic system is the damped pendulum above three magnets. Figure 294 shows
how regions of predictability (around the three magnet positions) gradually change into
a chaotic region, i.e., a region of effective unpredictability, for higher initial amplitudes.
The weather is also a chaotic system, as are dripping water-taps, the fall of dice, and many

* On the topic of chaos, see the beautiful book by Heinz-Otto P eitgen, Hartmut Jürgens &
Dietmar Saupe, Chaos and Fractals, Springer Verlag, 1992. It includes stunning pictures, the neces-
sary mathematical background, and some computer programs allowing personal exploration of the topic.
‘Chaos’ is an old word: according to Greek mythology, the first goddess, Gaia, i.e., the Earth, emerged from
the chaos existing at the beginning. She then gave birth to the other gods, the animals and the first humans.
the simplicity of complexity 425

state variable value
initial condition 1

initial condition 2

time

F I G U R E 293 Chaos as sensitivity to initial conditions.

The final position of the pendulum depends
on the exact initial position:

Motion Mountain – The Adventure of Physics
damped predictable
pendulum region
with metal
weight

chaotic
three colour-coded magnets region

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 294 A simple chaotic system: a metal pendulum over three magnets (fractal © Paul Nylander).

other everyday systems. For example, research on the mechanisms by which the heart
beat is generated has shown that the heart is not an oscillator, but a chaotic system with
irregular cycles. This allows the heart to be continuously ready for demands for changes
Ref. 331 in beat rate which arise once the body needs to increase or decrease its efforts.
There is chaotic motion also in machines: chaos appears in the motion of trains on the
rails, in gear mechanisms, and in fire-fighter’s hoses. The precise study of the motion in
Challenge 710 ny a zippo cigarette lighter will probably also yield an example of chaos. The mathematical
description of chaos – simple for some textbook examples, but extremely involved for
others – remains an important topic of research.
Incidentally, can you give a simple argument to show that the so-called butterfly effect
Challenge 711 s does not exist? This ‘effect’ is often cited in newspapers. The claim is that non-linearities
imply that a small change in initial conditions can lead to large effects; thus a butterfly
wing beat is alleged to be able to induce a tornado. Even though non-linearities do indeed
lead to growth of disturbances, the butterfly ‘effect’ has never been observed. Thus it does
not exist. This ‘effect’ exists only to sell books and to get funding.
All the steps from disorder to order, quasiperiodicity and finally to chaos, are ex-
amples of self-organization. These types of motion, illustrated in Figure 292, are observed
in many fluid systems. Their study should lead, one day, to a deeper understanding of the
426 14 self-organization and chaos

Ref. 332 mysteries of turbulence. Despite the fascination of this topic, we will not explore it fur-
ther, because it does not help solving the mystery of motion.

Emergence

“
From a drop of water a logician could predict

”
an Atlantic or a Niagara.
Arthur Conan Doyle, A Study in Scarlet

Self-organization is of interest also for a more general reason. It is sometimes said that
our ability to formulate the patterns or rules of nature from observation does not imply
the ability to predict all observations from these rules. According to this view, so-called
‘emergent’ properties exist, i.e., properties appearing in complex systems as something
new that cannot be deduced from the properties of their parts and their interactions.
(The ideological backdrop to this view is obvious; it is the latest attempt to fight the idea
of determinism.) The study of self-organization has definitely settled this debate. The
properties of water molecules do allow us to predict Niagara Falls.* Similarly, the diffu-

Motion Mountain – The Adventure of Physics
sion of signal molecules do determine the development of a single cell into a full human
being: in particular, cooperative phenomena determine the places where arms and legs
are formed; they ensure the (approximate) right–left symmetry of human bodies, prevent
mix-ups of connections when the cells in the retina are wired to the brain, and explain the
Ref. 334 fur patterns on zebras and leopards, to cite only a few examples. Similarly, the mechan-
isms at the origin of the heart beat and many other cycles have been deciphered. Several
cooperative phenomena in fluids have been simulated even down to the molecular level.
Self-organization provides general principles which allow us in principle to predict
the behaviour of complex systems of any kind. They are presently being applied to the
most complex system in the known universe: the human brain. The details of how it

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
learns to coordinate the motion of the body, and how it extracts information from the
images in the eye, are being studied intensely. The ongoing work in this domain is fas-
Ref. 335 cinating. (A neglected case of self-organization is laughter, but also humour itself.) If you
Challenge 713 e plan to become a scientist, consider taking this path.
Self-organization research provided the final arguments that confirmed what J. Of-
frey de la Mettrie stated and explored in his famous book L’homme machine in 1748:
humans are complex machines. Indeed, the lack of understanding of complex systems
in the past was due mainly to the restrictive teaching of the subject of motion, which
usually concentrated – as we do in this walk – on examples of motion in simple systems.
The concepts of self-organization allow us to understand and to describe what happens
during the functioning and the growth of organisms.
Even though the subject of self-organization provides fascinating insights, and will
do so for many years to come, we now leave it. We continue with our own adventure,

* Already small versions of Niagara Falls, namely dripping water taps, show a large range of cooperative
Ref. 333 phenomena, including the chaotic, i.e., non-periodic, fall of water drops. This happens when the water flow
Challenge 712 e rate has the correct value, as you can verify in your own kitchen.
the simplicity of complexity 427

water
pipe

pearls

finger
𝜆

Motion Mountain – The Adventure of Physics
F I G U R E 295 The wavy F I G U R E 296 Water F I G U R E 297 A
surface of icicles. pearls. braiding water stream
(© Vakhtang
Putkaradze).

namely to explore the basics of motion.

“
Ich sage euch: man muss noch Chaos in sich
haben, um einen tanzenden Stern gebären zu

”
euch.*
Friedrich Nietzsche, Also sprach Zarathustra.

Curiosities and fun challenges ab ou t self-organization
All icicles have a wavy surface, with a crest-to-crest distance of about 1 cm, as shown in
Ref. 336 Figure 295. The distance is determined by the interplay between water flow and surface
Challenge 714 ny cooling. How? (Indeed, stalactites do not show the effect.)
∗∗
When a fine stream of water leaves a water tap, putting a finger in the stream leads to a
Challenge 715 ny wavy shape, as shown in Figure 296. Why?
∗∗
The research on sand has shown that it is often useful to introduce the concept of granular
temperature, which quantifies how fast a region of sand moves. Research into this field is
still in full swing.

* ‘I tell you: one must have chaos inside oneself, in order to give birth to a dancing star. I tell you: you still
have chaos inside you.’
428 14 self-organization and chaos

F I G U R E 298 The Belousov-Zhabotinski
reaction: the liquid periodically changes
colour, both in space and time
(© Yamaguchi University).

∗∗

Motion Mountain – The Adventure of Physics
When water emerges from a oblong opening, the stream forms a braid pattern, as shown
in Figure 297. This effect results from the interplay and competition between inertia and
Ref. 337 surface tension: inertia tends to widen the stream, while surface tension tends to narrow
it. Predicting the distance from one narrow region to the next is still a topic of research.
If the experiment is done in free air, without a plate, one usually observes an additional
effect: there is a chiral braiding at the narrow regions, induced by the asymmetries of the
water flow. You can observe this effect in the toilet! Scientific curiosity knows no limits:
Challenge 716 e are you a right-turner or a left-turner, or both? On every day?
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
When wine is made to swirl in a wine glass, after the motion has calmed down, the wine
flowing down the glass walls forms little arcs. Can you explain in a few words what forms
Challenge 717 ny them?
∗∗
How does the average distance between cars parked along a street change over time,
Challenge 718 ny assuming a constant rate of cars leaving and arriving?
∗∗
A famous case of order appearance is the Belousov-Zhabotinski reaction. This mixture
of chemicals spontaneously produces spatial and temporal patterns. Thin layers pro-
duce slowly rotating spiral patterns, as shown in Figure 298; Large, stirred volumes
oscillate back and forth between two colours. A beautiful movie of the oscillations
can be found on www.uni-r.de/Fakultaeten/nat_Fak_IV/Organische_Chemie/Didaktik/
Keusch/D-oscill-d.htm. The exploration of this reaction led to the Nobel Prize in Chem-
istry for Ilya Prigogine in 1997.
∗∗
Gerhard Müller has discovered a simple but beautiful way to observe self-organization in
solids. His system also provides a model for a famous geological process, the formation of
the simplicity of complexity 429

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 299 A famous correspondence: on the left, hexagonal columns in starch, grown in a kitchen
pan (the red lines are 1 cm in length), and on the right, hexagonal columns in basalt, grown from lava in
Northern Ireland (top right, view of around 300 m, and middle right, view of around 40 m) and in
Iceland (view of about 30 m, bottom right) (© Gerhard Müller, Raphael Kessler - www.raphaelk.co.uk,
Bob Pohlad, and Cédric Hüsler).

hexagonal columns in basalt, such as the Giant’s Causeway in Northern Ireland. Similar
formations are found in many other places of the Earth. Just take some rice flour or
Ref. 338 corn starch, mix it with about half the same amount of water, put the mixture into a
Challenge 719 e pan and dry it with a lamp: hexagonal columns form. The analogy with basalt structures
is possible because the drying of starch and the cooling of lava are diffusive processes
governed by the same equations, because the boundary conditions are the same, and
because both materials respond to cooling with a small reduction in volume.
∗∗
Water flow in pipes can be laminar (smooth) or turbulent (irregular and disordered).
430 14 self-organization and chaos

The transition depends on the diameter 𝑑 of the pipe and the speed 𝑣 of the water. The
transition usually happens when the so-called Reynolds number – defined as Re = 𝑣𝑑/𝜂
becomes greater than about 2000. (The Reyonolds number is one of the few physical ob-
servables with a conventional abbreviation made of two letters.) Here, 𝜂 is the kinematic
viscosity of the water, around 1 mm2 /s; in contrast, the dynamic viscosity is defined as
Ref. 339 𝜇 = 𝜂𝜌, where 𝜌 is the density of the fluid. A high Reynolds number means a high ra-
tio between inertial and dissipative effects and specifies a turbulent flow; a low Reynolds
number is typical of viscous flow.
Modern, careful experiments show that with proper handling, laminar flows can be
produced up to Re = 100 000. A linear analysis of the equations of motion of the fluid, the
Navier–Stokes equations, even predicts stability of laminar flow for all Reynolds num-
bers. This riddle was solved only in the years 2003 and 2004. First, a complex mathem-
atical analysis showed that the laminar flow is not always stable, and that the transition
to turbulence in a long pipe occurs with travelling waves. Then, in 2004, careful experi-
ments showed that these travelling waves indeed appear when water is flowing through
Ref. 340 a pipe at large Reynolds numbers.

Motion Mountain – The Adventure of Physics
∗∗
For more beautiful pictures on self-organization in fluids, see the mentioned serve.me.
nus.edu.sg/limtt website.
∗∗
Chaos can also be observed in simple (and complicated) electronic circuits. If the elec-
tronic circuit that you have designed behaves erratically, check this option!
∗∗

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Also dance is an example of self-organization. This type of self-organization takes place in
the brain. Like for all complex movements, learning them is often a challenge. Nowadays
Ref. 341 there are beautiful books that tell how physics can help you improve your dancing skills
and the grace of your movements.
∗∗
Do you want to enjoy working on your PhD? Go into a scientific toy shop, and look for
any toy that moves in a complex way. There are high chances that the motion is chaotic;
explore the motion and present a thesis about it. For example, go to the extreme: explore
the motion of a hanging rope whose upper end is externally driven. This simple system
is fascinating in its range of complex motion behaviours.
∗∗
Self-organization is also observed in liquid corn starch–water mixtures. Enjoy the film
at www.youtube.com/watch?v=f2XQ97XHjVw and watch even more bizarre effects,
for humans walking over a pool filled with the liquid, on www.youtube.com/watch?
v=nq3ZjY0Uf-g.
∗∗
Snowflakes and snow crystals have already been mentioned as examples of self-
the simplicity of complexity 431

Motion Mountain – The Adventure of Physics
F I G U R E 300 How the shape of snow crystals depend on temperature and saturation (© Kenneth
Libbrecht).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
organization. Figure 300 shows the general connection. To learn more about this
fascinating topic, explore the wonderful website snowcrystals.com by Kenneth Lib-
Ref. 342 brecht. A complete classification of snow crystals has also been developed.
∗∗
A famous example of self-organization whose mechanisms are not well-known so far is
the hiccup. It is known that the vagus nerve plays a role in it. Like many other examples of
self-organization, it takes quite some energy to get rid of a hiccup. Modern experimental
research has shown that orgasms, which strongly stimulate the vagus nerve, are excel-
lent ways to overcome hiccups. One of these researchers won the 2006 IgNobel Prize for
medicine for his work.
∗∗
Another important example of self-organization is the weather. If you want to know more
about the known connections between the weather and the quality of human life on
Ref. 343 Earth, free of any ideology, read the wonderful book by Reichholf. It explains how the
weather between the continents is connected and describes how and why the weather
changed in the last one thousand years.
432 14 self-organization and chaos

Motion Mountain – The Adventure of Physics
F I G U R E 301 A
typical swarm
of starlings
that visitors in
Rome can
observe every
autumn
(© Andrea

∗∗
Does self-organization or biological evolution contradict the second principle of ther-
modynamics? Of course not.
Self-organizing systems are, by definition, open systems, and often far from equilib-
rium. The concept of entropy cannot be defined, in general, for such systems, and the
second principle of thermodynamics does not apply to them. However, entropy can be
defined for systems near equilibrium. In such systems the second principle of thermody-
namics must be modified; it is then possible to explore and confirm that self-organization
does not contradict but in fact follows from the modified second principle.
In particular, evolution does not contradict thermodynamics, as the Earth is not a
Ref. 344 closed thermodynamic system in equilibrium. Statements of the opposite are only made
by crooks.
∗∗
Ref. 345 In 2015, three physicists predicted that there is a largest Lyapunov exponent in nature,
the simplicity of complexity 433

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 302 An image of the south pole of Jupiter, taken in 2019 by the Juno space probe. The
cyclones have a typical diameter of 10 000 km (courtesy Msadler13, Wikimedia).

so that one always has
𝑘𝑇
𝜆 ⩽ 𝑒π (125)
ℏ
where 𝑇 is temperature, 𝑘 is Boltzmann’s constant, and ℏ is the quantum of action. The
growth of disorder is thus limited.
∗∗
Are systems that show self-organization the most complex ones that can be studied with
evolution equations? No. The most complex systems are those that consist of many in-
teracting self-organizing systems. The obvious example are swarms. Swarms of birds, as
Ref. 346 shown in Figure 301, of fish, of insects and of people – for example in a stadium or in
434 14 self-organization and chaos

Ref. 347 cars on a highway – have been studied extensively and are still a subject of research. Their
beauty is fascinating.
The other example of many interconnected self-organized systems is the brain; the
exploration of how the interconnected neurons work to produce our thoughts will occupy
researchers for many years. We will explore some aspects in the next volumes.
∗∗
Another example of the interplay between fluids, gravity, self-organisation and beauty is
shown in Figure 302.

Summary on self-organization and chaos
Appearance of order, in form of patterns, shapes and cycles, is not due to a decrease
in entropy, but to a competition between driving causes and dissipative effects in open
systems. Such appearance of order is a common process, is often automatic, and is pre-
dictable with (quite) simple equations. Also the growth of living systems and biological

Motion Mountain – The Adventure of Physics
evolution are examples of appearance of order.
Chaos, the sensitivity of evolution to initial conditions, is common in strongly driven
open systems. Chaos is at the basis of everyday chance, and often is described by simple
equations as well. In nature, complexity is apparent. Motion is simple.

F ROM T H E L I M I TAT ION S OF PH YSIC S
TO T H E L I M I T S OF MOT ION

“ ”
I only know that I know nothing.
Socrates, as cited by Plato

W
e have explored, in our environment, the concept of motion.
e called this exploration of moving objects and fluids Galilean physics.

Motion Mountain – The Adventure of Physics
e found that in everyday life, motion is predictable: nature shows no surprises
and no miracles. In particular, we have found six important aspects of this predictability:

1. Everyday motion is continuous. Motion allows us to define space and time.
2. Everyday motion conserves mass, momentum, energy and angular momentum.
Nothing appears out of nothing.
3. Everyday motion is relative: motion depends on the observer.
4. Everyday motion is reversible: everyday motion can occur backwards.
5. Everyday motion is mirror-invariant: everyday motion can occur in a mirror-reversed
way.

6. Everyday motion is lazy: motion happens in a way that minimizes change, i.e., phys-
ical action.

This Galilean description of nature made engineering possible: textile machines, steam
engines, combustion motors, kitchen appliances, watches, many children toys, fitness
machines, medical devices and all the progress in the quality of life that came with these
devices are due to the results of Galilean physics. But despite these successes, Socrates’
saying, cited above, still applies to Galilean physics: we still know almost nothing. Let us
see why.

R esearch topics in classical dynamics
Even though mechanics and thermodynamics are now several hundred years old, re-
search into its details is still ongoing. For example, we have already mentioned above
that it is unclear whether the Solar System is stable. The long-term future of the planets
is unknown! In general, the behaviour of few-body systems interacting through gravit-
Ref. 348 ation is still a research topic of mathematical physics. Answering the simple question of
how long a given set of bodies gravitating around each other will stay together is a for-
midable challenge. The history of this so-called many-body problem is long and involved.
436 15 from the limitations of physics

TA B L E 52 Examples of errors in state-of-the art measurements (numbers in brackets give one standard
deviation in the last digits), partly taken from physics.nist.gov/constants.

O b s e r va t i o n Measurement Precision /
accuracy
Highest precision achieved: ratio between −1.001 159 652 180 76(24) 2.6 ⋅ 10−13
the electron magnetic moment and the
Bohr magneton 𝜇𝑒 /𝜇B
High precision: Rydberg constant 10 973 731.568 539(55) m−1 5.0 ⋅ 10−12
High precision: astronomical unit 149 597 870.691(30) km 2.0 ⋅ 10−10
Industrial precision: typical part 2 μm of 20 cm 5 ⋅ 10−6
dimension tolerance in car engine
Low precision: gravitational constant 𝐺 6.674 28(67) ⋅ 10−11 Nm2 /kg2 1.0 ⋅ 10−4
Everyday precision: human circadian 15 h to 75 h 2
clock governing sleep

Motion Mountain – The Adventure of Physics
Interesting progress has been achieved, but the final answer still eludes us.
Many challenges remain in the fields of self-organization, of non-linear evolution
equations and of chaotic motion. In these fields, turbulence is a famous example: a pre-
cise description of turbulence has not yet been achieved, despite intense efforts. Large
prizes are offered for its solution.
Many other challenges motivate numerous researchers in mathematics, physics,
chemistry, biology, medicine and the other natural sciences. But apart from these re-
search topics, classical physics leaves unanswered several basic questions.

“
Democritus declared that there is a unique sort

”
of motion: that ensuing from collision.
Simplicius, Commentary on the Physics of
Ref. 349 Aristotle, 42, 10

Of the questions unanswered by classical physics, the details of contact and collisions are
among the most pressing. Indeed, we defined mass in terms of velocity changes during
Page 100 collisions. But why do objects change their motion in such instances? Why are collisions
between two balls made of chewing gum different from those between two stainless-steel
balls? What happens during those moments of contact?
Contact is related to material properties, which in turn influence motion in a complex
way. The complexity is such that the sciences of material properties developed independ-
ently from the rest of physics for a long time; for example, the techniques of metallurgy
(often called the oldest science of all), of chemistry and of cooking were related to the
properties of motion only in the twentieth century, after having been independently pur-
sued for thousands of years. Since material properties determine the essence of contact,
we need knowledge about matter and about materials to understand the notion of mass,
of contact and thus of motion. The parts of our adventure that deal with quantum theory
will reveal these connections.
to the limits of motion 437

What determines precision and accuracy?
Ref. 350 Precision has its own fascination. How many digits of π, the ratio between circumference
Challenge 720 e and diameter of a circle, do you know by heart? What is the largest number of digits of
π you have calculated yourself?
Is it possible to draw or cut a rectangle for which the ratio of lengths is a number, e.g. of
Challenge 721 s the form 0.131520091514001315211420010914..., whose digits encode a full book? (A
simple method would code a space as 00, the letter ‘a’ as 01, ‘b’ as 02, ‘c’ as 03, etc. Even
more interestingly, could the number be printed inside its own book?)
Why are so many measurement results, such as those of Table 52, of limited precision,
even if the available financial budget for the measurement apparatus is almost unlimited?
These are all questions about precision.
When we started climbing Motion Mountain, we explained that gaining height means
increasing the precision of our description of nature. To make even this statement itself
more precise, we distinguish between two terms: precision is the degree of reproducibil-
ity; accuracy is the degree of correspondence to the actual situation. Both concepts apply

Motion Mountain – The Adventure of Physics
to measurements,* to statement and to physical concepts.
Statements with false accuracy and false precision abound. What should we think
of a car company – Ford – who claim that the drag coefficient 𝑐w of a certain model
Challenge 722 s is 0.375? Or of the official claim that the world record in fuel consumption for cars is
2315.473 km/l? Or of the statement that 70.3 % of all citizens share a certain opinion?
One lesson we learn from investigations into measurement errors is that we should never
provide more digits for a result than we can put our hand into fire for.
Page 453 In short, precision and accuracy are limited. At present, the record number of reliable
digits ever measured for a physical quantity is 13. Why so few? Galilean physics doesn’t
provide an answer at all. What is the maximum number of digits we can expect in meas-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
urements; what determines it; and how can we achieve it? These questions are still open
at this point in our adventure. They will be covered in the parts on quantum theory.
In our walk we aim for highest possible precision and accuracy. Therefore, concepts
have mainly to be precise, and descriptions have to be accurate. Any inaccuracy is a proof
of lack of understanding. To put it bluntly, in our adventure, ‘inaccurate’ means wrong.
Increasing the accuracy and precision of our description of nature implies leaving behind
us all the mistakes we have made so far. This quest raises several issues.

C an all of nature be described in a b o ok?

“
Darum kann es in der Logik auch nie

”
Überraschungen geben.**
Ludwig Wittgenstein, Tractatus, 6.1251

Could the perfect physics publication, one that describes all of nature, exist? If it does,
it must also describe itself, its own production – including its readers and its author –
and most important of all, its own contents. Is such a book possible? Using the concept
of information, we can state that such a book must contain all information contained in

* For measurements, both precision and accuracy are best described by their standard deviation, as ex-
plained on page 459.
** ‘Hence there can never be surprises in logic.’
438 15 from the limitations of physics

the universe. Is this possible? Let us check the options.
If nature requires an infinitely long book to be fully described, such a publication ob-
viously cannot exist. In this case, only approximate descriptions of nature are possible
and a perfect physics book is impossible.
If nature requires a finite amount of information for its description, there are two op-
tions. One is that the information of the universe is so large that it cannot be summarized
in a book; then a perfect physics book is again impossible. The other option is that the
universe does contain a finite amount of information and that it can be summarized in
a few short statements. This would imply that the rest of the universe would not add to
the information already contained in the perfect physics book.
We note that the answer to this puzzle also implies the answer to another puzzle:
whether a brain can contain a full description of nature. In other words, the real ques-
tion is: can we understand nature? Can we reach our aim to understand motion? We
usually believe this. But the arguments just given imply that we effectively believe that
the universe does not contain more information than what our brain could contain or
Vol. VI, page 111 even contains already. What do you think? We will solve this puzzle later in our adven-

Motion Mountain – The Adventure of Physics
Challenge 723 e ture. Until then, do make up your own mind.

S omething is wrong ab ou t our description of motion

“
Je dis seulement qu’il n’y a point d’espace, où il
n’y a point de matière; et que l’espace lui-même

”
n’est point une réalité absolue. *
Leibniz

We described nature in a rather simple way. Objects are permanent and massive entities
localized in space-time. States are changing properties of objects, described by position

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in space and instant in time, by energy and momentum, and by their rotational equi-
valents. Time is the relation between events measured by a clock. Clocks are devices in
undisturbed motion whose position can be observed. Space and position is the relation
between objects measured by a metre stick. Metre sticks are devices whose shape is sub-
divided by some marks, fixed in an invariant and observable manner. Motion is change of
position with time (times mass); it is determined, does not show surprises, is conserved
(even in death), and is due to gravitation and other interactions.
Even though this description works rather well in practice, it contains a circular defin-
Challenge 724 s ition. Can you spot it? Each of the two central concepts of motion is defined with the
help of the other. Physicists worked for about 400 years on classical mechanics without
noticing or wanting to notice the situation. Even thinkers with an interest in discredit-
ing science did not point it out. Can an exact science be based on a circular definition?
Challenge 725 s Obviously yes, and physics has done quite well so far. Is the situation unavoidable in
principle?
Undoing the circular definition of Galilean physics is one of the aims of the rest of
our walk. The search for a solution is part of the last leg of our adventure. To achieve

* ‘I only say that there is no space where there is no matter; and that space itself is not an absolute reality.’
Gottfried Wilhelm Leibniz writes this already in 1716, in section 61 of his famous fifth letter to Clarke, the
assistant and spokesman of Newton. Newton, and thus Clarke, held the opposite view; and as usual, Leibniz
was right.
to the limits of motion 439

the solution, we need to increase substantially the level of precision in our description of
motion.
Whenever precision is increased, imagination is restricted. We will discover that many
types of motion that seem possible are not. Motion is limited. Nature limits speed, size,
acceleration, mass, force, power and many other quantities. Continue reading the other
parts of this adventure only if you are prepared to exchange fantasy for precision. It will
be no loss because exploring the precise working of nature will turn out to be more fas-
cinating than any fantasy.

Why is measurement possible?
In the description of gravity given so far, the one that everybody learns – or should learn
– at school, acceleration is connected to mass and distance via 𝑎 = 𝐺𝑀/𝑟2 . That’s all.
But this simplicity is deceiving. In order to check whether this description is correct,
we have to measure lengths and times. However, it is impossible to measure lengths and
time intervals with any clock or any ruler based on the gravitational interaction alone!

Motion Mountain – The Adventure of Physics
Challenge 726 s Try to conceive such an apparatus and you will inevitably be disappointed. You always
need a non-gravitational method to start and stop the stopwatch. Similarly, when you
measure length, e.g. of a table, you have to hold a ruler or some other device near it. The
interaction necessary to line up the ruler and the table cannot be gravitational.
A similar limitation applies even to mass measurements. Try to measure mass using
Challenge 727 s gravitation alone. Any scale or balance needs other – usually mechanical, electromag-
netic or optical – interactions to achieve its function. Can you confirm that the same
Challenge 728 s applies to speed and to angle measurements? In summary, whatever method we use,

⊳ In order to measure velocity, length, time, and mass, interactions other than

Our ability to measure shows that gravity is not all there is. And indeed, we still need to
understand charge and colours.
In short, Galilean physics does not explain our ability to measure. In fact, it does not
even explain the existence of measurement standards. Why do objects have fixed lengths?
Why do clocks work with regularity? Galilean physics cannot explain these observations;
we will need relativity and quantum physics to find out.

Is motion unlimited?
Galilean physics suggests that linear motion could go on forever. In fact, Galilean physics
tacitly assumes that the universe is infinite in space and time. Indeed, finitude of any
kind contradicts the Galilean description of motion. On the other hand, we know from
observation that the universe is not infinite: if it were infinite, the night would not be
Challenge 729 e dark.
Galilean physics also suggests that speeds can have any value. But the existence of
infinite speeds in nature would not allow us to define time sequences. Clocks would be
impossible. In other words, a description of nature that allows unlimited speeds is not
precise. Precision and measurements require limits.
Because Galilean physics disregards limits to motion, Galilean physics is inaccurate,
440 15 from the limitations of physics

and thus wrong.
To achieve the highest possible precision, and thus to find the correct description of
motion, we need to discover all of motion’s limits. So far, we have discovered one: there
is a smallest entropy in nature. We now turn to another, more striking limit: the speed
limit for energy, objects and signals. To observe and understand the speed limit, the next
volume explores the most rapid motion of energy, objects and signals that is known: the
motion of light.

NOTAT ION A N D C ON V E N T ION S

N
ewly introduced concepts are indicated, throughout this text, by italic typeface.
ew definitions are also referred to in the index. In this text,
aturally we use the international SI units; they are defined in Appendix B.
Page 453 Experimental results are cited with limited precision, usually only two digits, as this is
almost always sufficient for our purposes. High-precision reference values for important

Motion Mountain – The Adventure of Physics
quantities can also be found in Appendix B. Additional precision values on composite
Vol. V, page 342 physical systems are given in volume V.
But the information that is provided in this volume uses some additional conventions
that are worth a second look.

The L atin alphabet

“
What is written without effort is in general read

”
Ref. 351 without pleasure.
Samuel Johnson

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Books are collections of symbols. Writing was probably invented between 3400 and 3300
b ce by the Sumerians in Mesopotamia (though other possibilities are also discussed). It
then took over a thousand years before people started using symbols to represent sounds
instead of concepts: this is the way in which the first alphabet was created. This happened
between 2000 and 1600 b ce (possibly in Egypt) and led to the Semitic alphabet. The use
of an alphabet had so many advantages that it was quickly adopted in all neighbouring
cultures, though in different forms. As a result, the Semitic alphabet is the forefather of
all alphabets used in the world.
This text is written using the Latin alphabet. At first sight, this seems to imply that
its pronunciation cannot be explained in print, in contrast to the pronunciation of other
alphabets or of the International Phonetic Alphabet (IPA). (They can be explained using
the alphabet of the main text.) However, it is in principle possible to write a text that de-
scribes exactly how to move lips, mouth and tongue for each letter, using physical con-
cepts where necessary. The descriptions of pronunciations found in dictionaries make
indirect use of this method: they refer to the memory of pronounced words or sounds
found in nature.
Historically, the Latin alphabet was derived from the Etruscan, which itself was a de-
rivation of the Greek alphabet. An overview is given in Figure 303. There are two main
forms of the Latin alphabet.
442 a notation and conventions

Motion Mountain – The Adventure of Physics
F I G U R E 303 A summary of the history of the Latin alphabet (© Matt Baker at usefulcharts.com).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The ancient Latin alphabet,
used from the sixth century b ce onwards:
A B C D E F Z H I K L M N O P Q R S T V X

The classical Latin alphabet,
used from the second century b ce until the eleventh century:
A B C D E F G H I K L M N O P Q R S T V X Y Z

The letter G was added in the third century b ce by the first Roman to run a fee-paying
school, Spurius Carvilius Ruga. He added a horizontal bar to the letter C and substituted
the letter Z, which was not used in Latin any more, for this new letter. In the second
century b ce, after the conquest of Greece, the Romans included the letters Y and Z from
the Greek alphabet at the end of their own (therefore effectively reintroducing the Z) in
order to be able to write Greek words. This classical Latin alphabet was stable for the next
thousand years.*
The classical Latin alphabet was spread around Europe, Africa and Asia by the Ro-
mans during their conquests; due to its simplicity it began to be used for writing in nu-

* To meet Latin speakers and writers, go to www.alcuinus.net.
a notation and conventions 443

merous other languages. Most modern ‘Latin’ alphabets include a few other letters. The
letter W was introduced in the eleventh century in French and was then adopted in most
European languages.* The letter U was introduced in the mid-fifteenth century in Italy,
Ref. 352 the letter J at the end of that century in Spain, to distinguish certain sounds which had
previously been represented by V and I. The distinction proved a success and was already
common in most European languages in the sixteenth century. The contractions æ and
œ date from the Middle Ages. The German alphabet includes the sharp s, written ß, a
contraction of ‘ss’ or ‘sz’, and the Nordic alphabets added thorn, written Þ or þ, and eth,
Ref. 353 written Ð or ð, both taken from the futhorc,** and other signs.
Lower-case letters were not used in classical Latin; they date only from the Middle
Ages, from the time of Charlemagne. Like most accents, such as ê, ç or ä, which were
also first used in the Middle Ages, lower-case letters were introduced to save the then
expensive paper surface by shortening written words.

“
Outside a dog, a book is a man’s best friend.

”
Inside a dog, it’s too dark to read.
Groucho Marx

Motion Mountain – The Adventure of Physics
The Greek alphabet
The Greek alphabet is central to modern culture and civilization. It is at the origin of
the Etruscan alphabet, from which the Latin alphabet was derived. The Greek alphabet
was itself derived from the Phoenician or a similar northern Semitic alphabet in the
Ref. 355 tenth century b ce. The Greek alphabet, for the first time, included letters also for vowels,
which the Semitic alphabets lacked (and often still lack).
In the Phoenician alphabet and in many of its derivatives, such as the Greek alphabet,
each letter has a proper name. This is in contrast to the Etruscan and Latin alphabets.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The first two Greek letter names are, of course, the origin of the term alphabet itself.
In the tenth century b ce, the Ionian or ancient (eastern) Greek alphabet consisted of
the upper-case letters only. In the sixth century b ce several letters were dropped, while a
few new ones and the lower-case versions were added, giving the classical Greek alphabet.
Later, accents, subscripts and breathings were introduced. Table 53 also gives the values
signified by the letters when they were used as numbers. For this special use, the obsolete
ancient letters were kept during the classical period; thus they also acquired lower-case
forms.
The Latin correspondence in the table is the standard classical one, used for writing
Greek words. The question of the correct pronunciation of Greek has been hotly debated

* In Turkey, still in 2013, you can be convoked in front of a judge if you use the letters w, q or x in an official
letter; these letters only exist in the Kurdish language, not in Turkish. Using them is ‘unturkish’ behaviour,
support of terrorism and punishable by law. It is not generally known how physics and mathematics teachers
cope with this situation.
** The Runic script, also called Futhark or Futhorc, a type of alphabet used in the Middle Ages in Germanic,
Anglo–Saxon and Nordic countries, probably also derives from the Etruscan alphabet. The name derives
from the first six letters: f, u, th, a (or o), r, k (or c). The third letter is the letter thorn mentioned above; it is
often written ‘Y’ in Old English, as in ‘Ye Olde Shoppe.’ From the runic alphabet Old English also took the
Ref. 354 letter wyn to represent the ‘w’ sound, and the already mentioned eth. (The other letters used in Old English
– not from futhorc – were the yogh, an ancient variant of g, and the ligatures æ or Æ, called ash, and œ or
Œ, called ethel.)
444 a notation and conventions

TA B L E 53 The ancient and classical Greek alphabets, and the correspondence with Latin letters and
Indian digits.

Anc. Class. Name Corresp. Anc. Class. Name Corresp.
Α Α α alpha a 1 Ν Ν ν nu n 50
Β Β β beta b 2 Ξ Ξ ξ xi x 60
Γ Γ γ gamma g, n1 3 Ο Ο ο omicron o 70
Δ Δ δ delta d 4 Π Π π pi p 80
Ε Ε ε epsilon e 5 P Ϟ, qoppa3 q 90
F ϝ, Ϛ digamma, w 6 Ρ Ρ ρ rho r, rh 100
stigma2 Σ Σ σ, ς sigma4 s 200
Ζ Ζ ζ zeta z 7 Τ Τ τ tau t 300
Η Η η eta e 8 Υ υ upsilon y, u5 400
Θ Θ θ theta th 9 Φ φ phi ph, f 500
Ι Ι ι iota i, j 10 Χ χ chi ch 600

Motion Mountain – The Adventure of Physics
Κ Κ κ kappa k 20 Ψ ψ psi ps 700
Λ Λ λ lambda l 30 Ω ω omega o 800
Μ Μ μ mu m 40 Λ Ϡ
v sampi6 s 900

The regional archaic letters yot, sha and san are not included in the table. The letter san was the ancestor of
sampi.
1. Only if before velars, i.e., before kappa, gamma, xi and chi.
2. ‘Digamma’ is the name used for the F-shaped form. It was mainly used as a letter (but also sometimes,
in its lower-case form, as a number), whereas the shape and name ‘stigma’ is used only for the number.
Both names were derived from their respective shapes; in fact, the stigma is a medieval, uncial version

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of the digamma. The name ‘stigma’ is derived from the fact that the letter looks like a sigma with a tau
attached under it – though unfortunately not in all modern fonts. The original letter name, also giving its
pronunciation, was ‘waw’.
3. The version of qoppa that looks like a reversed and rotated z is still in occasional use in modern Greek.
Unicode calls this version ‘koppa’.
4. The second variant of sigma is used only at the end of words.
5. Uspilon corresponds to ‘u’ only as the second letter in diphthongs.
6. In older times, the letter sampi was positioned between pi and qoppa.

in specialist circles; the traditional Erasmian pronunciation does not correspond either to
the results of linguistic research nor to modern Greek. In classical Greek, the sound that
sheep make was βη–βη. (Erasmian pronunciation wrongly insists on a narrow η; modern
Greek pronunciation is different for β, which is now pronounced ‘v’, and for η, which is
now pronounced as ‘i:’ – a long ‘i’.) Obviously, the pronunciation of Greek varied from
region to region and over time. For Attic Greek, the main dialect spoken in the classical
period, the question is now settled. Linguistic research has shown that chi, phi and theta
were less aspirated than usually pronounced in English and sounded more like the initial
sounds of ‘cat’, ‘perfect’ and ‘tin’; moreover, the zeta seems to have been pronounced
more like ‘zd’ as in ‘buzzed’. As for the vowels, contrary to tradition, epsilon is closed
and short whereas eta is open and long; omicron is closed and short whereas omega is
wide and long, and upsilon is really a sound like a French ‘u’ or German ‘ü.’
a notation and conventions 445

The Greek vowels can have rough or smooth breathings, subscripts, and acute, grave,
circumflex or diaeresis accents. Breathings – used also on ρ – determine whether the
letter is aspirated. Accents, which were interpreted as stresses in the Erasmian pronunci-
ation, actually represented pitches. Classical Greek could have up to three of these added
signs per letter; modern Greek never has more than one.
Another descendant of the Greek alphabet* is the Cyrillic alphabet, which is used with
slight variations in many Slavic languages, such as Russian and Bulgarian. There is no
standard transcription from Cyrillic to Latin, so that often the same Russian name is
spelled differently in different countries or even in the same country on different occa-
sions.
TA B L E 54 The beginning of the Hebrew abjad.

L e t t e r N a m e s Correspondence
ℵ aleph a 1
ℶ beth b 2

Motion Mountain – The Adventure of Physics
ℷ gimel g 3
ℸ daleth d 4
etc.

The Hebrew alphabet and other scripts
The Phoenician alphabet is also the origin of the Hebrew consonant alphabet or abjad. Its
Ref. 356 first letters are given in Table 54. Only the letter aleph is commonly used in mathematics,
Vol. III, page 288 though others have been proposed.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Around one hundred writing systems are in use throughout the world. Experts clas-
sify them into five groups. Phonemic alphabets, such as Latin or Greek, have a sign for
each consonant and vowel. Abjads or consonant alphabets, such as Hebrew or Arabic,
have a sign for each consonant (sometimes including some vowels, such as aleph), and
do not write (most) vowels; most abjads are written from right to left. Abugidas, also
called syllabic alphabets or alphasyllabaries, such as Balinese, Burmese, Devanagari, Ta-
galog, Thai, Tibetan or Lao, write consonants and vowels; each consonant has an inherent
vowel which can be changed into the others by diacritics. Syllabaries, such as Hiragana
or Ethiopic, have a sign for each syllable of the language. Finally, complex scripts, such as
Chinese, Mayan, or Egyptian hieroglyphs, use signs which have both sound and mean-
ing. Writing systems can have text flowing from right to left, from bottom to top, and
can count book pages in the opposite sense to this book.
* The Greek alphabet is also the origin of the Gothic alphabet, which was defined in the fourth century by
Wulfila for the Gothic language, using also a few signs from the Latin and futhorc scripts.
The Gothic alphabet is not to be confused with the so-called Gothic letters, a style of the Latin alphabet
used all over Europe from the eleventh century onwards. In Latin countries, Gothic letters were replaced in
the sixteenth century by the Antiqua, the ancestor of the type in which this text is set. In other countries,
Gothic letters remained in use for much longer. They were used in type and handwriting in Germany until
1941, when the National Socialist government suddenly abolished them, in order to comply with popu-
lar demand. They remain in sporadic use across Europe. In many physics and mathematics books, Gothic
letters are used to denote vector quantities.
446 a notation and conventions

Even though there are about 6000 languages on Earth, there are only about one hun-
Ref. 357 dred writing systems in use today. About fifty other writing systems have fallen out of
use.*For physical and mathematical formulae, though, the sign system used in this text,
based on Latin and Greek letters, written from left to right and from top to bottom, is a
standard the world over. It is used independently of the writing system of the text con-
taining it.

Numbers and the Indian digits
Both the digits and the method used in this text to write numbers originated in India.
The Indian digits date from around 600 b ceand were brought to the Mediterranean by
Ref. 358 Arabic mathematicians in the Middle Ages. The number system used in this text is thus
much younger than the alphabet. The Indian numbers were made popular in Europe by
Leonardo of Pisa, called Fibonacci,** in his book Liber Abaci or ‘Book of Calculation’,
which he published in 1202. That book revolutionized mathematics. Anybody with paper
and a pen (the pencil had not yet been invented) was now able to calculate and write

Motion Mountain – The Adventure of Physics
down numbers as large as reason allowed, or even larger, and to perform calculations
with them. Fibonacci’s book started:

Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1. Cum his itaque novem figu-
ris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet
numerus, ut inferius demonstratur.***

The Indian method of writing numbers, the Indian number system, introduced two in-
novations: a large one, the positional system, and a small one, the digit zero. **** The pos-
itional system, as described by Fibonacci, was so much more efficient that it completely

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
replaced the previous Roman number system, which writes 1996 as IVMM or MCMIVC or
MCMXCVI, as well as the Greek number system, in which the Greek letters were used for
numbers in the way shown in Table 53, thus writing 1996 as ͵αϠϞϚʹ. Compared to these
systems, the Indian numbers are a much better technology.
The Indian number system proved so practical that calculations done on paper com-
pletely eliminated the need for the abacus, which therefore fell into disuse. The abacus is
still in use in countries, for example in Asia, America or Africa, and by people who do
not use a positional system to write numbers. It is also useful for the blind.
The Indian number system also eliminated the need for systems to represent numbers
with fingers. Such ancient systems, which could show numbers up to 10 000 and more,

* A well-designed website on the topic is www.omniglot.com. The main present and past writing systems
are encoded in the Unicode standard, which at present contains 52 writing systems. See www.unicode.org.
An eccentric example of a writing system is the alphabet proposed by George Bernard Shaw, presented at
en.wikipedia.org/wiki/Shavian_alphabet.
** Leonardo di Pisa, called Fibonacci (b. c. 1175 Pisa, d. 1250 Pisa), was the most important mathematician
of his time.
*** ‘The nine figures of the Indians are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which
in Arabic is called zephirum, any number can be written, as will be demonstrated below.’
**** It is thus not correct to call the digits 0 to 9 Arabic. Both the digits used in Arabic texts and the digits
used in Latin texts such as this one derive from the Indian digits. You can check this by yourself: only the
digits 0, 2, 3 and 7 resemble those used in Arabic writing, and then only if they are turned clockwise by 90°.
a notation and conventions 447

have left only one trace: the term ‘digit’ itself, which derives from the Latin word for
finger.
The power of the positional number system is often forgotten. For example, only a
positional number system allows mental calculations and makes calculating prodigies
possible.*

The symb ols used in the text

“
To avoide the tediouse repetition of these
woordes: is equalle to: I will sette as I doe often
in woorke use, a paire of paralleles, or Gemowe
lines of one lengthe, thus: = , bicause noe .2.

”
thynges, can be moare equalle
Robert Recorde**

Besides text and numbers, physics books contain other symbols. Most symbols have been
developed over hundreds of years, so that only the clearest and simplest are now in use.
In this adventure, the symbols used as abbreviations for physical quantities are all taken

Motion Mountain – The Adventure of Physics
from the Latin or Greek alphabets and are always defined in the context where they are
used. The symbols designating units, constants and particles are defined in Appendix B
Page 453 and in Appendix B of volume V. The symbols used in this text are those in common use
Ref. 356 in the practice and the teaching of physics.
There is even an international standard for the symbols in physical formulae – ISO
EN 80000, formerly ISO 31 – but it is shamefully expensive, virtually inaccessible and
incredibly useless: the symbols listed in it are those in common use anyway, and their use
is not binding anywhere, not even in the standard itself! ISO 80000 is a prime example
of bureaucracy gone wrong.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The mathematical symbols used in this text, in particular those for mathematical op-
erations and relations, are given in the following list, together with their historical origin.
Ref. 360 The details of their history have been extensively studied in by scholars.

TA B L E 55 The history of mathematical notation and symbols.

Symbol Meaning Origin

+, − plus, minus Johannes Widmann 1489; the plus sign
is derived from Latin ‘et’.
√ read as ‘square root’ used by Christoff Rudolff in 1525; the
sign evolved from a point.
= equal to Robert Recorde 1557

* Currently, the shortest time for finding the thirteenth (integer) root of a hundred-digit (integer) number,
a result with 8 digits, is below 4 seconds, and 70.2 seconds for a 200 digit number, for which the result
has sixteen digits. Both records are held by Alexis Lemaire. For more about the stories and the methods of
Ref. 359 calculating prodigies, see the bibliography.
** Robert Recorde (b. c. 1510 Tenby, d. 1558 London), mathematician and physician; he died in prison be-
cause of debts. The quotation is from his The Whetstone of Witte, 1557. An image showing the quote can be
found at en.wikipedia.org/wiki/Equals_sign. It is usually suggested that the quote is the first introduction
of the equal sign; claims that Italian mathematicians used the equal sign before Recorde are not backed up
Ref. 360 by convincing examples.
448 a notation and conventions

TA B L E 55 (Continued) The history of mathematical notation and symbols.

Symbol Meaning Origin

{ }, [ ], ( ) grouping symbols use starts in the sixteenth century
>, < larger than, smaller than Thomas Harriot 1631
× multiplied with, times England c. 1600, made popular by Wil-
liam Oughtred 1631
𝑎𝑛 𝑎 to the power 𝑛, 𝑎 ⋅ ... ⋅ 𝑎 (𝑛 factors) René Descartes 1637
𝑥, 𝑦, 𝑧 coordinates, unknowns René Descartes 1637
𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 constants and equations for unknowns René Descartes 1637
∞ infinity John Wallis 1655
d/d𝑥, d𝑥, derivative, differential, integral Gottfried Wilhelm Leibniz 1675
∫ 𝑦 d𝑥
: divided by Gottfried Wilhelm Leibniz 1684
⋅ multiplied with, times Gottfried Wilhelm Leibniz c. 1690

Motion Mountain – The Adventure of Physics
𝑎1 , 𝑎𝑛 indices Gottfried Wilhelm Leibniz c. 1690
∼ similar to Gottfried Wilhelm Leibniz c. 1690
π circle number, 4 arctan 1 William Jones 1706
𝜑𝑥 function of 𝑥 Johann Bernoulli 1718
𝑓𝑥, 𝑓(𝑥) function of 𝑥 Leonhard Euler 1734
e ∑∞ 1
𝑛=0 𝑛! = lim 𝑛→∞ (1 + 1/𝑛) 𝑛
Leonhard Euler 1736
󸀠
𝑓 (𝑥) derivative of function at 𝑥 Giuseppe Lagrangia 1770
Δ𝑥, ∑ difference, sum Leonhard Euler 1755
∏ product Carl Friedrich Gauss 1812

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i imaginary unit, +√−1 Leonhard Euler 1777
≠ is different from Leonhard Euler eighteenth century
∂/∂𝑥 partial derivative, read like ‘d/d𝑥’ it was derived from a cursive form of
‘d’ or of the letter ‘dey’ of the Cyrillic
alphabet by Adrien-Marie Legendre in
1786 and made popular by Carl Gustav
Jacobi in 1841
𝑛! factorial, 1 ⋅ 2 ⋅ ...⋅ Christian Kramp 1808
Δ Laplace operator Robert Murphy 1833
|𝑥| absolute value Karl Weierstrass 1841
∇ read as ‘nabla’ (or ‘del’) introduced by William Hamilton in
1853 and Peter Tait in 1867, named after
the shape of an old Egyptian musical in-
strument
⊂, ⊃ set inclusion Ernst Schröder in 1890
∪, ∩ set union and intersection Giuseppe Peano 1888
∈ element of Giuseppe Peano 1888
⊗ dyadic product or tensor product or unknown
outer product
⟨𝜓|, |𝜓⟩ bra and ket state vectors Paul Dirac 1930
a notation and conventions 449

TA B L E 55 (Continued) The history of mathematical notation and symbols.

Symbol Meaning Origin

⌀ empty set André Weil as member of the Nicolas
Bourbaki group in the early twentieth
century
[𝑥] the measurement unit of a quantity 𝑥 twentieth century

Other signs used here have more complicated origins. The & sign is a contraction of Latin
Ref. 361 et meaning ‘and’, as is often more clearly visible in its variations, such as &, the common
italic form.
Each of the punctuation signs used in sentences with modern Latin alphabets, such
as , . ; : ! ? ‘ ’ » « – ( ) ... has its own history. Many are from ancient Greece, but the
Ref. 362 question mark is from the court of Charlemagne, and exclamation marks appear first in
the sixteenth century.* The @ or at-sign probably stems from a medieval abbreviation

Motion Mountain – The Adventure of Physics
Ref. 363 of Latin ad, meaning ‘at’, similarly to how the & sign evolved from Latin et. In recent
years, the smiley :-) and its variations have become popular. The smiley is in fact a new
version of the ‘point of irony’ which had been proposed in 1899, without success, by the
poet Alcanter de Brahm (b. 1868 Mulhouse, d. 1942 Paris).
The section sign § dates from the thirteenth century in northern Italy, as was shown by
Ref. 364 the palaeographer Paul Lehmann. It was derived from ornamental versions of the capital
letter C for capitulum, i.e., ‘little head’ or ‘chapter.’ The sign appeared first in legal texts,
where it is still used today, and then spread into other domains.
The paragraph sign ¶ was derived from a simpler ancient form looking like the Greek
letter Γ, a sign which was used in manuscripts from ancient Greece until well into the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Middle Ages to mark the start of a new text paragraph. In the Middle Ages it took the
modern form, probably because a letter c for caput was added in front of it.
Ref. 365 One of the most important signs of all, the white space separating words, was due to
Celtic and Germanic influences when these people started using the Latin alphabet. It
became commonplace between the ninth and the thirteenth century, depending on the
language in question.

C alendars
The many ways to keep track of time differ greatly from civilization to civilization. The
most common calendar, and the one used in this text, is also one of the most absurd, as
it is a compromise between various political forces who tried to shape it.
In ancient times, independent localized entities, such as tribes or cities, preferred
lunar calendars, because lunar timekeeping is easily organized locally. This led to the
use of the month as a calendar unit. Centralized states imposed solar calendars, based
on the year. Solar calendars require astronomers, and thus a central authority to finance
them. For various reasons, farmers, politicians, tax collectors, astronomers, and some,
but not all, religious groups wanted the calendar to follow the solar year as precisely as
possible. The compromises necessary between days and years are the origin of leap days.
* On the parenthesis see the beautiful book by J. Lennard, But I Digress, Oxford University Press, 1991.
450 a notation and conventions

The compromises necessary between months and year led to the varying lengths of the
months; they are different in different calendars. The most commonly used year–month
structure was organized over 2000 years ago by Caesar, and is thus called the Julian cal-
endar.
The system was destroyed only a few years later: August was lengthened to 31 days
when it was named after Augustus. Originally, the month was only 30 days long; but in
order to show that Augustus was as important as Caesar, after whom July is named, all
month lengths in the second half of the year were changed, and February was shortened
by one additional day.
Ref. 366 The week is an invention of the Babylonians. One day in the Babylonian week was
‘evil’ or ‘unlucky’, so it was better to do nothing on that day. The modern week cycle
with its resting day descends from that superstition. (The way astrological superstition
Page 221 and astronomy cooperated to determine the order of the weekdays is explained in the
section on gravitation.) Although about three thousand years old, the week was fully in-
cluded into the Julian calendar only around the year 300, towards the end of the Western
Roman Empire. The final change in the Julian calendar took place between 1582 and 1917

Motion Mountain – The Adventure of Physics
(depending on the country), when more precise measurements of the solar year were
used to set a new method to determine leap days, a method still in use today. Together
with a reset of the date and the fixation of the week rhythm, this standard is called the
Gregorian calendar or simply the modern calendar. It is used by a majority of the world’s
population.
Despite its complexity, the modern calendar does allow you to determine the day of
the week of a given date in your head. Just execute the following six steps:
1. take the last two digits of the year, and divide by 4, discarding any fraction;
2. add the last two digits of the year;

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
3. subtract 1 for January or February of a leap year;
4. add 6 for 2000s or 1600s, 4 for 1700s or 2100s,
2 for 1800s and 2200s, and 0 for 1900s or 1500s;
5. add the day of the month;
6. add the month key value, namely 144 025 036 146 for JFM AMJ JAS OND.
The remainder after division by 7 gives the day of the week, with the correspondence 1-2-
3-4-5-6-0 meaning Sunday-Monday-Tuesday-Wednesday-Thursday-Friday-Saturday.*
When to start counting the years is a matter of choice. The oldest method not attached
to political power structures was that used in ancient Greece, when years were coun-
ted from the first Olympic games. People used to say, for example, that they were born
in the first year of the twenty-third Olympiad. Later, political powers always imposed
the counting of years from some important event onwards.** Maybe reintroducing the

* Remembering the intermediate result for the current year can simplify things even more, especially since
the dates 4.4, 6.6, 8.8, 10.10, 12.12, 9.5, 5.9, 7.11, 11.7 and the last day of February all fall on the same day
of the week, namely on the year’s intermediate result plus 4.
** The present counting of years was defined in the Middle Ages by setting the date for the foundation of
Rome to the year 753 bce, or 753 before the Common Era, and then counting backwards, so that the bce
years behave almost like negative numbers. However, the year 1 follows directly after the year 1 bce: there
was no year 0.
Some other standards set by the Roman Empire explain several abbreviations used in the text:
- c. is a Latin abbreviation for circa and means ‘roughly’;
a notation and conventions 451

Olympic counting is worth considering?

People Names
In the Far East, such as Corea*, Japan or China, family names are put in front of the
given name. For example, the first Japanese winner of the Nobel Prize in Physics was
Yukawa Hideki. In India, often, but not always, there is no family name; in those cases,
the father’s first name is used. In Russia, the family name is rarely used in conversa-
tion; instead, the first name of the father is. For example, Lev Landau was addressed as
Lev Davidovich (‘son of David’). In addition, Russian transliteration is not standard-
ized; it varies from country to country and from tradition to tradition. For example, one
finds the spellings Dostojewski, Dostoevskij, Dostoïevski and Dostoyevsky for the same
person. In the Netherlands, the official given names are never used; every person has a
semi-official first name by which he is called. For example, Gerard ’t Hooft’s official given
name is Gerardus. In Germany, some family names have special pronunciations. For ex-
ample, Voigt is pronounced ‘Fohgt’. In Italy, during the Middle Age and the Renaissance,

Motion Mountain – The Adventure of Physics
people were called by their first name only, such as Michelangelo or Galileo, or often by
first name plus a personal surname that was not their family name, but was used like one,
such as Niccolò Tartaglia or Leonardo Fibonacci. In ancient Rome, the name by which
people are known is usually their surname. The family name was the middle name. For
example, Cicero’s family name was Tullius. The law introduced by Cicero was therefore
known as ‘lex Tullia’. In ancient Greece, there were no family names. People had only
one name. In the English language, the Latin version of the Greek name is used, such as
Democritus.

Abbreviations and eponyms or concepts?

The EPR paradox in the Bohm formulation can perhaps be resolved using the GRW
approach, with the help of the WKB approximation of the Schrödinger equation.

- i.e. is a Latin abbreviation for id est and means ‘that is’;
- e.g. is a Latin abbreviation for exempli gratia and means ‘for the sake of example’;
- ibid. is a Latin abbreviation for ibidem and means ‘at that same place’;
- inf. is a Latin abbreviation for infra and means ‘(see) below’;
- op. cit. is a Latin abbreviation for opus citatum and means ‘the cited work’;
- et al. is a Latin abbreviation for et alii and means ‘and others’.
By the way, idem means ‘the same’ and passim means ‘here and there’ or ‘throughout’. Many terms used in
physics, like frequency, acceleration, velocity, mass, force, momentum, inertia, gravitation and temperature,
are derived from Latin. In fact, it is arguable that the language of science has been Latin for over two thou-
sand years. In Roman times it was Latin vocabulary with Latin grammar, in modern times it switched to
Latin vocabulary with French grammar, then for a short time to Latin vocabulary with German grammar,
after which it changed to Latin vocabulary with British/American grammar.
Many units of measurement also date from Roman times, as explained in the next appendix. Even the
Ref. 367 infatuation with Greek technical terms, as shown in coinages such as ‘gyroscope’, ‘entropy’ or ‘proton’,
dates from Roman times.
* Corea was temporarily forced to change its spelling to ‘Korea’ by the Japanese Army because the generals
could not bear the fact that Corea preceded Japan in the alphabet. This is not a joke.
452 a notation and conventions

Using such vocabulary is the best way to make language unintelligible to outsiders. (In
fact, the sentence is nonsense anyway, because the ‘GRW approach’ is false.) First of all,
the sentence uses abbreviations, which is a shame. On top of this, the sentence uses
people’s names to characterize concepts, i.e., it uses eponyms. Originally, eponyms were
intended as tributes to outstanding achievements. Today, when formulating radical new
laws or variables has become nearly impossible, the spread of eponyms intelligible to a
steadily decreasing number of people simply reflects an increasingly ineffective drive to
fame.
Eponyms are a proof of scientist’s lack of imagination. We avoid them as much as
possible in our walk and give common names to mathematical equations or entities
wherever possible. People’s names are then used as appositions to these names. For ex-
ample, ‘Newton’s equation of motion’ is never called ‘Newton’s equation’; ‘Einstein’s
field equations’ is used instead of ‘Einstein’s equations’; and ‘Heisenberg’s equation of
motion’ is used instead of ‘Heisenberg’s equation’.
However, some exceptions are inevitable: certain terms used in modern physics have
no real alternatives. The Boltzmann constant, the Planck scale, the Compton wavelength,

Motion Mountain – The Adventure of Physics
the Casimir effect and Lie groups are examples. In compensation, the text makes sure that
you can look up the definitions of these concepts using the index. In addition, the text
Ref. 368 tries to provide pleasurable reading.

U N I T S , M E A SU R E M E N T S A N D
C ON STA N T S

M
easurements are comparisons with standards. Standards are based on units.
any different systems of units have been used throughout the world.
ost of these standards confer power to the organization in charge of them.
Such power can be misused; this is the case today, for example in the computer in-
dustry, and was so in the distant past. The solution is the same in both cases: organize

Motion Mountain – The Adventure of Physics
an independent and global standard. For measurement units, this happened in the
eighteenth century: in order to avoid misuse by authoritarian institutions, to eliminate
problems with differing, changing and irreproducible standards, and – this is not a joke
– to simplify tax collection and to make it more just, a group of scientists, politicians
and economists agreed on a set of units. It is called the Système International d’Unités,
abbreviated SI, and is defined by an international treaty, the ‘Convention du Mètre’.
The units are maintained by an international organization, the ‘Conférence Générale
des Poids et Mesures’, and its daughter organizations, the ‘Commission Internationale
des Poids et Mesures’ and the ‘Bureau International des Poids et Mesures’ (BIPM). All
originated in the times just before the French revolution.

SI units
All SI units are built from seven base units. Their simplest definitions, translated from
French into English, are the following ones, together with the dates of their formulation
and a few comments:
‘The second is the duration of 9 192 631 770 periods of the radiation corresponding
to the transition between the two hyperfine levels of the ground state of the caesium 133
atom.’ (1967) The 2019 definition is equivalent, but much less clear.*
‘The metre is the length of the path travelled by light in vacuum during a time inter-
val of 1/299 792 458 of a second.’ (1983) The 2019 definition is equivalent, but much less
clear.*
‘The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed
numerical value of the Planck constant h to be 6.626 070 15 ⋅ 10−34 when expressed in the
unit J ⋅ s, which is equal to kg ⋅ m2 ⋅ s−1 .’ (2019)*
‘The ampere, symbol A, is the SI unit of electric current. It is defined by taking the
fixed numerical value of the elementary charge e to be 1.602 176 634 ⋅ 10−19 when ex-
pressed in the unit C, which is equal to A ⋅ s.’ (2019)* This definition is equivalent to:
One ampere is 6.241 509 074... ⋅ 1018 elementary charges per second.
‘The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by
454 b units, measurements and constants

taking the fixed numerical value of the Boltzmann constant 𝑘 to be 1.380649 ⋅10−23 when
expressed in the unit J ⋅ K−1 .’ (2019)*
‘The mole, symbol mol, is the SI unit of amount of substance. One mole contains
exactly 6.02214076 ⋅ 1023 elementary entities.’ (2019)*
‘The candela is the luminous intensity, in a given direction, of a source that emits
monochromatic radiation of frequency 540 ⋅ 1012 hertz and has a radiant intensity in
that direction of (1/683) watt per steradian.’ (1979) The 2019 definition is equivalent, but
much less clear.*
We note that both time and length units are defined as certain properties of a standard
example of motion, namely light. In other words, also the Conférence Générale des Poids
et Mesures makes the point that the observation of motion is a prerequisite for the defin-
ition and construction of time and space. Motion is the fundament of every observation
and of all measurement. By the way, the use of light in the definitions had been proposed
already in 1827 by Jacques Babinet.**
From these basic units, all other units are defined by multiplication and division. Thus,

Motion Mountain – The Adventure of Physics
all SI units have the following properties:
SI units form a system with state-of-the-art precision: all units are defined with a pre-
cision that is higher than the precision of commonly used measurements. Moreover, the
precision of the definitions is regularly being improved. The present relative uncertainty
of the definition of the second is around 10−14 , for the metre about 10−10 , for the kilo-
gram about 10−9 , for the ampere 10−7 , for the mole less than 10−6 , for the kelvin 10−6 and
for the candela 10−3 .
SI units form an absolute system: all units are defined in such a way that they can
be reproduced in every suitably equipped laboratory, independently, and with high pre-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
cision. This avoids as much as possible any error or misuse by the standard-setting or-
ganization. In fact, the SI units are as now as near as possible to Planck’s natural units,
which are presented below. In practice, the SI is now an international standard defining
the numerical values of the seven constants Δ𝜈Cs , 𝑐, ℏ, 𝑒, 𝑘, 𝑁A and 𝐾cd . After over 200
years of discussions, the CGPM has little left to do.
SI units form a practical system: the base units are quantities of everyday magnitude.
Frequently used units have standard names and abbreviations. The complete list includes
the seven base units just given, the supplementary units, the derived units and the ad-
mitted units.
The supplementary SI units are two: the unit for (plane) angle, defined as the ratio
of arc length to radius, is the radian (rad). For solid angle, defined as the ratio of the
subtended area to the square of the radius, the unit is the steradian (sr).
The derived units with special names, in their official English spelling, i.e., without
capital letters and accents, are:

* The symbols of the seven units are s, m, kg, A, K, mol and cd. The full official definitions are found at
Ref. 370 www.bipm.org. For more details about the levels of the caesium atom, consult a book on atomic physics.
The Celsius scale of temperature 𝜃 is defined as: 𝜃/°C = 𝑇/K − 273.15; note the small difference with the
number appearing in the definition of the kelvin. In the definition of the candela, the frequency of the light
corresponds to 555.5 nm, i.e., green colour, around the wavelength to which the eye is most sensitive.
** Jacques Babinet (1794–1874), French physicist who published important work in optics.
b units, measurements and constants 455

Name A bbre v iat i o n Name A b b r e v i at i o n

hertz Hz = 1/s newton N = kg m/s2
pascal Pa = N/m2 = kg/m s2 joule J = Nm = kg m2 /s2
watt W = kg m2 /s3 coulomb C = As
volt V = kg m2 /As3 farad F = As/V = A2 s4 /kg m2
ohm Ω = V/A = kg m2 /A2 s3 siemens S = 1/Ω
weber Wb = Vs = kg m2 /As2 tesla T = Wb/m2 = kg/As2 = kg/Cs
henry H = Vs/A = kg m2 /A2 s2 degree Celsius °C (see definition of kelvin)
lumen lm = cd sr lux lx = lm/m2 = cd sr/m2
becquerel Bq = 1/s gray Gy = J/kg = m2 /s2
sievert Sv = J/kg = m2 /s2 katal kat = mol/s

We note that in all definitions of units, the kilogram only appears to the powers of 1,
Challenge 730 s 0 and −1. Can you try to formulate the reason?

Motion Mountain – The Adventure of Physics
The admitted non-SI units are minute, hour, day (for time), degree 1° = π/180 rad,
minute 1 󸀠 = π/10 800 rad, second 1 󸀠󸀠 = π/648 000 rad (for angles), litre, and tonne. All
other units are to be avoided.
All SI units are made more practical by the introduction of standard names and ab-
breviations for the powers of ten, the so-called prefixes:*

Power Name Power Name Power Name Power Name
101 deca da 10−1 deci d 1018 Exa E 10−18 atto a
102 hecto h 10−2 centi c 1021 Zetta Z 10−21 zepto z

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
103 kilo k 10−3 milli m 1024 Yotta Y 10−24 yocto y
106 Mega M 10−6 micro μ unofficial: Ref. 371
109 Giga G 10−9 nano n 1027 Xenta X 10−27 xenno x
1012 Tera T 10−12 pico p 1030 Wekta W 10−30 weko w
1015 Peta P 10−15 femto f 1033 Vendekta V 10−33 vendeko v
1036 Udekta U 10−36 udeko u

SI units form a complete system: they cover in a systematic way the full set of ob-
servables of physics. Moreover, they fix the units of measurement for all other sciences
as well.

* Some of these names are invented (yocto to sound similar to Latin octo ‘eight’, zepto to sound similar
to Latin septem, yotta and zetta to resemble them, exa and peta to sound like the Greek words ἑξάκις and
πεντάκις for ‘six times’ and ‘five times’, the unofficial ones to sound similar to the Greek words for nine,
ten, eleven and twelve); some are from Danish/Norwegian (atto from atten ‘eighteen’, femto from femten
‘fifteen’); some are from Latin (from mille ‘thousand’, from centum ‘hundred’, from decem ‘ten’, from
nanus ‘dwarf’); some are from Italian (from piccolo ‘small’); some are Greek (micro is from μικρός ‘small’,
deca/deka from δέκα ‘ten’, hecto from ἑκατόν ‘hundred’, kilo from χίλιοι ‘thousand’, mega from μέγας
‘large’, giga from γίγας ‘giant’, tera from τέρας ‘monster’).
Translate: I was caught in such a traffic jam that I needed a microcentury for a picoparsec and that my
Challenge 731 e car’s average fuel consumption was huge: it was two tenths of a square millimetre.
456 b units, measurements and constants

SI units form a universal system: they can be used in trade, in industry, in commerce,
at home, in education and in research. They could even be used by extraterrestrial civil-
izations, if they existed.
SI units form a self-consistent system: the product or quotient of two SI units is also
an SI unit. This means that in principle, the same abbreviation, e.g. ‘SI’, could be used
for every unit.
The SI units are not the only possible set that could fulfil all these requirements, but they
are the only existing system that does so.*

The meaning of measurement
Every measurement is a comparison with a standard. Therefore, any measurement re-
Challenge 732 e quires matter to realize the standard (even for a speed standard), and radiation to achieve
the comparison. The concept of measurement thus assumes that matter and radiation ex-
ist and can be clearly separated from each other.
Every measurement is a comparison. Measuring thus implies that space and time ex-

Motion Mountain – The Adventure of Physics
ist, and that they differ from each other.
Every measurement produces a measurement result. Therefore, every measurement
implies the storage of the result. The process of measurement thus implies that the situ-
ation before and after the measurement can be distinguished. In other terms, every meas-
urement is an irreversible process.
Every measurement is a process. Thus every measurement takes a certain amount of
time and a certain amount of space.
All these properties of measurements are simple but important. Beware of anybody
who denies them.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Curiosities and fun challenges ab ou t units
Not using SI units can be expensive. In 1999, NASA lost a satellite on Mars because some
software programmers had used provincial units instead of SI units in part of the code.
As a result of using feet instead of meters, the Mars Climate Orbiter crashed into the
planet, instead of orbiting it; the loss was around 100 million euro.**
∗∗
The second does not correspond to 1/86 400th of the day any more, though it did in the
year 1900; the Earth now takes about 86 400.002 s for a rotation, so that the International
Earth Rotation Service must regularly introduce a leap second to ensure that the Sun is at

* Apart from international units, there are also provincial units. Most provincial units still in use are of
Roman origin. The mile comes from milia passum, which used to be one thousand (double) strides of about
1480 mm each; today a nautical mile, once defined as minute of arc on the Earth’s surface, is defined as
exactly 1852 m. The inch comes from uncia/onzia (a twelfth – now of a foot). The pound (from pondere ‘to
weigh’) is used as a translation of libra – balance – which is the origin of its abbreviation lb. Even the habit
of counting in dozens instead of tens is Roman in origin. These and all other similarly funny units – like
the system in which all units start with ‘f’, and which uses furlong/fortnight as its unit of velocity – are now
officially defined as multiples of SI units.
** This story revived an old but false urban legend claiming that only three countries in the world do not
use SI units: Liberia, the USA and Myanmar.
b units, measurements and constants 457

the highest point in the sky at 12 o’clock sharp.* The time so defined is called Universal
Time Coordinate. The speed of rotation of the Earth also changes irregularly from day to
day due to the weather; the average rotation speed even changes from winter to summer
because of the changes in the polar ice caps; in addition that average decreases over time,
because of the friction produced by the tides. The rate of insertion of leap seconds is
therefore higher than once every 500 days, and not constant in time.
∗∗
The most precise clock ever built, using microwaves, had a stability of 10−16 during a
Ref. 372 running time of 500 s. For longer time periods, the record in 1997 was about 10−15 ; but
Ref. 373 values around 10−17 seem within technological reach. The precision of clocks is limited
for short measuring times by noise, and for long measuring times by drifts, i.e., by sys-
tematic effects. The region of highest stability depends on the clock type; it usually lies
between 1 ms for optical clocks and 5000 s for masers. Pulsars are the only type of clock
for which this region is not known yet; it certainly lies at more than 20 years, the time
elapsed at the time of writing since their discovery.

Motion Mountain – The Adventure of Physics
∗∗
The least precisely measured of the fundamental constants of physics are the gravitational
constant 𝐺 and the strong coupling constant 𝛼s . Even less precisely known are the age of
Page 466 the universe and its density (see Table 60).
∗∗
The precision of mass measurements of solids is limited by such simple effects as the
adsorption of water. Can you estimate the mass of a monolayer of water – a layer with

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 733 s thickness of one molecule – on a metal weight of 1 kg?
∗∗
In the previous millennium, thermal energy used to be measured using the unit calorie,
written as cal. 1 cal is the energy needed to heat 1 g of water by 1 K. To confuse matters,
1 kcal was often written 1 Cal. (One also spoke of a large and a small calorie.) The value
of 1 kcal is 4.1868 kJ.
∗∗
SI units are adapted to humans: the values of heartbeat, human size, human weight, hu-
man temperature and human substance are no more than a couple of orders of mag-
nitude near the unit value. SI units thus (roughly) confirm what Protagoras said 25 cen-
turies ago: ‘Man is the measure of all things.’
∗∗
Some units systems are particularly badly adapted to humans. The most infamous is shoe

* Their website at hpiers.obspm.fr gives more information on the details of these insertions, as does maia.
usno.navy.mil, one of the few useful military websites. See also www.bipm.fr, the site of the BIPM.
458 b units, measurements and constants

size 𝑆. It is a pure number calculated as

𝑆France = 1.5 cm−1 (𝑙 + (1 ± 1) cm)
𝑆central Europe = 1.5748 cm−1 (𝑙 + (1 ± 1) cm)
𝑆Anglo−saxon men = 1.181 cm−1 (𝑙 + (1 ± 1) cm) − 22 (126)

where 𝑙 is the length of a foot and the correction length depends on the manufacturing
company. In addition, the Anglo-Saxon formula is not valid for women and children,
where the first factor depends, for marketing reasons, both on the manufacturer and on
size itself. The ISO standard for shoe size requires, unsurprisingly, to use foot length in
millimetres.
∗∗
The table of SI prefixes covers 72 orders of magnitude. How many additional prefixes will
be needed? Even an extended list will include only a small part of the infinite range of

Motion Mountain – The Adventure of Physics
possibilities. Will the Conférence Générale des Poids et Mesures have to go on forever,
Challenge 734 s defining an infinite number of SI prefixes? Why?
∗∗
In the 21st century, the textile industry uses three measurement systems to express how
fine a fibre is. All three use the linear mass density 𝑚/𝑙. The international system uses
the unit 1 tex = 1 g/km. Another system is particular to silk and used the Denier as unit:
1 den = 1/9 g/km. The third system uses the unit Number English, defined as the number
of hanks of cotton that weigh one pound. Here, a hank is 7 leas. A lea is 120 yards and a
yard is three feet. In addition, the defining number of hanks differs for linnen and differs

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
again for wool, and in addition it depends on the treatment method of the wool. Reading
about textile units makes every comedy show feel like a boring lullaby.
∗∗
The French philosopher Voltaire, after meeting Newton, publicized the now famous story
that the connection between the fall of objects and the motion of the Moon was dis-
covered by Newton when he saw an apple falling from a tree. More than a century later,
just before the French Revolution, a committee of scientists decided to take as the unit
of force precisely the force exerted by gravity on a standard apple, and to name it after
the English scientist. After extensive study, it was found that the mass of the standard
apple was 101.9716 g; its weight was called 1 newton. Since then, visitors to the museum
in Sèvres near Paris have been able to admire the standard metre, the standard kilogram
and the standard apple.*

* To be clear, this is a joke; no standard apple exists. It is not a joke however, that owners of several apple
trees in Britain and in the US claim descent, by rerooting, from the original tree under which Newton had
Ref. 374 his insight. DNA tests have even been performed to decide if all these derive from the same tree. The result
was, unsurprisingly, that the tree at MIT, in contrast to the British ones, is fake.
b units, measurements and constants 459

N
number of measurements

standard deviation

full width at half maximum
(FWHM)

limit curve for a large number
of measurements: the
Gaussian distribution

x x
average value measured values

Motion Mountain – The Adventure of Physics
F I G U R E 304 A precision experiment and its measurement distribution. The precision is high if the
width of the distribution is narrow; the accuracy is high if the centre of the distribution agrees with the
actual value.

Precision and accuracy of measurements
Measurements are the basis of physics. Every measurement has an error. Errors are due
to lack of precision or to lack of accuracy. Precision means how well a result is reproduced
when the measurement is repeated; accuracy is the degree to which a measurement cor-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
responds to the actual value.
Lack of precision is due to accidental or random errors; they are best measured by the
standard deviation, usually abbreviated 𝜎; it is defined through
𝑛
𝜎2 = 1
𝑛−1
∑(𝑥𝑖 − 𝑥)̄ 2 , (127)
𝑖=1

where 𝑥̄ is the average of the measurements 𝑥𝑖 . (Can you imagine why 𝑛 − 1 is used in
Challenge 735 s the formula instead of 𝑛?)
For most experiments, the distribution of measurement values tends towards a nor-
mal distribution, also called Gaussian distribution, whenever the number of measure-
ments is increased. The distribution, shown in Figure 304, is described by the expression

(𝑥−𝑥)̄ 2
𝑁(𝑥) ≈ e− 2𝜎2 . (128)

The square 𝜎2 of the standard deviation is also called the variance. For a Gaussian distri-
Challenge 736 e bution of measurement values, 2.35𝜎 is the full width at half maximum.
Lack of accuracy is due to systematic errors; usually these can only be estimated. This
estimate is often added to the random errors to produce a total experimental error, some-
Ref. 375 times also called total uncertainty. The relative error or uncertainty is the ratio between
460 b units, measurements and constants

the error and the measured value.
For example, a professional measurement will give a result such as 0.312(6) m. The
number between the parentheses is the standard deviation 𝜎, in units of the last digits.
As above, a Gaussian distribution for the measurement results is assumed. Therefore, a
Challenge 737 e value of 0.312(6) m implies that the actual value is expected to lie
— within 1𝜎 with 68.3 % probability, thus in this example within 0.312 ± 0.006 m;
— within 2𝜎 with 95.4 % probability, thus in this example within 0.312 ± 0.012 m;
— within 3𝜎 with 99.73 % probability, thus in this example within 0.312 ± 0.018 m;
— within 4𝜎 with 99.9937 % probability, thus in this example within 0.312 ± 0.024 m;
— within 5𝜎 with 99.999 943 % probability, thus in this example within 0.312 ± 0.030 m;
— within 6𝜎 with 99.999 999 80 % probability, thus in this example within
0.312 ± 0.036 m;
— within 7𝜎 with 99.999 999 999 74 % probability, thus in this example within
0.312 ± 0.041 m.
Challenge 738 s (Do the latter numbers make sense?)

Motion Mountain – The Adventure of Physics
Note that standard deviations have one digit; you must be a world expert to use two,
and a fool to use more. If no standard deviation is given, a (1) is assumed. As a result,
among professionals, 1 km and 1000 m are not the same length!
What happens to the errors when two measured values 𝐴 and 𝐵 are added or subtrac-
ted? If all measurements are independent – or uncorrelated – the standard deviation of
the sum and that of difference is given by 𝜎 = √𝜎𝐴2 + 𝜎𝐵2 . For both the product or ratio of
two measured and uncorrelated values 𝐶 and 𝐷, the result is 𝜌 = √𝜌𝐶2 + 𝜌𝐷2 , where the
𝜌 terms are the relative standard deviations.
Challenge 739 s Assume you measure that an object moves 1 m in 3 s: what is the measured speed

Limits to precision
What are the limits to accuracy and precision? There is no way, even in principle, to
measure a length 𝑥 to a precision higher than about 61 digits, because in nature, the ratio
between the largest and the smallest measurable length is Δ𝑥/𝑥 > 𝑙Pl/𝑑horizon = 10−61 .
Challenge 740 e (Is this ratio valid also for force or for volume?) In the final volume of our text, studies
Vol. VI, page 94 of clocks and metre bars strengthen this theoretical limit.
But it is not difficult to deduce more stringent practical limits. No imaginable machine
can measure quantities with a higher precision than measuring the diameter of the Earth
within the smallest length ever measured, about 10−19 m; that is about 26 digits of preci-
sion. Using a more realistic limit of a 1000 m sized machine implies a limit of 22 digits.
If, as predicted above, time measurements really achieve 17 digits of precision, then they
are nearing the practical limit, because apart from size, there is an additional practical
restriction: cost. Indeed, an additional digit in measurement precision often means an
additional digit in equipment cost.

Physical constants
In physics, general observations are deduced from more fundamental ones. As a con-
sequence, many measurements can be deduced from more fundamental ones. The most
b units, measurements and constants 461

fundamental measurements are those of the physical constants.
The following tables give the world’s best values of the most important physical con-
stants and particle properties – in SI units and in a few other common units – as pub-
Ref. 376 lished in the standard references. The values are the world averages of the best measure-
ments made up to the present. As usual, experimental errors, including both random
and estimated systematic errors, are expressed by giving the standard deviation in the
last digits. In fact, behind each of the numbers in the following tables there is a long
Ref. 377 story which is worth telling, but for which there is not enough room here.
In principle, all quantitative properties of matter can be calculated with quantum the-
Vol. V, page 261 ory – more precisely, equations of the standard model of particle – and a set of basic
physical constants that are given in the next table. For example, the colour, density and
elastic properties of any material can be predicted, in principle, in this way.

TA B L E 57 Basic physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎

Motion Mountain – The Adventure of Physics
Constants that define the SI measurement units
Vacuum speed of light 𝑐 𝑐 299 792 458 m/s 0
𝑐
Original Planck constant ℎ 6.626 070 15 ⋅ 10−34 Js 0
Reduced Planck constant, ℏ 1.054 571 817 ... ⋅ 10−34 Js 0
quantum of action
Positron charge 𝑐 𝑒 0.160 217 6634 aC 0
𝑐
Boltzmann constant 𝑘 1.380 649 ⋅ 10−23 J/K 0
Avogadro’s number 𝑁A 6.022 140 76 ⋅ 1023 1/mol 0

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Constant that should define the SI measurement units
Gravitational constant 𝐺 6.674 30(15) ⋅ 10−11 Nm2 /kg2 2.2 ⋅ 10−5
Other fundamental constants
Number of space-time dimensions 3+1 0𝑏
2
Fine-structure constant 𝑑 or 𝛼 = 4π𝜀𝑒 ℏ𝑐 1/137.035 999 084(21) 1.5 ⋅ 10−10
0

e.m. coupling constant = 𝑔em (𝑚2e 𝑐2 ) = 0.007 297 352 5693(11) 1.5 ⋅ 10−10
Fermi coupling constant 𝑑 or 𝐺F /(ℏ𝑐)3 1.166 3787(6) ⋅ 10−5 GeV−2 5.1 ⋅ 10−7
weak coupling constant 𝛼w (𝑀Z ) = 𝑔w2 /4π 1/30.1(3) 1 ⋅ 10−2
Strong coupling constant 𝑑 𝛼s (𝑀Z ) = 𝑔s2 /4π 0.1179(10) 8.5 ⋅ 10−3
Weak mixing angle sin2 𝜃W (𝑀𝑆) 0.231 22(4) 1.7 ⋅ 10−4
sin2 𝜃W (on shell) 0.222 90(30) 1.3 ⋅ 10−3
= 1 − (𝑚W /𝑚Z )2
0.97383(24) 0.2272(10) 0.00396(9)
CKM quark mixing matrix |𝑉| ( 0.2271(10) 0.97296(24) 0.04221(80) )
0.00814(64) 0.04161(78) 0.999100(34)
Jarlskog invariant 𝐽 3.08(18) ⋅ 10−5
0.82(2) 0.55(4) 0.150(7)
PMNS neutrino mixing m. |𝑃| (0.37(13) 0.57(11) 0.71(7) )
0.41(13) 0.59(10) 0.69(7)
462 b units, measurements and constants

TA B L E 57 (Continued) Basic physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎

Electron mass 𝑚e 9.109 383 7015(28) ⋅ 10−31 kg 3.0 ⋅ 10−10
5.485 799 090 65(16) ⋅ 10−4 u 2.9 ⋅ 10−11
0.510 998 950 00(15) MeV 3.0 ⋅ 10−10
Muon mass 𝑚μ 1.883 531 627(42) ⋅ 10−28 kg 2.2 ⋅ 10−8
105.658 3755(23) MeV 2.2 ⋅ 10−8
2 −5
Tau mass 𝑚𝜏 1.776 82(12) GeV/𝑐 6.8 ⋅ 10
El. neutrino mass 𝑚𝜈e < 2 eV/𝑐2
Muon neutrino mass 𝑚𝜈𝜇 < 2 eV/𝑐2
Tau neutrino mass 𝑚𝜈𝜏 < 2 eV/𝑐2
Up quark mass 𝑢 21.6(+0.49/ − 0.26) MeV/𝑐2
Down quark mass 𝑑 4.67(+0.48/ − 0.17) MeV/𝑐2
Strange quark mass 𝑠 93(+11/ − 5) MeV/𝑐2

Motion Mountain – The Adventure of Physics
Charm quark mass 𝑐 1.27(2) GeV/𝑐2
Bottom quark mass 𝑏 4.18(3) GeV/𝑐2
Top quark mass 𝑡 172.9(0.4) GeV/𝑐2
Photon mass γ < 2 ⋅ 10−54 kg
W boson mass 𝑊± 80.379(12) GeV/𝑐2
Z boson mass 𝑍0 91.1876(21) GeV/𝑐2
Higgs mass H 125.10(14) GeV/𝑐2
Gluon mass g1...8 c. 0 MeV/𝑐2

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
𝑎. Uncertainty: standard deviation of measurement errors.
𝑏. Measured from to 10−19 m to 1026 m.
𝑐. Defining constant.
𝑑. All coupling constants depend on the 4-momentum transfer, as explained in the section on
Page 131 renormalization. Fine-structure constant is the traditional name for the electromagnetic coup-
ling constant 𝑔em in the case of a 4-momentum transfer of 𝑄2 = 𝑚2e 𝑐2 , which is the smallest
one possible. At higher momentum transfers it has larger values, e.g., 𝑔em (𝑄2 = 𝑀W
2 2
𝑐 ) ≈ 1/128.
In contrast, the strong coupling constant has lover values at higher momentum transfers; e.g.,
𝛼s (34 GeV) = 0.14(2).

Why do all these basic constants have the values they have? For any basic constant with
a dimension, such as the quantum of action ℏ, the numerical value has only historical
meaning. It is 1.054 ⋅ 10−34 Js because of the SI definition of the joule and the second.
The question why the value of a dimensional constant is not larger or smaller therefore
always requires one to understand the origin of some dimensionless number giving the
ratio between the constant and the corresponding natural unit that is defined with 𝑐, 𝐺,
Vol. IV, page 208 𝑘, 𝑁A and ℏ. Details and values for the natural units are given in the dedicated section.
In other words, understanding the sizes of atoms, people, trees and stars, the duration
of molecular and atomic processes, or the mass of nuclei and mountains, implies under-
standing the ratios between these values and the corresponding natural units. The key to
understanding nature is thus the understanding of all measurement ratios, and thus of
b units, measurements and constants 463

all dimensionless constants. This quest, including the understanding of the fine-structure
constant 𝛼 itself, is completed only in the final volume of our adventure.
The basic constants yield the following useful high-precision observations.

TA B L E 58 Derived physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t.

Vacuum permeability 𝜇0 1.256 637 062 12(19) μH/m 1.5 ⋅ 10−10
Vacuum permittivity 𝜀0 = 1/𝜇0 𝑐2 8.854 187 8128(13) pF/m 1.5 ⋅ 10−10
Vacuum impedance 𝑍0 = √𝜇0 /𝜀0 376.730 313 668(57) Ω 1.5 ⋅ 10−10
Loschmidt’s number 𝑁L 2.686 780 111... ⋅ 1025 1/m3 0
at 273.15 K and 101 325 Pa
Faraday’s constant 𝐹 = 𝑁A 𝑒 96 485.332 12... C/mol 0
Universal gas constant 𝑅 = 𝑁A 𝑘 8.314 462 618... J/(mol K) 0
Molar volume of an ideal gas 𝑉 = 𝑅𝑇/𝑝 22.413 969 54... l/mol 0

Motion Mountain – The Adventure of Physics
at 273.15 K and 101 325 Pa
Rydberg constant 𝑎 𝑅∞ = 𝑚e 𝑐𝛼2 /2ℎ 10 973 731.568 160(21) m−1 1.9 ⋅ 10−12
Conductance quantum 𝐺0 = 2𝑒2 /ℎ 77.480 917 29... μS 0
Magnetic flux quantum 𝜑0 = ℎ/2𝑒 2.067 833 848... fWb 0
Josephson frequency ratio 2𝑒/ℎ 483.597 8484... THz/V 0
Von Klitzing constant ℎ/𝑒2 = 𝜇0 𝑐/2𝛼 25 812.807 45... Ω 0
Bohr magneton 𝜇B = 𝑒ℏ/2𝑚e 9.274 010 0783(28) yJ/T 3.0 ⋅ 10−10
Classical electron radius 𝑟e = 𝑒2 /4π𝜀0 𝑚e 𝑐2 2.817 940 3262(13) f m 4.5 ⋅ 10−10
Compton wavelength 𝜆 C = ℎ/𝑚e 𝑐 2.426 310 238 67(73) pm 3.0 ⋅ 10−10

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
of the electron 𝜆c = ℏ/𝑚e 𝑐 = 𝑟e /𝛼 0.386 159 267 96(12) pm 3.0 ⋅ 10−10
Bohr radius 𝑎 𝑎∞ = 𝑟e /𝛼2 52.917 721 0903(80) pm 1.5 ⋅ 10−10
Quantum of circulation ℎ/2𝑚e 3.636 947 5516(11) cm2 /s 3.0 ⋅ 10−10
Specific positron charge 𝑒/𝑚e 175.882 001 076(55) GC/kg 3.0 ⋅ 10−10
Cyclotron frequency 𝑓c /𝐵 = 𝑒/2π𝑚e 27.992 489 872(9) GHz/T 3.0 ⋅ 10−10
of the electron
Electron magnetic moment 𝜇e −9.284 764 7043(28) yJ/T 3.0 ⋅ 10−10
𝜇e /𝜇B −1.001 159 652 181 28(18) 1.7 ⋅ 10−13
𝜇e /𝜇N −1 838.281 971 88(11) ⋅ 103 6.0 ⋅ 10−11
Electron g-factor 𝑔e −2.002 319 304 362 56(35) 1.7 ⋅ 10−13
Muon–electron mass ratio 𝑚μ /𝑚e 206.768 2830(46) 2.2 ⋅ 10−8
Muon magnetic moment 𝜇μ −4.490 448 30(10) ⋅ 10−26 J/T 2.2 ⋅ 10−8
Muon g-factor 𝑔μ −2.002 331 8418(13) 6.3 ⋅ 10−10
Atomic mass unit 1 u = 𝑚12C /12 1.660 539 066 60(50) ⋅ 10−27 kg 3.0 ⋅ 10−10
Proton mass 𝑚p 1.672 621 923 69(51) ⋅ 10−27 kg 3.1 ⋅ 10−10
1.007 276 466 621(53) u 5.3 ⋅ 10−11
938.272 088 16(29) MeV 3.1 ⋅ 10−10
Proton–electron mass ratio 𝑚p /𝑚e 1 836.152 673 43(11) 6.0 ⋅ 10−11
Specific proton charge 𝑒/𝑚p 9.578 833 1560(29) ⋅ 107 C/kg 3.1 ⋅ 10−10
464 b units, measurements and constants

TA B L E 58 (Continued) Derived physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t.

Proton Compton wavelength 𝜆 C,p = ℎ/𝑚p 𝑐 1.321 409 855 39(40) f m 3.1 ⋅ 10−10
Nuclear magneton 𝜇N = 𝑒ℏ/2𝑚p 5.050 783 7461(15) ⋅ 10 J/T 3.1 ⋅ 10−10
−27

Proton magnetic moment 𝜇p 1.410 606 797 36(60) ⋅ 10−26 J/T 4.2 ⋅ 10−10
𝜇p /𝜇B 1.521 032 202 30(46) ⋅ 10−3 3.0 ⋅ 10−10
𝜇p /𝜇N 2.792 847 344 63(82) 2.9 ⋅ 10−10
Proton gyromagnetic ratio 𝛾p = 2𝜇𝑝 /ℎ 42.577 478 518(18) MHz/T 4.2 ⋅ 10−10
Proton g factor 𝑔p 5.585 694 6893(16) 2.9 ⋅ 10−10
Neutron mass 𝑚n 1.674 927 498 04(95) ⋅ 10−27 kg 5.7 ⋅ 10−10
1.008 664 915 95(43) u 4.8 ⋅ 10−10
939.565 420 52(54) MeV 5.7 ⋅ 10−10
Neutron–electron mass ratio 𝑚n /𝑚e 1 838.683 661 73(89) 4.8 ⋅ 10−10
Neutron–proton mass ratio 𝑚n /𝑚p 1.001 378 419 31(49) 4.9 ⋅ 10−10

Motion Mountain – The Adventure of Physics
Neutron Compton wavelength 𝜆 C,n = ℎ/𝑚n 𝑐 1.319 590 905 81(75) f m 5.7 ⋅ 10−10
Neutron magnetic moment 𝜇n −0.966 236 51(23) ⋅ 10−26 J/T 2.4 ⋅ 10−7
𝜇n /𝜇B −1.041 875 63(25) ⋅ 10−3 2.4 ⋅ 10−7
𝜇n /𝜇N −1.913 042 73(45) 2.4 ⋅ 10−7
Stefan–Boltzmann constant 𝜎 = π2 𝑘4 /60ℏ3 𝑐2 56.703 744 19... nW/m K 2 4
0
Wien’s displacement constant 𝑏 = 𝜆 max 𝑇 2.897 771 955... mmK 0
58.789 257 57... GHz/K 0
Electron volt eV 0.160 217 6634... aJ 0
Bits to entropy conversion const. 𝑘 ln 2 1023 bit = 0.956 994... J/K 0

𝑎. For infinite mass of the nucleus.

Some useful properties of our local environment are given in the following table.

TA B L E 59 Astronomical constants.

Q ua nt i t y Symbol Va l u e

Tropical year 1900 𝑎 𝑎 31 556 925.974 7 s
Tropical year 1994 𝑎 31 556 925.2 s
Mean sidereal day 𝑑 23ℎ 56󸀠 4.090 53󸀠󸀠
Average distance Earth–Sun 𝑏 149 597 870.691(30) km
Astronomical unit 𝑏 AU 149 597 870 691 m
Light year, based on Julian year 𝑏 al 9.460 730 472 5808 Pm
Parsec pc 30.856 775 806 Pm = 3.261 634 al
Earth’s mass 𝑀♁ 5.973(1) ⋅ 1024 kg
Geocentric gravitational constant 𝐺𝑀 3.986 004 418(8) ⋅ 1014 m3 /s2
2
Earth’s gravitational length 𝑙♁ = 2𝐺𝑀/𝑐 8.870 056 078(16) mm
b units, measurements and constants 465

TA B L E 59 (Continued) Astronomical constants.

Q ua nt i t y Symbol Va l u e

Earth’s equatorial radius 𝑐 𝑅♁eq 6378.1366(1) km
𝑐
Earth’s polar radius 𝑅♁p 6356.752(1) km
𝑐
Equator–pole distance 10 001.966 km (average)
Earth’s flattening 𝑐 𝑒♁ 1/298.25642(1)
Earth’s av. density 𝜌♁ 5.5 Mg/m3
Earth’s age 𝑇♁ 4.50(4) Ga = 142(2) Ps
Earth’s normal gravity 𝑔 9.806 65 m/s2
Earth’s standard atmospher. pressure 𝑝0 101 325 Pa
Moon’s radius 𝑅v 1738 km in direction of Earth
Moon’s radius 𝑅h 1737.4 km in other two directions
Moon’s mass 𝑀 7.35 ⋅ 1022 kg
Moon’s mean distance 𝑑 𝑑 384 401 km

Motion Mountain – The Adventure of Physics
Moon’s distance at perigee 𝑑 typically 363 Mm, historical minimum
359 861 km
Moon’s distance at apogee 𝑑 typically 404 Mm, historical maximum
406 720 km
Moon’s angular size 𝑒 average 0.5181° = 31.08 󸀠 , minimum
0.49°, maximum 0.55°
Moon’s average density 𝜌 3.3 Mg/m3
Moon’s surface gravity 𝑔 1.62 m/s2
Moon’s atmospheric pressure 𝑝 from 10−10 Pa (night) to 10−7 Pa (day)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Jupiter’s mass 𝑀 1.90 ⋅ 1027 kg
Jupiter’s radius, equatorial 𝑅 71.398 Mm
Jupiter’s radius, polar 𝑅 67.1(1) Mm
Jupiter’s average distance from Sun 𝐷 778 412 020 km
Jupiter’s surface gravity 𝑔 24.9 m/s2
Jupiter’s atmospheric pressure 𝑝 from 20 kPa to 200 kPa
Sun’s mass 𝑀⊙ 1.988 43(3) ⋅ 1030 kg
Sun’s gravitational length 2𝐺𝑀⊙ /𝑐2 2.953 250 08(5) km
Heliocentric gravitational constant 𝐺𝑀⊙ 132.712 440 018(8) ⋅ 1018 m3 /s2
Sun’s luminosity 𝐿⊙ 384.6 YW
Solar equatorial radius 𝑅⊙ 695.98(7) Mm
Sun’s angular size 0.53∘ average; minimum on fourth of July
(aphelion) 1888 󸀠󸀠 , maximum on fourth of
January (perihelion) 1952 󸀠󸀠
Sun’s average density 𝜌⊙ 1.4 Mg/m3
Sun’s average distance AU 149 597 870.691(30) km
Sun’s age 𝑇⊙ 4.6 Ga
Solar velocity 𝑣⊙g 220(20) km/s
around centre of galaxy
466 b units, measurements and constants

TA B L E 59 (Continued) Astronomical constants.

Q ua nt i t y Symbol Va l u e

Solar velocity 𝑣⊙b 370.6(5) km/s
against cosmic background
Sun’s surface gravity 𝑔⊙ 274 m/s2
Sun’s lower photospheric pressure 𝑝⊙ 15 kPa
Distance to Milky Way’s centre 8.0(5) kpc = 26.1(1.6) kal
Milky Way’s age 13.6 Ga
Milky Way’s size c. 1021 m or 100 kal
Milky Way’s mass 1012 solar masses, c. 2 ⋅ 1042 kg
Most distant galaxy cluster known SXDF-XCLJ 9.6 ⋅ 109 al
0218-0510

Motion Mountain – The Adventure of Physics
𝑎. Defining constant, from vernal equinox to vernal equinox; it was once used to define the
second. (Remember: π seconds is about a nanocentury.) The value for 1990 is about 0.7 s less,
Challenge 741 s corresponding to a slowdown of roughly 0.2 ms/a. (Watch out: why?) There is even an empirical
Ref. 378 formula for the change of the length of the year over time.
𝑏. The truly amazing precision in the average distance Earth–Sun of only 30 m results from time
averages of signals sent from Viking orbiters and Mars landers taken over a period of over twenty
years. Note that the International Astronomical Union distinguishes the average distance Earth–
Sun from the astronomical unit itself; the latter is defined as a fixed and exact length. Also the
light year is a unit defined as an exact number by the IAU. For more details, see www.iau.org/
public/measuring.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
𝑐. The shape of the Earth is described most precisely with the World Geodetic System. The last
edition dates from 1984. For an extensive presentation of its background and its details, see the
www.wgs84.com website. The International Geodesic Union refined the data in 2000. The radii
and the flattening given here are those for the ‘mean tide system’. They differ from those of the
‘zero tide system’ and other systems by about 0.7 m. The details constitute a science in itself.
𝑑. Measured centre to centre. To find the precise position of the Moon in the sky at a given
date, see the www.fourmilab.ch/earthview/moon_ap_per.html page. For the planets, see the page
www.fourmilab.ch/solar/solar.html and the other pages on the same site.
𝑒. Angles are defined as follows: 1 degree = 1∘ = π/180 rad, 1 (first) minute = 1 󸀠 = 1°/60, 1 second
(minute) = 1 󸀠󸀠 = 1 󸀠 /60. The ancient units ‘third minute’ and ‘fourth minute’, each 1/60th of the
preceding, are not in use any more. (‘Minute’ originally means ‘very small’, as it still does in
modern English.)

Some properties of nature at large are listed in the following table. (If you want a chal-
Challenge 742 s lenge, can you determine whether any property of the universe itself is listed?)

TA B L E 60 Cosmological constants.

Q ua nt i t y Symbol Va l u e

Cosmological constant Λ c. 1 ⋅ 10−52 m−2
Age of the universe 𝑎 𝑡0 4.333(53) ⋅ 1017 s = 13.8(0.1) ⋅ 109 a
(determined from space-time, via expansion, using general relativity)
b units, measurements and constants 467

TA B L E 60 (Continued) Cosmological constants.

Q ua nt i t y Symbol Va l u e

Age of the universe 𝑎 𝑡0 over 3.5(4) ⋅ 1017 s = 11.5(1.5) ⋅ 109 a
(determined from matter, via galaxies and stars, using quantum theory)
Hubble parameter 𝑎 𝐻0 2.3(2) ⋅ 10−18 s−1 = 0.73(4) ⋅ 10−10 a−1
= ℎ0 ⋅ 100 km/s Mpc = ℎ0 ⋅ 1.0227 ⋅ 10−10 a−1
𝑎
Reduced Hubble parameter ℎ0 0.71(4)
Deceleration parameter 𝑎 ̈ 0 /𝐻02 −0.66(10)
𝑞0 = −(𝑎/𝑎)
Universe’s horizon distance 𝑎 𝑑0 = 3𝑐𝑡0 40.0(6) ⋅ 1026 m = 13.0(2) Gpc
Universe’s topology trivial up to 1026 m
Number of space dimensions 3, for distances up to 1026 m
Critical density 𝜌c = 3𝐻02 /8π𝐺 ℎ20 ⋅ 1.878 82(24) ⋅ 10−26 kg/m3
of the universe = 0.95(12) ⋅ 10−26 kg/m3
(Total) density parameter 𝑎 Ω0 = 𝜌0 /𝜌c 1.02(2)

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𝑎
Baryon density parameter ΩB0 = 𝜌B0 /𝜌c 0.044(4)
Cold dark matter density parameter 𝑎 ΩCDM0 = 𝜌CDM0 /𝜌c 0.23(4)
Neutrino density parameter 𝑎 Ω𝜈0 = 𝜌𝜈0 /𝜌c 0.001 to 0.05
𝑎
Dark energy density parameter ΩX0 = 𝜌X0 /𝜌c 0.73(4)
Dark energy state parameter 𝑤 = 𝑝X /𝜌X −1.0(2)
Baryon mass 𝑚b 1.67 ⋅ 10−27 kg
Baryon number density 0.25(1) /m3
Luminous matter density 3.8(2) ⋅ 10−28 kg/m3
Stars in the universe 𝑛s 1022±1

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Baryons in the universe 𝑛b 1081±1
𝑏
Microwave background temperature 𝑇0 2.725(1) K
Photons in the universe 𝑛𝛾 1089
2 4 4
Photon energy density 𝜌𝛾 = π 𝑘 /15𝑇0 4.6 ⋅ 10−31 kg/m3
Photon number density 410.89 /cm3 or 400 /cm3 (𝑇0 /2.7 K)3
Density perturbation amplitude √𝑆 5.6(1.5) ⋅ 10−6
Gravity wave amplitude √𝑇 < 0.71√𝑆
Mass fluctuations on 8 Mpc 𝜎8 0.84(4)
Scalar index 𝑛 0.93(3)
Running of scalar index d𝑛/d ln 𝑘 −0.03(2)
Planck length 𝑙Pl = √ℏ𝐺/𝑐3 1.62 ⋅ 10−35 m
Planck time 𝑡Pl = √ℏ𝐺/𝑐5 5.39 ⋅ 10−44 s
Planck mass 𝑚Pl = √ℏ𝑐/𝐺 21.8 μg
𝑎
Instants in history 𝑡0 /𝑡Pl 8.7(2.8) ⋅ 1060
Space-time points 𝑁0 = (𝑅0 /𝑙Pl )3 ⋅ 10244±1
inside the horizon 𝑎 (𝑡0 /𝑡Pl )
Mass inside horizon 𝑀 1054±1 kg
468 b units, measurements and constants

𝑎. The index 0 indicates present-day values.
𝑏. The radiation originated when the universe was 380 000 years old and had a temperature of
about 3000 K; the fluctuations Δ𝑇0 which led to galaxy formation are today about 16 ± 4 μK =
Vol. II, page 231 6(2) ⋅ 10−6 𝑇0 .

Useful numbers

e 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 699959
π 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 375105
π2 9.86960 44010 89358 61883 44909 99876 15113 53136 99407 240790
Ref. 350 γ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 939923
ln 2 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 360255
ln 10 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 628772
√10 3.16227 76601 68379 33199 88935 44432 71853 37195 55139 325216

S OU RC E S OF I N F OR M AT ION ON
MOT ION

“
No place affords a more striking conviction of

”
the vanity of human hopes than a public library.
Samuel Johnson

“
In a consumer society there are inevitably two
kinds of slaves: the prisoners of addiction and

”
the prisoners of envy.

Motion Mountain – The Adventure of Physics
Ivan Illich**

I
n the text, good books that introduce neighbouring domains are presented
n the bibliography. The bibliography also points to journals and websites,
n order to satisfy more intense curiosity about what is encountered in this ad-
venture. All citations can also be found by looking up the author in the name index. To
find additional information, either libraries or the internet can help.
In a library, review articles of recent research appear in journals such as Reviews of
Modern Physics, Reports on Progress in Physics, Contemporary Physics and Advances in
Physics. Good pedagogical introductions are found in the American Journal of Physics,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
the European Journal of Physics and Physik in unserer Zeit.
Overviews on research trends occasionally appear in magazines such as Physics
World, Physics Today, Europhysics Journal, Physik Journal and Nederlands tijdschrift
voor natuurkunde. For coverage of all the sciences together, the best sources are the
magazines Nature, New Scientist, Naturwissenschaften, La Recherche and Science News.
Research papers on the foundations of motion appear mainly in Physics Letters B,
Nuclear Physics B, Physical Review D, Physical Review Letters, Classical and Quantum
Gravity, General Relativity and Gravitation, International Journal of Modern Physics and
Modern Physics Letters. The newest results and speculative ideas are found in conference
proceedings, such as the Nuclear Physics B Supplements. Research articles also appear
in Fortschritte der Physik, European Physical Journal, La Rivista del Nuovo Cimento,
Europhysics Letters, Communications in Mathematical Physics, Journal of Mathematical
Physics, Foundations of Physics, International Journal of Theoretical Physics and Journal
of Physics G.
There are only a few internet physics journals of quality: one is Living Reviews in Re-
lativity, found at www.livingreviews.org, the other is the New Journal of Physics, which
can be found at the www.njp.org website. There are, unfortunately, also many internet
physics journals that publish incorrect research. They are easy to spot: they ask for money

** Ivan Illich (b. 1926 Vienna, d. 2002 Bremen), theologian and social and political thinker.
470 c sources of information on motion

to publish a paper.
By far the simplest way to keep in touch with ongoing research on motion and modern
physics is to use the internet, the international computer network. To start using it, ask a
friend who knows.*
In the last decade of the twentieth century, the internet expanded into a combination
of library, business tool, discussion platform, media collection, garbage collection and,
above all, addiction provider. Do not use it too much. Commerce, advertising and –
unfortunately – addictive material for children, youth and adults, as well as crime of all
kind are also an integral part of the web. With a personal computer, a modem and free
browser software, you can look for information in millions of pages of documents or
destroy your professional career through addiction. The various parts of the documents
are located in various computers around the world, but the user does not need to be
aware of this.**
Most theoretical physics papers are available free of charge, as preprints, i.e., before
official publication and checking by referees, at the arxiv.org website. A service for finding
subsequent preprints that cite a given one is also available.

Motion Mountain – The Adventure of Physics
Research papers on the description of motion appear after this text is published can
also be found via www.webofknowledge.com a site accessible only from libraries. It al-
lows one to search for all publications which cite a given paper.
Searching the web for authors, organizations, books, publications, companies or
simple keywords using search engines can be a rewarding experience or an episode of ad-
diction, depending entirely on yourself. A selection of interesting servers about motion
is given below.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
* It is also possible to use the internet and to download files through FTP with the help of email only. But
the tools change too often to give a stable guide here. Ask your friend.
** Several decades ago, the provocative book by Ivan Illich, Deschooling Society, Harper & Row, 1971,
listed four basic ingredients for any educational system:
1. access to resources for learning, e.g. books, equipment, games, etc. at an affordable price, for everybody,
at any time in their life;
2. for all who want to learn, access to peers in the same learning situation, for discussion, comparison,
cooperation and competition;
3. access to elders, e.g. teachers, for their care and criticism towards those who are learning;
4. exchanges between students and performers in the field of interest, so that the latter can be models for
the former. For example, there should be the possibility to listen to professional musicians and reading the
works of specialist writers. This also gives performers the possibility to share, advertise and use their skills.
Illich develops the idea that if such a system were informal – he then calls it a ‘learning web’ or
‘opportunity web’ – it would be superior to formal, state-financed institutions, such as conventional schools,
for the development of mature human beings. These ideas are deepened in his following works, Deschooling
Our Lives, Penguin, 1976, and Tools for Conviviality, Penguin, 1973.
Today, any networked computer offers email (electronic mail), FTP (file transfers to and from another
computer), access to discussion groups on specific topics, such as particle physics, and the world-wide web.
In a rather unexpected way, all these facilities of the internet have transformed it into the backbone of the
‘opportunity web’ discussed by Illich. However, as in any school, it strongly depends on the user’s discipline
whether the internet actually does provide a learning web or an entry into addiction.
c sources of information on motion 471

TA B L E 61 Some interesting sites on the world-wide web.

To p i c We b s i t e a d d r e s s

General knowledge
Innovation in science and www.innovations-report.de
technology
Book collections www.ulib.org
books.google.com
Entertaining science education www.popsci.com/category/popsci-authors/theodore-gray
by Theodore Gray
Entertaining and professional thehappyscientist.com
science education by Robert
Krampf
Science Frontiers www.science-frontiers.com
Science Daily News www.sciencedaily.com

Motion Mountain – The Adventure of Physics
Science News www.sciencenews.org
Encyclopedia of Science www.daviddarling.info
Interesting science research www.max-wissen.de
Quality science videos www.vega.org.uk
ASAP Science videos plus.google.com/101786231119207015313/posts
Physics
Learning physics with toys www.arvindguptatoys.com
from rubbish
Official SI unit website www.bipm.fr

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Unit conversion www.chemie.fu-berlin.de/chemistry/general/units.html
Particle data pdg.web.cern.ch
Engineering data and formulae www.efunda.com
Information on relativity math.ucr.edu/home/baez/relativity.html
Research preprints arxiv.org
www.slac.stanford.edu/spires
Abstracts of papers in physics www.osti.gov
journals
Many physics research papers sci-hub.tv, sci-hub.la
libgen.pw, libgen.io
Physics news, weekly www.aip.org/physnews/update
Physics news, daily phys.org
Physics problems by Yacov www.tau.ac.il/~kantor/QUIZ/
KantorKantor, Yacov
Physics problems by Henry www.phy.duke.edu/~hsg/physics-challenges/challenges.html
Greenside
Physics ‘question of the week’ www.physics.umd.edu/lecdem/outreach/QOTW/active
Physics ‘miniproblem’ www.nyteknik.se/miniproblemet
Physikhexe physik-verstehen-mit-herz-und-hand.de/html/de-6.html
472 c sources of information on motion

To p i c We b s i t e a d d r e s s

Magic science tricks www.sciencetrix.com
Physics stack exchange physics.stackexchange.com
‘Ask the experts’ www.sciam.com/askexpert_directory.cfm
Nobel Prize winners www.nobel.se/physics/laureates
Videos of Nobel Prize winner www.mediatheque.lindau-nobel.org
talks
Pictures of physicists www.if.ufrj.br/famous/physlist.html
Physics organizations www.cern.ch
www.hep.net
www.nikhef.nl
www.het.brown.edu/physics/review/index.html
Physics textbooks on the web www.physics.irfu.se/CED/Book
www.biophysics.org/education/resources.htm

Motion Mountain – The Adventure of Physics
www.lightandmatter.com
www.physikdidaktik.uni-karlsruhe.de/index_en.html
www.feynmanlectures.info
hyperphysics.phy-astr.gsu.edu/hbase/hph.html
www.motionmountain.net
Three beautiful French sets of feynman.phy.ulaval.ca/marleau/notesdecours.htm
notes on classical mechanics
and particle theory
The excellent Radical www.physics.nmt.edu/~raymond/teaching.html
Freshman Physics by David

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Raymond
Physics course scripts from ocw.mit.edu/courses/physics/
MIT
Physics lecture scripts in www.akleon.de
German and English
‘World lecture hall’ wlh.webhost.utexas.edu
Optics picture of the day www.atoptics.co.uk/opod.htm
Living Reviews in Relativity www.livingreviews.org
Wissenschaft in die Schulen www.wissenschaft-schulen.de
Videos of Walter ocw.mit.edu/courses/physics/
Lewin’sIndexLewin, Walter 8-01-physics-i-classical-mechanics-fall-1999/
physics lectures
Physics videos of Matt Carlson www.youtube.com/sciencetheater
Physics videos by the www.sixtysymbols.com
University of Nottingham
Physics lecture videos www.coursera.org/courses?search=physics
www.edx.org/course-list/allschools/physics/allcourses
Mathematics
c sources of information on motion 473

To p i c We b s i t e a d d r e s s

‘Math forum’ internet mathforum.org/library
resource collection
Biographies of mathematicians www-history.mcs.st-andrews.ac.uk/BiogIndex.html
Purdue math problem of the www.math.purdue.edu/academics/pow
week
Macalester College maths mathforum.org/wagon
problem of the week
Mathematical formulae dlmf.nist.gov
Weisstein’s World of mathworld.wolfram.com
Mathematics
Functions functions.wolfram.com
Symbolic integration www.integrals.com
Algebraic surfaces www.mathematik.uni-kl.de/~hunt/drawings.html
Math lecture videos, in www.j3l7h.de/videos.html

Motion Mountain – The Adventure of Physics
German
Gazeta Matematica, in www.gazetamatematica.net
Romanian
Astronomy
ESA sci.esa.int
NASA www.nasa.gov
Hubble space telescope hubble.nasa.gov
Sloan Digital Sky Survey skyserver.sdss.org

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The ‘cosmic mirror’ www.astro.uni-bonn.de/~dfischer/mirror
Solar System simulator space.jpl.nasa.gov
Observable satellites liftoff.msfc.nasa.gov/RealTime/JPass/20
Astronomy picture of the day antwrp.gsfc.nasa.gov/apod/astropix.html
The Earth from space www.visibleearth.nasa.gov
From Stargazers to Starships www.phy6.org/stargaze/Sintro.htm
Current solar data www.n3kl.org/sun
Specific topics
Sonic wonders to visit in the www.sonicwonders.org
world
Encyclopedia of photonics www.rp-photonics.com
Chemistry textbook, online chemed.chem.wisc.edu/chempaths/GenChem-Textbook
Minerals webmineral.com
www.mindat.org
Geological Maps onegeology.org
Optical illusions www.sandlotscience.com
Rock geology sandatlas.org
Petit’s science comics www.jp-petit.org
Physical toys www.e20.physik.tu-muenchen.de/~cucke/toylinke.htm
474 c sources of information on motion

To p i c We b s i t e a d d r e s s

Physics humour www.dctech.com/physics/humor/biglist.php
Literature on magic www.faqs.org/faqs/magic-faq/part2
music library, searchable by imslp.org
tune
Making paper aeroplanes www.pchelp.net/paper_ac.htm
www.ivic.qc.ca/~aleexpert/aluniversite/klinevogelmann.html
Small flying helicopters pixelito.reference.be
Science curiosities www.wundersamessammelsurium.info
Ten thousand year clock www.longnow.org
Gesellschaft Deutscher www.gdnae.de
Naturforscher und Ärzte
Pseudoscience suhep.phy.syr.edu/courses/modules/PSEUDO/pseudo_main.
html
Crackpots www.crank.net

Motion Mountain – The Adventure of Physics
Periodic table with videos for www.periodicvideos.com
each element
Mathematical quotations math.furman.edu/mwoodard/~mquot.html
The ‘World Question Center’ www.edge.org/questioncenter.html
Plagiarism www.plagiarized.com
Hoaxes www.museumofhoaxes.com
Encyclopedia of Earth www.eoearth.org
This is colossal thisiscolossal.com

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Do you want to study physics without actually going to university? Nowadays it is pos-
sible to do so via email and internet, in German, at the University of Kaiserslautern.*
In the near future, a nationwide project in Britain should allow the same for English-
speaking students. As an introduction, use the latest update of this physics text!

“
Das Internet ist die offenste Form der

”
geschlossenen Anstalt.**
Matthias Deutschmann

“ ”
Si tacuisses, philosophus mansisses.***
After Boethius.

* See the www.fernstudium-physik.de website.
** ‘The internet is the most open form of a closed institution.’
*** ‘If you had kept quiet, you would have remained a philosopher.’ After the story Boethius (c. 480–c. 525)
tells in De consolatione philosophiae, 2.7, 67 ff.
C HA L L E NG E H I N T S A N D S OLU T ION S

“
Never make a calculation before you know the

”
answer.
John Wheeler’s motto

John Wheeler wanted people to estimate, to try and to guess; but not saying the guess out
loud. A correct guess reinforces the physics instinct, whereas a wrong one leads to the
pleasure of surprise. Guessing is thus an important first step in solving every problem.

Motion Mountain – The Adventure of Physics
Teachers have other criteria to keep in mind. Good problems can be solved on differ-
ent levels of difficulty, can be solved with words or with images or with formulae, activate
knowledge, concern real-world applications, and are open.

Challenge 1, page 10: Do not hesitate to be demanding and strict. The next edition of the text
will benefit from it.
Challenge 2, page 16: There are many ways to distinguish real motion from an illusion of mo-
tion: for example, only real motion can be used to set something else into motion. In addition,
the motion illusions of the figures show an important failure; nothing moves if the head and the

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
paper remain fixed with respect to each other. In other words, the illusion only amplifies existing
motion, it does not create motion from nothing.
Challenge 3, page 17: Without detailed and precise experiments, both sides can find examples
to prove their point. Creation is supported by the appearance of mould or bacteria in a glass
of water; creation is also supported by its opposite, namely traceless disappearance, such as the
disappearance of motion. However, conservation is supported and creation falsified by all those
investigations that explore assumed cases of appearance or disappearance in full detail.
Challenge 4, page 19: The amount of water depends on the shape of the bucket. The system
chooses the option (tilt or straight) for which the centre of gravity is lowest.
Challenge 5, page 20: To simplify things, assume a cylindrical bucket. If you need help, do the
experiment at home. For the reel, the image is misleading: the rim on which the reel advances
has a larger diameter than the section on which the string is wound up. The wound-up string
does not touch the floor, like for the reel shown in Figure 305.
Challenge 6, page 19: Political parties, sects, helping organizations and therapists of all kinds are
typical for this behaviour.
Challenge 7, page 24: The issue is not yet completely settled for the motion of empty space, such
as in the case of gravitational waves. Thus, the motion of empty space might be an exception. In
any case, empty space is not made of small particles of finite size, as this would contradict the
transversality of gravity waves.
Challenge 8, page 26: Holes are not physical systems, because in general they cannot be tracked.
Challenge 9, page 26: The circular definition is: objects are defined as what moves with respect
476 challenge hints and solutions

F I G U R E 305 The assumed shape for the reel puzzle.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 306
A soap
bubble while
bursting
(© Peter
Wienerr-
oither).

to the background, and the background is defined as what stays when objects change. We shall
Page 438 return to this important issue several times in our adventure. It will require a certain amount of
patience to solve it, though.
Challenge 10, page 28: No, the universe does not have a state. It is not measurable, not even in
Vol. IV, page 169 principle. See the discussion on the issue in volume IV, on quantum theory.
Challenge 11, page 28: The final list of intrinsic properties for physical systems found in nature
Vol. V, page 262 is given in volume V, in the section on particle physics. And of course, the universe has no in-
trinsic, permanent properties. None of them are measurable for the universe as a whole, not even
in principle.
Challenge 12, page 31: Hint: yes, there is such a point.
challenge hints and solutions 477

Challenge 13, page 31: See Figure 306 for an intermediate step. A bubble bursts at a point, and
then the rim of the hole increases rapidly until it disappears on the antipodes. During that process
the remaining of the bubble keeps its spherical shape, as shown in the figure. For a film of the
process, see www.youtube.com/watch?v=dIZwQ24_OU0 (or search for ‘bursting soap bubble’).
In other words, the final droplets that are ejected stem from the point of the bubble which is
opposite to the point of puncture; they are never ejected from the centre of the bubble.
Challenge 14, page 31: A ghost can be a moving image; it cannot be a moving object, as objects
Vol. IV, page 136 cannot interpenetrate.
Challenge 15, page 31: If something could stop moving, motion could disappear into nothing.
For a precise proof, one would have to show that no atom moves any more. So far, this has never
been observed: motion is conserved. (Nothing in nature can disappear into nothing.)
Challenge 16, page 31: This would indeed mean that space is infinite; however, it is impossible
to observe that something moves ‘forever’: nobody lives that long. In short, there is no way to
prove that space is infinite in this way. In fact, there is no way to prove that space is infinite in
any other way either.
Challenge 17, page 31: The necessary rope length is 𝑛ℎ, where 𝑛 is the number of wheels/pulleys.

Motion Mountain – The Adventure of Physics
And yes, the farmer is indeed doing something sensible.
Challenge 19, page 31: How would you measure this?
Challenge 20, page 31: The number of reliable digits of a measurement result is a simple quanti-
fication of precision. More details can be found by looking up ‘standard deviation’ in the index.
Challenge 21, page 31: No; memory is needed for observation and measurements. This is the
case for humans and measurement apparatus. Quantum theory will make this particularly clear.
Challenge 22, page 31: Note that you never have observed zero speed. There is always some
measurement error which prevents one to say that something is zero. No exceptions!
Challenge 23, page 32: (264 − 1) = 18 446 744 073 700 551 615 grains of wheat, with a grain

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
weight of 40 mg, are 738 thousand million tons. Given a world harvest in 2006 of 606 million
tons, the grains amount to about 1200 years of the world’s wheat harvests.
The grain number calculation is simplified by using the formula 1 + 𝑚 + 𝑚2 + 𝑚3 + ...𝑚𝑛 =
𝑛+1
(𝑚 − 1)/(𝑚 − 1), that gives the sum of the so-called geometric sequence. The name is historical
and is used as a contrast to the arithmetic sequence 1 + 2 + 3 + 4 + 5 + ...𝑛 = 𝑛(𝑛 + 1)/2. Can you
prove the two expressions?
The chess legend is mentioned first by Ibn Khallikan (b. 1211 Arbil, d. 1282 Damascus). King
Shiram and king Balhait, also mentioned in the legend, are historical figures that lived between
the second and fourth century CE. The legend appears to have combined two different stories.
Indeed, the calculation of grains appears already in the year 947, in the famous text Meadows of
Gold and Mines of Precious Stones by Al-Masudi (b. c. 896 Baghdad, d. 956 Cairo).
Challenge 24, page 32: In clean experiments, the flame leans forward. But such experiments are
not easy, and sometimes the flame leans backward. Just try it. Can you explain both observations?
Challenge 25, page 32: Accelerometers are the simplest motion detectors. They exist in form of
piezoelectric devices that produce a signal whenever the box is accelerated and can cost as little as
one euro. Another accelerometer that might have a future is an interference accelerometer that
makes use of the motion of an interference grating; this device might be integrated in silicon.
Other, more precise accelerometers use gyroscopes or laser beams running in circles.
Velocimeters and position detectors can also detect motion; they need a wheel or at least an
optical way to look out of the box. Tachographs in cars are examples of velocimeters, computer
mice are examples of position detectors.
478 challenge hints and solutions

A cheap enough device would be perfect to measure the speed of skiers or skaters. No such
device exists yet.
Challenge 26, page 32: The ball rolls (or slides) towards the centre of the table, as the table centre
is somewhat nearer to the centre of the Earth than the border; then the ball shoots over, perform-
ing an oscillation around the table centre. The period is 84 min, as shown in challenge 405. (This
has never been observed, so far. Why?)
Challenge 27, page 32: Only if the acceleration never vanishes. Accelerations can be felt. Accel-
erometers are devices that measure accelerations and then deduce the position. They are used in
aeroplanes when flying over the Atlantic. If the box does not accelerate, it is impossible to say
whether it moves or sits still. It is even impossible to say in which direction one moves. (Close
your eyes in a train at night to confirm this.)
Challenge 28, page 32: The block moves twice as fast as the cylinders, independently of their
radius.
Challenge 29, page 32: This method is known to work with other fears as well.
Challenge 30, page 33: Three couples require 11 passages. Two couples require 5. For four or
more couples there is no solution. What is the solution if there are 𝑛 couples and 𝑛 − 1 places

Motion Mountain – The Adventure of Physics
on the boat?
Challenge 31, page 33: Hint: there is an infinite number of such shapes. These curves are called
also Reuleaux curves. Another hint: The 20 p and 50 p coins in the UK have such shapes. And yes,
other shapes than cylinders are also possible: take a twisted square bar, for example.
Challenge 32, page 33: If you do not know, ask your favourite restorer of old furniture.
Challenge 33, page 33: For this beautiful puzzle, see arxiv.org/abs/1203.3602.
Challenge 34, page 33: Conservation, relativity and minimization are valid generally. In some
rare processes in nuclear physics, motion invariance (reversibility) is broken, as is mirror invari-
ance. Continuity is known not to be valid at smallest length and time intervals, but no experiment
has yet probed those domains, so that continuity is still valid in practice.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 35, page 34: In everyday life, this is correct; what happens when quantum effects are
taken into account?
Challenge 36, page 36: Take the average distance change of two neighbouring atoms in a piece
of quartz over the last million years. Do you know something still slower?
Challenge 37, page 37: There is only one way: compare the velocity to be measured with the
speed of light – using cleverly placed mirrors. In fact, almost all physics textbooks, both for
schools and for university, start with the definition of space and time. Otherwise excellent re-
lativity textbooks have difficulties avoiding this habit, even those that introduce the now standard
k-calculus (which is in fact the approach mentioned here). Starting with speed is the most logical
and elegant approach. But it is possible to compare speeds without metre sticks and clocks. Can
you devise a method?
Challenge 38, page 37: There is no way to sense your own motion if you are in a vacuum. No
Page 156 way in principle. This result is often called the principle of relativity.
In fact, there is a way to measure your motion in space (though not in vacuum): measure your
speed with respect to the cosmic background radiation. So we have to be careful about what is
implied by the question.
Challenge 39, page 37: The wing load 𝑊/𝐴, the ratio between weight 𝑊 and wing area 𝐴, is
obviously proportional to the third root of the weight. (Indeed, 𝑊 ∼ 𝑙3 , 𝐴 ∼ 𝑙2 , 𝑙 being the
dimension of the flying object.) This relation gives the green trend line.
The wing load 𝑊/𝐴, the ratio between weight 𝑊 and wing area 𝐴, is, like all forces in fluids,
proportional to the square of the cruise speed 𝑣: we have 𝑊/𝐴 = 𝑣2 0.38 kg/m3 . The unexplained
challenge hints and solutions 479

Motion Mountain – The Adventure of Physics
factor contains the density of air and a general numerical coefficient that is difficult to calculate.
This relation connects the upper and lower horizontal scales in the graph.
As a result, the cruise speed scales as the sixth root of weight: 𝑣 ∼ 𝑊1/6 . In other words, an
Airbus A380 is 750 000 million times heavier than a fruit fly, but only a hundred times as fast.
Challenge 41, page 41: Equivalently: do points in space exist? The final part of our adventure
Vol. VI, page 65 explores this issue in detail.
Challenge 42, page 42: All electricity sources must use the same phase when they feed electric
power into the net. Clocks of computers on the internet must be synchronized.
Challenge 43, page 42: Note that the shift increases quadratically with time, not linearly.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 44, page 43: Galileo measured time with a scale (and with other methods). His stop-
watch was a water tube that he kept closed with his thumb, pointing into a bucket. To start
the stopwatch, he removed his thumb, to stop it, he put it back on. The volume of water in
the bucket then gave him a measure of the time interval. This is told in his famous book Ga-
lileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla
mecanica e i movimenti locali, usually simply called the ‘Discorsi’, which he published in 1638
with Louis Elsevier in Leiden, in the Netherlands.
Challenge 45, page 44: Natural time is measured with natural motion. Natural motion is the
motion of light. Natural time is thus defined with the help of the motion of light.
Challenge 46, page 48: There is no way to define a local time at the poles that is consistent with
all neighbouring points. (For curious people, check the website www.arctic.noaa.gov/gallery_np.
html.)
Challenge 48, page 50: The forest is full of light and thus of light rays: they are straight, as shown
by the sunbeams in Figure 307.
Challenge 49, page 50: One pair of muscles moves the lens along the third axis by deforming
the eye from prolate to spherical to oblate.
Challenge 50, page 50: You can solve this problem by trying to think in four dimensions. (Train
using the well-known three-dimensional projections of four-dimensional cubes.) Try to imagine
how to switch the sequence when two pieces cross. Note: it is usually not correct, in this domain,
to use time instead of a fourth spatial dimension!
Challenge 51, page 52: Measure distances using light.
480 challenge hints and solutions

Challenge 54, page 56: It is easier to work with the unit torus. Take the unit interval [0, 1] and
equate the endpoints. Define a set 𝐵 in which the elements are a given real number 𝑏 from the
interval plus all those numbers that differ from that real by a rational number. The unit circle
can be thought as the union of all the sets 𝐵. (In fact, every set 𝐵 is a shifted copy of the rational
numbers ℚ.) Now build a set 𝐴 by taking one element from each set 𝐵. Then build the set family
consisting of the set 𝐴 and its copies 𝐴 𝑞 shifted by a rational 𝑞. The union of all these sets is
the unit torus. The set family is countably infinite. Then divide it into two countably infinite set
families. It is easy to see that each of the two families can be renumbered and its elements shifted
in such a way that each of the two families forms a unit torus.
Ref. 44 Mathematicians say that there is no countably infinitely additive measure of ℝ𝑛 or that sets
such as 𝐴 are non-measurable. As a result of their existence, the ‘multiplication’ of lengths is
possible. Later on, we shall explore whether bread or gold can be multiplied in this way.
Challenge 55, page 56: Hint: start with triangles.
Challenge 56, page 56: An example is the region between the x-axis and the function which as-
signs 1 to every transcendental and 0 to every non-transcendental number.
Challenge 57, page 57: We use the definition of the function of the text. The dihedral angle of a

Motion Mountain – The Adventure of Physics
regular tetrahedron is an irrational multiple of π, so the tetrahedron has a non-vanishing Dehn
invariant. The cube has a dihedral angle of π/2, so the Dehn invariant of the cube is 0. Therefore,
the cube is not equidecomposable with the regular tetrahedron.
Challenge 58, page 58: If you think you can show that empty space is continuous, you are wrong.
Check your arguments. If you think you can prove the opposite, you might be right – but only if
you already know what is explained in the final part of the text. If that is not the case, check your
arguments. In fact, time is neither discrete nor continuous.
Challenge 60, page 59: Obviously, we use light to check that the plumb line is straight, so the two
definitions must be the same. This is the case because the field lines of gravity are also possible
paths for the motion of light. However, this is not always the case; can you spot the exceptions?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Another way to check straightness is along the surface of calm water.
A third, less precise way is to make use of the straightness sensors on the brain. The human
brain has a built-in faculty to determine whether an object seen with the eyes is straight. There
are special cells in the brain that fire when this is the case. Any book on vision perception tells
more about this topic.
Challenge 61, page 60: The hollow Earth theory is correct if the distance formula is used con-
sistently. In particular, one has to make the assumption that objects get smaller as they approach
the centre of the hollow sphere. Good explanations of all events are found on www.geocities.
com/inversedearth. Quite some material can be found on the internet, also under the names of
celestrocentric system, inner world theory or concave Earth theory. There is no way to prefer one
description over the other, except possibly for reasons of simplicity or intellectual laziness.
Challenge 63, page 61: A hint is given in Figure 308. For the measurement of the speed of light
with almost the same method, see volume II, on page 20.
Challenge 64, page 61: A fast motorbike is faster: a motorbike driver can catch an arrow, a stunt
that was shown on the German television show ‘Wetten dass’ in the year 2001.
Challenge 65, page 61: The ‘only’ shape that prevents a cover to fall into the hole beneath is a
circular shape. Actually, slight deviations from the circular shape are also allowed.
Challenge 68, page 61: The walking speed of older men depends on their health. If people walk
Ref. 379 faster than 1.4 m/s, they are healthy. The study concluded that the grim reaper walks with a pre-
ferred speed of 0.82 m/s, and with a maximum speed of 1.36 m/s.
Challenge 69, page 62: 72 stairs.
challenge hints and solutions 481

paper discs
F I G U R E 308 A simple way to measure bullet speeds.

do not cut

cut first,

Motion Mountain – The Adventure of Physics
through cut last
both sides

do not cut
F I G U R E 309 How to make a hole
in a postcard that allows stepping
through it.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 73, page 62: See Figure 309 for a way to realize the feat.
Challenge 74, page 62: Within 1 per cent, one fifth of the height must be empty, and four fifths
must be filled; the exact value follows from √3 2 = 1.25992...
Challenge 75, page 62: One pencil draws a line of between 20 and 80 km, if no lead is lost when
sharpening. Numbers for the newly invented plastic, flexible pencils are not available.
Challenge 79, page 63: The bear is white, because the obvious spot of the house is at the North
pole. But there are infinitely many additional spots (without bears) near the South pole: can you
find them?
Challenge 80, page 63: We call 𝐿 the initial length of the rubber band, 𝑣 the speed of the snail
relative to the band and 𝑉 the speed of the horse relative to the floor. The speed of the snail
relative to the floor is given as
d𝑠 𝑠
=𝑣+𝑉 . (129)
d𝑡 𝐿 + 𝑉𝑡
This is a so-called differential equation for the unknown snail position 𝑠(𝑡). You can check – by
simple insertion – that its solution is given by
𝑣
𝑠(𝑡) = (𝐿 + 𝑉𝑡) ln(1 + 𝑉𝑡/𝐿) . (130)
𝑉
482 challenge hints and solutions

rope ℎ = Δ𝑙/2π rope 𝐻

𝑅 𝑅

Earth Earth

𝑅
Δ𝑙 = 2(√(𝑅 + 𝐻)2 − 𝑅2 − 𝑅 arccos 𝑅+𝐻
)

F I G U R E 310 Two ways to lengthen a rope around the Earth.

Motion Mountain – The Adventure of Physics
Therefore, the snail reaches the horse at a time
𝐿 𝑉/𝑣
𝑡reaching = (𝑒 − 1) (131)
𝑉
which is finite for all values of 𝐿, 𝑉 and 𝑣. You can check, however, that the time is very large
indeed, if realistic speed values are used.
Challenge 81, page 63: Colour is a property that applies only to objects, not to boundaries. In
the mentioned case, only spots and backgrounds have colours. The question shows that it is easy
to ask questions that make no sense also in physics.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 82, page 63: You can do this easily yourself. You can even find websites on the topic.
Challenge 84, page 64: Clocks with two hands: 22 times. Clocks with three hands: 2 times.
Challenge 85, page 64: 44 times.
Challenge 86, page 64: For two hands, the answer is 143 times.
Challenge 87, page 64: The Earth rotates with a speed of 15 minutes per minute.
Challenge 88, page 64: You might be astonished, but no reliable data exist on this question. The
highest speed of a throw measured so far seems to be a 45 m/s cricket bowl. By the way, much
more data are available for speeds achieved with the help of rackets. The c. 70 m/s of fast badmin-
ton smashes seem to be a good candidate for record racket speed; similar speeds are achieved by
golf balls.
Challenge 89, page 64: A spread out lengthening by 1 m allows even many cats to slip through,
as shown on the left side of Figure 310. But the right side of the figure shows a better way to
use the extra rope length, as Dimitri Yatsenko points out: a localized lengthening by 1 mm then
already yields a height of 1.25 m, allowing a child to walk through. In fact, a lengthening by 1 m
performed in this way yields a peak height of 121 m!
Challenge 90, page 64: 1.8 km/h or 0.5 m/s.
Challenge 92, page 64: The question makes sense, especially if we put our situation in relation
to the outside world, such as our own family history or the history of the universe. The different
usage reflects the idea that we are able to determine our position by ourselves, but not the time
Page 238 in which we are. The section on determinism will show how wrong this distinction is.
challenge hints and solutions 483

𝑑 𝑏

𝑤

𝐿
𝑅

F I G U R E 311 Leaving a parking
space – the outer turning radius.

Challenge 93, page 64: Yes, there is. However, this is not obvious, as it implies that space and
time are not continuous, in contrast to what we learn in primary school. The answer will be
found in the final part of this text.

Motion Mountain – The Adventure of Physics
Challenge 94, page 64: For a curve, use, at each point, the curvature radius of the circle approx-
imating the curve in that point; for a surface, define two directions in each point and use two
such circles along these directions.
Challenge 95, page 65: It moves about 1 cm in 50 ms.
Challenge 96, page 65: The surface area of the lung is between 100 and 200 m2 , depending on
the literature source, and that of the intestines is between 200 and 400 m2 .
Challenge 97, page 65: A limit does not exist in classical physics; however, there is one in nature
which appears as soon as quantum effects are taken into account.
Challenge 98, page 65: The final shape is a full cube without any hole.

𝑑 = √(𝐿 − 𝑏)2 − 𝑤2 + 2𝑤√𝑅2 − (𝐿 − 𝑏)2 − 𝐿 + 𝑏 , (132)

as deduced from Figure 311. See also R. Hoyle, Requirements for a perfect s-shaped parallel park-
ing maneuvre in a simple mathematical model, 2003. In fact, the mathematics of parallel park-
ing is beautiful and interesting. See, for example, the web page rigtriv.wordpress.com/2007/10/
01/parallel-parking/ or the explanation in Edward Nelson, Tensor Analysis, Princeton Uni-
versity Press, 1967, pp. 33–36. Nelson explains how to define vector fields that change the four-
dimensional configuration of a car, and how to use their algebra to show that a car can leave
parking spaces with arbitrarily short distances to the cars in front and in the back.
Challenge 100, page 66: A smallest gap does not exist: any value will do! Can you show this?
Challenge 101, page 66: The following solution was proposed by Daniel Hawkins.
Assume you are sitting in car A, parked behind car B, as shown in Figure 312. There are two
basic methods for exiting a parking space that requires the reverse gear: rotating the car to move
the centre of rotation away from (to the right of) car B, and shifting the car downward to move
the centre of rotation away from (farther below) car B. The first method requires car A to be
partially diagonal, which means that the method will not work for 𝑑 less than a certain value,
essentially the value given above, when no reverse gear is needed. We will concern ourselves
with the second method (pictured), which will work for an infinitesimal 𝑑.
In the case where the distance 𝑑 is less than the minimum required distance to turn out of the
parking space without using the reverse gear for a given geometry 𝐿, 𝑤, 𝑏, 𝑅, an attempt to turn
484 challenge hints and solutions

𝑑
Assumed motion (schematic)
of car A:
𝑚

second point of rotation
(straightening phase)

car B car A
𝑤
𝑇

Motion Mountain – The Adventure of Physics
𝑅
𝑥

𝑑

𝑏 𝑏

out of the parking space will result in the corner of car A touching car B at a distance 𝑇 away
from the edge of car B, as shown in Figure 312. This distance 𝑇 is the amount by which car A
must be translated downward in order to successfully turn out of the parking space.
The method to leave the parking space, shown in the top left corner of Figure 312, requires
two phases to be successful: the initial turning phase, and the straightening phase. By turning
and straightening out, we achieve a vertical shift downward and a horizontal shift left, while
preserving the original orientation. That last part is key because if we attempted to turn until
the corner of car A touched car B, car A would be rotated, and any attempt to straighten out
would just follow the same arc backward to the initial position, while turning the wheel the other
direction would rotate the car even more, as in the first method described above.
Our goal is to turn as far as we can and still be able to completely straighten out by time car A
touches car B. To analyse just how much this turn should be, we must first look at the properties
of a turning car.
Ackermann steering is the principle that in order for a car to turn smoothly, all four wheels
must rotate about the same point. This was patented by Rudolph Ackermann in 1817. Some prop-
erties of Ackermann steering in relation to this problem are as follows:
challenge hints and solutions 485

• The back wheels stay in alignment, but the front wheels (which we control), must turn
different amounts to rotate about the same centre.

• The centres of rotation for left and right turns are on opposite sides of the car

• For equal magnitudes of left and right turns, the centres of rotation are equidistant from
the nearest edge of the car. Figure 312 makes this much clearer.

• All possible centres of rotation are on the same line, which also always passes through the
back wheels.

Motion Mountain – The Adventure of Physics
• When the back wheels are ‘straight’ (straight will always mean in the same orientation as
the initial position), they will be vertically aligned with the centres of rotation.

• When the car is turning about one centre, say the one associated with the maximum left
turn, then the potential centre associated with the maximum right turn will rotate along
with the car. Similarly, when the cars turns about the right centre, the left centre rotates.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Now that we know the properties of Ackermann steering, we can say that in order to maximize
the shift downward while preserving the orientation, we must turn left about the 1st centre such
that the 2nd centre rotates a horizontal distance 𝑑, as shown in Figure 312. When this is achieved,
we brake, and turn the steering wheel the complete opposite direction so that we are now turning
right about the 2nd centre. Because we shifted leftward 𝑑, we will straighten out at the exact
moment car A comes in contact with car B. This results in our goal, a downward shift 𝑚 and
leftward shift 𝑑 while preserving the orientation of car A. A similar process can be performed in
reverse to achieve another downward shift 𝑚 and a rightward shift 𝑑, effectively moving car A
from its initial position (before any movement) downward 2𝑚 while preserving its orientation.
This can be done indefinitely, which is why it is possible to get out of a parking space with an
infinitesimal 𝑑 between car A and car B. To determine how many times this procedure (both
sets of turning and straightening) must be performed, we must only divide 𝑇 (remember 𝑇 is
the amount by which car A must be shifted downward in order to turn out of the parking spot
normally) by 2𝑚, the total downward shift for one iteration of the procedure. Symbolically,

𝑇
𝑛= . (133)
2𝑚

In order to get an expression for 𝑛 in terms of the geometry of the car, we must solve for 𝑇 and
486 challenge hints and solutions

F I G U R E 313 A simple drawing – one of F I G U R E 314 The trajectory of the
the many possible one – that allows middle point between the two ends
proving Pythagoras’ theorem. of the hands of a clock.

Motion Mountain – The Adventure of Physics
2𝑚. To simplify the derivations we define a new length 𝑥, also shown in Figure 312.

𝑥 = √𝑅2 − (𝐿 − 𝑏)2

𝑇 = √𝑅2 − (𝐿 − 𝑏 + 𝑑)2 − 𝑥 + 𝑤

= √𝑅2 − (𝐿 − 𝑏 + 𝑑)2 − √𝑅2 − (𝐿 − 𝑏)2 + 𝑤

= 2√𝑅2 − (𝐿 − 𝑏)2 − 𝑤 − √(2√𝑅2 − (𝐿 − 𝑏)2 − 𝑤)2 − 𝑑2

= 2√𝑅2 − (𝐿 − 𝑏)2 − 𝑤 − √4(𝑅2 − (𝐿 − 𝑏)2 ) − 4𝑤√𝑅2 − (𝐿 − 𝑏)2 + 𝑤2 − 𝑑2

= 2√𝑅2 − (𝐿 − 𝑏)2 − 𝑤 − √4𝑅2 − 4(𝐿 − 𝑏)2 − 4𝑤√𝑅2 − (𝐿 − 𝑏)2 + 𝑤2 − 𝑑2

We then get

𝑇 √𝑅2 − (𝐿 − 𝑏 + 𝑑)2 − √𝑅2 − (𝐿 − 𝑏)2 + 𝑤
𝑛= = .
2𝑚
4√𝑅 − (𝐿 − 𝑏) − 2𝑤 − 2√4𝑅2 − 4(𝐿 − 𝑏)2 − 4𝑤√𝑅2 − (𝐿 − 𝑏)2 + 𝑤2 − 𝑑2
2 2

The value of 𝑛 must always be rounded up to the next integer to determine how many times one
must go backward and forward to leave the parking spot.
Challenge 102, page 66: Nothing, neither a proof nor a disproof.
Challenge 103, page 67: See volume II, on page 20. On extreme shutters, see also the discussion
in Volume VI, on page 120.
challenge hints and solutions 487

1°
4°
3° 6°

3° 2° 3°

10°

F I G U R E 315 The angles deﬁned by the hands against the sky, when the arms are extended.

Motion Mountain – The Adventure of Physics
Challenge 104, page 67: A hint for the solution is given in Figure 313.
Challenge 105, page 67: Because they are or were liquid.
Challenge 106, page 67: The shape is shown in Figure 314; it has eleven lobes.
Challenge 107, page 67: The cone angle 𝜑, the angle between the cone axis and the cone bor-
der (or equivalently, half the apex angle of the cone) is related to the solid angle Ω through the
relation Ω = 2π(1 − cos 𝜑). Use the surface area of a spherical cap to confirm this result.
Challenge 109, page 68: See Figure 315.
Challenge 113, page 69: Hint: draw all objects involved.
Challenge 114, page 69: The curve is obviously called a catenary, from Latin ‘catena’ for chain.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
The formula for a catenary is 𝑦 = 𝑎 cosh(𝑥/𝑎). If you approximate the chain by short straight
segments, you can make wooden blocks that can form an arch without any need for glue. The
St. Louis arch is in shape of a catenary. A suspension bridge has the shape of a catenary before
it is loaded, i.e., before the track is attached to it. When the bridge is finished, the shape is in
between a catenary and a parabola.
Challenge 115, page 70: The inverse radii, or curvatures, obey 𝑎2 +𝑏2 +𝑐2 +𝑑2 = (1/2)(𝑎+𝑏+𝑐+𝑑)2 .
This formula was discovered by René Descartes. If one continues putting circles in the remaining
spaces, one gets so-called circle packings, a pretty domain of recreational mathematics. They have
many strange properties, such as intriguing relations between the coordinates of the circle centres
and their curvatures.
Challenge 116, page 70: One option: use the three-dimensional analogue of Pythagoras’s the-
orem. The answer is 9.
Challenge 117, page 70: There are two solutions. (Why?) They are the two positive solutions of
𝑙2 = (𝑏 + 𝑥)2 + (𝑏 + 𝑏2 /𝑥)2 ; the height is then given as ℎ = 𝑏 + 𝑥. The two solutions are 4.84 m and
1.26 m. There are closed formulas for the solutions; can you find them?
Challenge 118, page 71: The best way is to calculate first the height 𝐵 at which the blue ladder
touches the wall. It is given as a solution of 𝐵4 − 2ℎ𝐵3 − (𝑟2 − 𝑏2 )𝐵2 + 2ℎ(𝑟2 − 𝑏2 )𝐵 − ℎ2 (𝑟2 − 𝑏2 ) = 0.
Integer-valued solutions are discussed in Martin Gardner, Mathematical Circus, Spectrum,
1996.
Challenge 119, page 71: Draw a logarithmic scale, i.e., put every number at a distance corres-
ponding to its natural logarithm. Such a device, called a slide rule, is shown in Figure 316. Slide
488 challenge hints and solutions

Motion Mountain – The Adventure of Physics
rules were the precursors of electronic calculators; they were used all over the world in prehistoric
times, i.e., until around 1970. See also the web page www.oughtred.org.
Challenge 120, page 71: Two more days. Build yourself a model of the Sun and the Earth to
verify this. In fact, there is a small correction to the value 2, for the same reason that the makes
the solar day shorter than 24 hours.
Challenge 121, page 71: The Sun is exactly behind the back of the observer; it is setting, and the
rays are coming from behind and reach deep into the sky in the direction opposite to that of the
Sun.
1
Challenge 123, page 71: The volume is given by 𝑉 = ∫ 𝐴d𝑥 = ∫−1 4(1 − 𝑥2 )d𝑥 = 16/3.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 124, page 71: Yes. Try it with a paper model.
Challenge 125, page 72: Problems appear when quantum effects are added. A two-dimensional
universe would have no matter, since matter is made of spin 1/2 particles. But spin 1/2 particles
do not exist in two dimensions. Can you find additional reasons?
Challenge 126, page 72: Two dimensions of time do not allow ordering of events and obser-
vations. To say ‘before’ and ‘afterwards’ becomes impossible. In everyday life and all domains
accessible to measurement, time is definitely one-dimensional.
Challenge 127, page 72: No experiment has ever found any hint. Can this be nevertheless? Prob-
ably not, as argued in the last volume of this book series.
Challenge 130, page 73: The best solution seems to be 23 extra lines. Can you deduce it? To avoid
spoiling the fun of searching, no solution is given here. You can find solutions on blog.vixra.org/
2010/12/26/a-christmas-puzzle.
Challenge 131, page 74: If you solve this so-called ropelength problem, you will become a fam-
ous mathematician. The length is known only with about 6 decimals of precision. No exact for-
mula is known, and the exact shape of such ideal knots is unknown for all non-trivial knots. The
problem is also unsolved for all non-trivial ideal closed knots, for which the two ends are glued
together.
Challenge 132, page 76: From 𝑥 = 𝑔𝑡2 /2 you get the following rule: square the number of
seconds, multiply by five and you get the depth in metres.
Challenge 133, page 76: Just experiment.
challenge hints and solutions 489

Challenge 134, page 77: The Academicians suspended one cannon ball with a thin wire just in
front of the mouth of the cannon. When the shot was released, the second, flying cannon ball
flew through the wire, thus ensuring that both balls started at the same time. An observer from
far away then tried to determine whether both balls touched the Earth at the same time. The
experiment is not easy, as small errors in the angle and air resistance confuse the results.
Challenge 135, page 77: A parabola has a so-called focus or focal point. All light emitted from
that point and reflected exits in the same direction: all light rays are emitted in parallel. The
name ‘focus’ – Latin for fireplace – expresses that it is the hottest spot when a parabolic mirror
is illuminated. Where is the focus of the parabola 𝑦 = 𝑥2 ? (Ellipses have two foci, with a slightly
different definition. Can you find it?)
Challenge 136, page 78: The long jump record could surely be increased by getting rid of the
sand stripe and by measuring the true jumping distance with a photographic camera; this would
allow jumpers to run more closely to their top speed. The record could also be increased by a
small inclined step or by a spring-suspended board at the take-off location, to increase the take-
off angle.
Challenge 137, page 78: It may be held by Roald Bradstock, who threw a golf ball over 155 m.
Records for throwing mobile phones, javelins, people and washing machines are shorter.

Motion Mountain – The Adventure of Physics
Challenge 138, page 79: Walk or run in the rain, measure your own speed 𝑣 and the angle from
the vertical 𝛼 with which the rain appears to fall. Then the speed of the rain is 𝑣rain = 𝑣/ tan 𝛼.
Challenge 139, page 79: In ice skating, quadruple jumps are now state of the art. In dance, no
such drive for records exists.
Challenge 140, page 79: Neglecting air resistance and approximating the angle by 45°, we get
𝑣 = √𝑑𝑔 , or about 3.8 m/s. This speed is created by a steady pressure build-up, using blood
pressure, which is suddenly released with a mechanical system at the end of the digestive canal.
The cited reference tells more about the details.
Challenge 141, page 79: On horizontal ground, for a speed 𝑣 and an angle from the horizontal

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
𝛼, neglecting air resistance and the height of the thrower, the distance 𝑑 is 𝑑 = 𝑣2 sin 2𝛼/𝑔.
Challenge 142, page 79: Astonishingly, the answer is not clear. In 2012, the human record is el-
even balls. For robots, the present record is three balls, as performed by the Sarcoman robot. The
internet is full of material and videos on the topic. It is a challenge for people and robots to reach
the maximum possible number of balls.
Challenge 143, page 79: It is said so, as raindrops would then be ice spheres and fall with high
speed.
Challenge 144, page 80: Yes! People have gone to hospital and even died because a falling bul-
let went straight through their head. See S. Mirsky, It is high, it is far, Scientific American
p. 86, February 2004, or C. Tuijn, Vallende kogels, Nederlands tijdschrift voor natuurkunde 71,
pp. 224–225, 2005. Firing a weapon into the air is a crime.
Challenge 145, page 80: This is a true story. The answer can only be given if it is known whether
the person had the chance to jump while running or not. In the case described by R. Cross,
Forensic physics 101: falls from a height, American Journal of Physics 76, pp. 833–837, 2008, there
was no way to run, so that the answer was: murder.
Challenge 146, page 80: For jumps of an animal of mass 𝑚 the necessary energy 𝐸 is given as
𝐸 = 𝑚𝑔ℎ, and the work available to a muscle is roughly speaking proportional to its mass 𝑊 ∼ 𝑚.
Thus one gets that the height ℎ is independent of the mass of the animal. In other words, the
specific mechanical energy of animals is around 1.5 ± 0.7 J/kg.
Challenge 147, page 80: Stones never follow parabolas: when studied in detail, i.e., when the
change of 𝑔 with height is taken into account, their precise path turns out to be an ellipse. This
490 challenge hints and solutions

shape appears most clearly for long throws, such as throws around a sizeable part of the Earth,
or for orbiting objects. In short, stones follow parabolas only if the Earth is assumed to be flat. If
its curvature is taken into account, they follow ellipses.
Challenge 148, page 81: The set of all rotations around a point in a plane is indeed a vector
space. What about the set of all rotations around all points in a plane? And what about the three-
dimensional cases?
Challenge 151, page 81: The scalar product between two vectors 𝑎 and 𝑏 is given by

𝑎𝑏 = 𝑎𝑏 cos ∢(𝑎, 𝑏) . (134)

How does this differ from the vector product?
Challenge 154, page 84: One candidate for the lowest practical acceleration of a physical system
are the accelerations measured by gravitational wave detectors. They are below 10−13 m/s2 . But
these low values are beaten by the acceleration of the continental drift after the continents ‘snap’
apart: they accelerate from 7 mm/a to 40 mm/a in a ‘mere’ 3 million years. This corresponds to
a value of 10−23 m/s2 . Is there a theoretical lowest limit to acceleration?
Challenge 155, page 85: In free fall (when no air is present) or inside a space station orbiting

Motion Mountain – The Adventure of Physics
the Earth, you are accelerated but do not feel anything. However, the issue is not so simple. On
the one hand, constant and homogeneous accelerations are indeed not felt if there is no non-
accelerated reference. This indistinguishability or equivalence between acceleration and ‘feeling
nothing’ was an essential step for Albert Einstein in his development of general relativity. On the
other hand, if our senses were sensitive enough, we would feel something: both in the free fall
and in the space station, the acceleration is neither constant nor homogeneous. So we can indeed
say that accelerations found in nature can always be felt.
Challenge 156, page 85: Professor to student: What is the derivative of velocity? Acceleration!
What is the derivative of acceleration? I don’t know. Jerk! The fourth, fifth and sixth derivatives

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
of position are sometimes called snap, crackle and pop.
Challenge 158, page 87: One can argue that any source of light must have finite size.
Challenge 160, page 89: What the unaided human eye perceives as a tiny black point is usually
about 50 μm in diameter.
Challenge 161, page 89: See volume III, page 170.
Challenge 162, page 89: One has to check carefully whether the conceptual steps that lead us to
extract the concept of point from observations are correct. It will be shown in the final part of
the adventure that this is not the case.
Challenge 163, page 89: One can rotate the hand in a way that the arm makes the motion de-
scribed. See also volume IV, page 132.
Challenge 164, page 89: The number of cables has no limit. A visualization of tethered rotation
with 96 connections is found in volume VI, on page 182.
Challenge 165, page 90: The blood and nerve supply is not possible if the wheel has an axle. The
method shown to avoid tangling up connections only works when the rotating part has no axle:
the ‘wheel’ must float or be kept in place by other means. It thus becomes impossible to make
a wheel axle using a single piece of skin. And if a wheel without an axle could be built (which
might be possible), then the wheel would periodically run over the connection. Could such a
axle-free connection realize a propeller?
By the way, it is still thinkable that animals have wheels on axles, if the wheel is a ‘dead’
object. Even if blood supply technologies like continuous flow reactors were used, animals could
not make such a detached wheel grow in a way tuned to the rest of the body and they would have
challenge hints and solutions 491

difficulties repairing a damaged wheel. Detached wheels cannot be grown on animals; they must
be dead.
Challenge 166, page 92: The brain in the skull, the blood factories inside bones or the growth of
the eye are examples.
Challenge 167, page 92: In 2007, the largest big wheels for passengers are around 150 m in dia-
meter. The largest wind turbines are around 125 m in diameter. Cement kilns are the longest
wheels: they can be over 300 m along their axis.
Challenge 168, page 92: Air resistance reduces the maximum distance achievable with the soc-
cer ball – which is realized for an angle of about π/4 = 45° – from around 𝑣2 /𝑔 = 91.7 m down
to around 50 m.
Challenge 173, page 98: One can also add the Sun, the sky and the landscape to the list.
Challenge 174, page 99: There is no third option. Ghosts, hallucinations, Elvis sightings, or ex-
traterrestrials must all be objects or images. Also shadows are only special types of images.
Challenge 175, page 99: The issue was hotly discussed in the seventeenth century; even Galileo
argued for them being images. However, they are objects, as they can collide with other objects,
as the spectacular collision between Jupiter and the comet Shoemaker-Levy 9 in 1994 showed. In

Motion Mountain – The Adventure of Physics
the meantime, satellites have been made to collide with comets and even to shoot at them (and
hitting).
Challenge 176, page 100: The minimum speed is roughly the one at which it is possible to ride
without hands. If you do so, and then gently push on the steering wheel, you can make the ex-
perience described above. Watch out: too strong a push will make you fall badly.
The bicycle is one of the most complex mechanical systems of everyday life, and it is still a
subject of research. And obviously, the world experts are Dutch. An overview of the behaviour
of a bicycle is given in Figure 317. The main result is that the bicycle is stable in the upright
position at a range of medium speeds. Only at low and at large speeds must the rider actively
steer to ensure upright position of the bicycle.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
For more details, see the paper J. P. Meijaard, J. M. Papadopoulos, A. Ruina
& A. L. Schwab, Linearized dynamics equations for the balance and steer of a bicycle: a
benchmark and review, Proceedings of the Royal Society A 463, pp. 1955–1982, 2007, and
J. D. G. Kooijman, A. L. Schwab & J. P. Meijaard, Experimental validation of a model
of an uncontrolled bicycle, Multibody System Dynamics 19, pp. 115–132, 2008. See also the
audiophile.tam.cornell.edu/~als93/Bicycle/index.htm website.
Challenge 177, page 102: The total weight decreased slowly, due to the evaporated water lost by
sweating and, to a minor degree, due to the exhaled carbon bound in carbon dioxide.
Challenge 178, page 102: This is a challenge where the internet can help a lot. For a general in-
troduction, see the book by Lee Siegel, Net of Magic – Wonders and Deception in India, Uni-
versity of Chicago Press, 1991.
Challenge 179, page 103: If the moving ball is not rotating, after the collision the two balls will
depart with a right angle between them.
Challenge 180, page 103: As the block is heavy, the speed that it acquires from the hammer is
small and easily stopped by the human body. This effect works also with an anvil instead of a
concrete block. In another common variation the person does not lie on nails, but on air: he just
keeps himself horizontal, with head and shoulders on one chair, and the feet on a second one.
Challenge 181, page 104: Yes, the definition of mass works also for magnetism, because the pre-
cise condition is not that the interaction is central, but that the interaction realizes a more general
condition that includes accelerations such as those produced by magnetism. Can you deduce the
condition from the definition of mass as that quantity that keeps momentum conserved?
492 challenge hints and solutions

Bicycle motion
6

(stable at negative values, unstable at positive values)
Imaginary part
4
Weave
Magnitude of eigenvalues [1/s]

2
measured
values
Real part
0

–2
Capsize
Real part

Motion Mountain – The Adventure of Physics
–4
Castering

–6
0 1 2 3 5 6 7 8 9
speed [m/s]
Unstable speeds Stable speed range Unstable
speeds

F I G U R E 317 The measured (black bars) and calculated behaviour (coloured lines) – more precisely, the
dynamical eigenvalues – of a bicycle as a function of its speed (© Arend Schwab).

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 182, page 106: Rather than using the inertial effects of the Earth, it is easier to deduce
Page 185 its mass from its gravitational effects.
Challenge 187, page 107: At first sight, relativity implies that tachyons have imaginary mass;
however, the imaginary factor can be extracted from the mass–energy and mass–momentum
relation, so that one can define a real mass value for tachyons. As a result, faster tachyons have
smaller energy and smaller momentum than slower ones. In fact, both tachyon momentum and
tachyon energy can be a negative number of any size.
Challenge 188, page 109: The leftmost situation has a tiny effect, the second situation makes the
car roll forward and backwards, the right two pictures show ways to open wine bottles without
an bottle opener.
Challenge 189, page 109: Legs are never perfectly vertical; they would immediately glide away.
Once the cat or the person is on the floor, it is almost impossible to stand up again.
Challenge 190, page 109: Momentum (or centre of mass) conservation would imply that the en-
vironment would be accelerated into the opposite direction. Energy conservation would imply
that a huge amount of energy would be transferred between the two locations, melting everything
in between. Teleportation would thus contradict energy and momentum conservation.
Challenge 191, page 110: The part of the tides due to the Sun, the solar wind, and the interac-
tions between both magnetic fields are examples of friction mechanisms between the Earth and
the Sun.
challenge hints and solutions 493

Challenge 192, page 110: With the factor 1/2, increase of (physical) kinetic energy is equal to
the (physical) work performed on a system: total energy is thus conserved only if the factor 1/2
is added.
Challenge 194, page 112: It is a smart application of momentum conservation.
Challenge 195, page 112: Neither. With brakes on, the damage is higher, but still equal for both
cars.
Challenge 196, page 113: Heating systems, transport engines, engines in factories, steel plants,
electricity generators covering the losses in the power grid, etc. By the way, the richest countries
in the world, such as Sweden or Switzerland, consume only half the energy per inhabitant as the
USA. This waste is one of the reasons for the lower average standard of living in the USA.
Challenge 202, page 117: Just throw the brick into the air and compare the dexterity needed to
make it turn around various axes.
Challenge 203, page 118: Use the definition of the moment of inertia and Pythagoras’ theorem
for every mass element of the body.
Challenge 204, page 119: Hang up the body, attaching the rope at two different points of the

Motion Mountain – The Adventure of Physics
body. The crossing point of the prolonged rope lines is the centre of mass.
Challenge 205, page 119: See Tables 19 and 20.
Challenge 206, page 119: Spheres have an orientation, because we can always add a tiny spot
on their surface. This possibility is not given for microscopic objects, and we shall study this
situation in the part on quantum theory.
Challenge 209, page 120: Yes, the ape can reach the banana. The ape just has to turn around its
own axis. For every turn, the plate will rotate a bit towards the banana. Of course, other methods,
like blowing at a right angle to the axis, peeing, etc., are also possible.
Challenge 210, page 121: Self-propelled linear motion contradicts the conservation of mo-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
mentum; self-propelled change of orientation (as long as the motion stops again) does not con-
tradict any conservation law. But the deep, underlying reason for the difference will be unveiled
in the final part of our adventure.
Challenge 212, page 121: The points that move exactly along the radial direction of the wheel
form a circle below the axis and above the rim. They are the points that are sharp in Figure 79 of
page 121.
Challenge 213, page 122: Use the conservation of angular momentum around the point of con-
tact. If all the wheel’s mass is assumed in the rim, the final rotation speed is half the initial one;
it is independent of the friction coefficient.
Challenge 215, page 123: Probably the ‘rest of the universe’ was meant by the writer. Indeed, a
moving a part never shifts the centre of gravity of a closed system. But is the universe closed? Or
a system? The last part of our adventure addresses these issues.
Challenge 219, page 124: Hint: energy and momentum conservation yield two equations; but
in the case of three balls there are three variables. What else is needed? See F. Herrmann &
M. Seitz, How does the ball-chain work?, American Journal of Physics 50, pp. 977–981, 1982. The
bigger challenge is to build a high precision ball-chain, in which the balls behave as expected,
minimizing spurious motion. Nobody seems to have built one yet, as the internet shows. Can
you?
Challenge 220, page 124: The method allowed Phileas Fogg to win the central bet in the well-
known adventure novel by Jules Verne, Around the World in Eighty Days, translated from Le
tour du monde en quatre-vingts jours, first published in 1872.
494 challenge hints and solutions

Challenge 221, page 125: The human body is more energy-efficient at low and medium power
output. The topic is still subject of research, as detailed in the cited reference. The critical slope
is estimated to be around 16° for uphill walkers, but should differ for downhill walkers.
Challenge 223, page 125: Hint: an energy per distance is a force.
Challenge 224, page 126: The conservation of angular momentum saves the glass. Try it.
Challenge 225, page 126: First of all, MacDougall’s experimental data is flawed. In the six cases
MacDougall examined, he did not know the exact timing of death. His claim of a mass decrease
cannot be deduced from his own data. Modern measurements on dying sheep, about the same
mass as humans, have shown no mass change, but clear weight pulses of a few dozen grams
when the heart stopped. This temporary weight decrease could be due to the expelling of air or
moisture, to the relaxing of muscles, or to the halting of blood circulation. The question is not
settled.
Challenge 227, page 126: Assuming a square mountain, the height ℎ above the surrounding
crust and the depth 𝑑 below are related by
ℎ 𝜌m − 𝜌c
= (135)
𝑑 𝜌c

Motion Mountain – The Adventure of Physics
where 𝜌c is the density of the crust and 𝜌m is the density of the mantle. For the density values
given, the ratio is 6.7, leading to an additional depth of 6.7 km below the mountain.
Challenge 229, page 127: The can filled with liquid. Videos on the internet show the experiment.
Why is this the case?
Challenge 232, page 127: The matter in the universe could rotate – but not the universe itself.
Measurements show that within measurement errors there is no mass rotation.
Challenge 233, page 128: The behaviour of the spheres can only be explained by noting that
elastic waves propagate through the chain of balls. Only the propagation of these elastic waves,
in particular their reflection at the end of the chain, explains that the same number of balls that

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
hit on one side are lifted up on the other. For long times, friction makes all spheres oscillate in
phase. Can you confirm this?
Challenge 234, page 128: When the short cylinder hits the long one, two compression waves
start to run from the point of contact through the two cylinders. When each compression wave
arrives at the end, it is reflected as an expansion wave. If the geometry is well chosen, the expan-
sion wave coming back from the short cylinder can continue into the long one (which is still in
his compression phase). For sufficiently long contact times, waves from the short cylinder can
thus depose much of their energy into the long cylinder. Momentum is conserved, as is energy;
the long cylinder is oscillating in length when it detaches, so that not all its energy is transla-
tional energy. This oscillation is then used to drive nails or drills into stone walls. In commercial
hammer drills, length ratios of 1:10 are typically used.
Challenge 235, page 128: The momentum transfer to the wall is double when the ball rebounds
perfectly.
Challenge 236, page 129: If the cork is in its intended position: take the plastic cover off the
cork, put the cloth around the bottle or the bottle in the shoe (this is for protection reasons only)
and repeatedly hit the bottle on the floor or a fall in an inclined way, as shown in Figure 72 on
page 109. With each hit, the cork will come out a bit.
If the cork has fallen inside the bottle: put half the cloth inside the bottle; shake until the cork
falls unto the cloth. Pull the cloth out: first slowly, until the cloth almost surrounds the cork, and
then strongly.
Challenge 237, page 129: Indeed, the lower end of the ladder always touches the floor. Why?
challenge hints and solutions 495

Challenge 238, page 129: The atomic force microscope.
Challenge 240, page 129: Running man: 𝐸 ≈ 0.5 ⋅ 80 kg ⋅ (5 m/s)2 = 1 kJ; rifle bullet: 𝐸 ≈ 0.5 ⋅
0.04 kg ⋅ (500 m/s)2 = 5 kJ.
Challenge 241, page 130: It almost doubles in size.
Challenge 242, page 130: At the highest point, the acceleration is 𝑔 sin 𝛼, where 𝛼 is the angle
of the pendulum at the highest point. At the lowest point, the acceleration is 𝑣2 /𝑙, where 𝑙 is
the length of the pendulum. Conservation of energy implies that 𝑣2 = 2𝑔𝑙(1 − cos 𝛼). Thus the
problem requires that sin 𝛼 = 2(1 − cos 𝛼). This results in cos 𝛼 = 3/5.
Challenge 243, page 130: One needs the mass change equation d𝑚/d𝑡 = π𝜌vapour 𝑟2 |𝑣| due to the
mist and the drop speed evolution 𝑚 d𝑣/d𝑡 = 𝑚𝑔 − 𝑣 d𝑚/d𝑡. These two equations yield
d𝑣2 2𝑔 𝑣2
= −6 (136)
d𝑟 𝐶 𝑟
where 𝐶 = 𝜌vapour /4𝜌water . The trick is to show that this can be rewritten as

d 𝑣2 2𝑔 𝑣2
𝑟 = −7 . (137)
d𝑟 𝑟 𝐶 𝑟

Motion Mountain – The Adventure of Physics
For large times, all physically sensible solutions approach 𝑣2 /𝑟 = 2𝑔/7𝐶; this implies that for
large times,
d𝑣 𝑣2 𝑔 𝑔𝐶 2
= and 𝑟 = 𝑡 . (138)
d𝑡 𝑟 7 14
About this famous problem, see for example, B. F. Edwards, J. W. Wilder & E. E. Scime,
Dynamics of falling raindrops, European Journal of Physics 22, pp. 113–118, 2001, or A. D. Sokal,
The falling raindrop, revisited, preprint at arxiv.org/abs/0908.0090.
Challenge 244, page 130: One is faster, because the moments of inertia differ. Which one?
Challenge 245, page 130: There is no simple answer, as aerodynamic drag plays an important

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
role. There are almost no studies on the topic. By the way, competitive rope jumping is challen-
ging; for example, a few people in the world are able to rotate the rope 5 times under their feet
during a single jump. Can you do better?
Challenge 246, page 130: Weigh the bullet and shoot it against a mass hanging from the ceiling.
From the mass and the angle it is deflected to, the momentum of the bullet can be determined.
Challenge 248, page 130: The curve described by the midpoint of a ladder sliding down a wall
is a circle.
Challenge 249, page 131: The switches use the power that is received when the switch is pushed
and feeds it to a small transmitter that acts a high frequency remote control to switch on the
light.
Challenge 250, page 131: A clever arrangement of bimetals is used. They move every time the
temperature changes from day to night – and vice versa – and wind up a clock spring. The clock
itself is a mechanical clock with low energy consumption.
Challenge 251, page 131: The weight of the lift does not change at all when a ship enters it. A
twin lift, i.e., a system in which both lifts are mechanically or hydraulically connected, needs no
engine at all: it is sufficient to fill the upper lift with a bit of additional water every time a ship
enters it. Such ship lifts without engines at all used to exist in the past.
Challenge 254, page 133: This is not easy; a combination of friction and torques play a role. See
for example the article J. Sauer, E. Schörner & C. Lennerz, Real-time rigid body simula-
tion of some classical mechanical toys, 10th European Simulation and Symposium and Exhibition
(ESS ’98) 1998, pp. 93–98, or www.lennerz.de/paper_ess98.pdf.
496 challenge hints and solutions

Page 166 Challenge 257, page 135: See Figure 123 for an example. The pole is not at the zenith.
Challenge 258, page 135: Robert Peary had forgotten that on the date he claimed to be at the
North Pole, 6th of April 1909, the Sun is very low on the horizon, casting very long shadows,
about ten times the height of objects. But on his photograph the shadows are much shorter. (In
fact, the picture is taken in such a way to hide all shadows as carefully as possible.) Interestingly,
he had even convinced the US congress to officially declare him the first man on the North Pole
in 1911. (A rival crook had claimed to have reached it before Peary, but his photograph has the
same mistake.) Peary also cheated on the travelled distances of the last few days; he also failed
to mention that the last days he was pulled by his partner, Matthew Henson, because he was not
able to walk any more. In fact Matthew Henson deserves more credit for that adventure than
Peary. Henson, however, did not know that Peary cheated on the position they had reached.
Challenge 260, page 136: Laplace and Gauss showed that the eastward deflection 𝑑 of a falling
object is given by
𝑑 = 2/3Ω cos 𝜑√2ℎ3 /𝑔 . (139)
Here Ω = 72.92 μrad/s is the angular velocity of the Earth, 𝜑 is the latitude, 𝑔 the gravitational
acceleration and ℎ is the height of the fall.

Motion Mountain – The Adventure of Physics
Challenge 261, page 139: The Coriolis effect can be seen as the sum two different effects of equal
magnitude. The first effect is the following: on a rotating background, velocity changes over time.
What an inertial (non-rotating) observer sees as a constant velocity will be seen a velocity that
changes over time by the rotating observer. The acceleration seen by the rotating observer is neg-
ative, and is proportional to the angular velocity and to the velocity.
The second effect is change of velocity in space. In a rotating frame of reference, different
points have different velocities. The effect is negative, and proportional to the angular velocity
and to the velocity.
In total, the Coriolis acceleration (or Coriolis effect) is thus 𝑎C = −2𝜔 × 𝑣.
Challenge 262, page 140: A short pendulum of length 𝐿 that swings in two dimensions (with

𝜌2 𝑙2 𝜌2
L = 𝑇 − 𝑉 = 12 𝑚𝜌2̇ (1 + 2
) + 𝑧 2 − 12 𝑚𝜔02 𝜌2 (1 + ) (140)
𝐿 2𝑚𝜌 4 𝐿2

where as usual the basic frequency is 𝜔02 = 𝑔/𝐿 and the angular momentum is 𝑙𝑧 = 𝑚𝜌2 𝜑.̇ The
two additional terms disappear when 𝐿 → ∞; in that case, if the system oscillates in an ellipse
with semiaxes 𝑎 and 𝑏, the ellipse is fixed in space, and the frequency is 𝜔0 . For finite pendulum
length 𝐿, the frequency changes to

𝑎2 + 𝑏2
𝜔 = 𝜔0 (1 − ) . (141)
16 𝐿2
The ellipse turns with a frequency
3 𝑎𝑏
Ω=𝜔 . (142)
8 𝐿2
These formulae can be derived using the least action principle, as shown by C. G. Gray,
G. Karl & V. A. Novikov, Progress in classical and quantum variational principles, arxiv.org/
abs/physics/0312071. In other words, a short pendulum in elliptical motion shows a precession
even without the Coriolis effect. Since this precession frequency diminishes with 1/𝐿2 , the effect
is small for long pendulums, where only the Coriolis effect is left over. To see the Coriolis effect
in a short pendulum, one thus has to avoid that it starts swinging in an elliptical orbit by adding
a mechanism that suppresses elliptical motion.
challenge hints and solutions 497

light
source
Ω𝑡

F I G U R E 318
𝑡=0 𝑡 = 2π𝑅/𝑐 Deducing the
expression for the
Sagnac effect.

Challenge 263, page 141: The Coriolis acceleration is the reason for the deviation from the
straight line. The Coriolis acceleration is due to the change of speed with distance from the rota-

Motion Mountain – The Adventure of Physics
tion axis. Now think about a pendulum, located in Paris, swinging in the North-South direction
with amplitude 𝐴. At the Southern end of the swing, the pendulum is further from the axis by
𝐴 sin 𝜑, where 𝜑 is the latitude. At that end of the swing, the central support point overtakes the
pendulum bob with a relative horizontal speed given by 𝑣 = 2π𝐴 sin 𝜑/23 h56 min. The period
of precession is given by 𝑇F = 𝑣/2π𝐴, where 2π𝐴 is the circumference 2π𝐴 of the envelope of
the pendulum’s path (relative to the Earth). This yields 𝑇F = 23 h56 min/ sin 𝜑. Why is the value
that appears in the formula not 24 h, but 23 h56 min?
Challenge 264, page 141: Experiments show that the axis of the gyroscope stays fixed with re-
spect to distant stars. No experiment shows that it stays fixed with respect to absolute space,
because this kind of ‘‘absolute space’’ cannot be defined or observed at all. It is a useless concept.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 265, page 141: Rotation leads to a small frequency and thus colour changes of the
circulating light.
Challenge 266, page 141: The weight changes when going east or when moving west due to the
Coriolis acceleration. If the rotation speed is tuned to the oscillation frequency of the balance,
the effect is increased by resonance. This trick was also used by Eötvös.
Challenge 267, page 141: The Coriolis acceleration makes the bar turn, as every moving body is
deflected to the side, and the two deflections add up in this case. The direction of the deflection
depends on whether the experiments is performed on the northern or the southern hemisphere.
Challenge 268, page 141: When rotated by π around an east–west axis, the Coriolis force pro-
duces a drift velocity of the liquid around the tube. It has the value
𝑣 = 2𝜔𝑟 sin 𝜃, (143)
as long as friction is negligible. Here 𝜔 is the angular velocity of the Earth, 𝜃 the latitude and 𝑟
the (larger) radius of the torus. For a tube with 1 m diameter in continental Europe, this gives a
speed of about 6.3 ⋅ 10−5 m/s.
The measurement can be made easier if the tube is restricted in diameter at one spot, so that
the velocity is increased there. A restriction by an area factor of 100 increases the speed by the
same factor. When the experiment is performed, one has to carefully avoid any other effects that
lead to moving water, such as temperature gradients across the system.
Challenge 269, page 142: Imagine a circular light path (for example, inside a circular glass fibre)
and two beams moving in opposite directions along it, as shown in Figure 318. If the fibre path
498 challenge hints and solutions

rotates with rotation frequency Ω, we can deduce that, after one turn, the difference Δ𝐿 in path
length is
4π𝑅2 Ω
Δ𝐿 = 2𝑅Ω𝑡 = . (144)
𝑐
The phase difference is thus
8π2 𝑅2
Δ𝜑 = Ω (145)
𝑐𝜆
if the refractive index is 1. This is the required formula for the main case of the Sagnac effect.
It is regularly suggested that the Sagnac effect can only be understood with help of general
relativity; this is wrong. As just done, the effect is easily deduced from the invariance of the speed
of light 𝑐. The effect is a consequence of special relativity.
Challenge 270, page 145: The metal rod is slightly longer on one side of the axis. When the wire
keeping it up is burned with a candle, its moment of inertia decreases by a factor of 104 ; thus it
starts to rotate with (ideally) 104 times the rotation rate of the Earth, a rate which is easily visible
by shining a light beam on the mirror and observing how its reflection moves on the wall.
Challenge 272, page 152: The original result by Bessel was 0.3136 󸀠󸀠 , or 657.7 thousand orbital

Motion Mountain – The Adventure of Physics
radii, which he thought to be 10.3 light years or 97.5 Pm.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 319 How the night sky, and our galaxy in particular, looks in the near-infrared (NASA false
colour image).

Challenge 274, page 155: The galaxy forms a stripe in the sky. The galaxy is thus a flattened
structure. This is even clearer in the infrared, as shown more clearly in Figure 319. From the flat-
tening (and its circular symmetry) we can deduce that the galaxy must be rotating. Thus other
matter must exist in the universe.
Challenge 276, page 158: See page 189.
Challenge 278, page 158: The scale reacts to your heartbeat. The weight is almost constant over
time, except when the heart beats: for a short duration of time, the weight is somewhat lowered
at each beat. Apparently it is due to the blood hitting the aortic arch when the heart pumps it
upwards. The speed of the blood is about 0.3 m/s at the maximum contraction of the left ventricle.
The distance to the aortic arch is a few centimetres. The time between the contraction and the
reversal of direction is about 15 ms. And the measured weight is not even constant for a dead
person, as air currents disturb the measurement.
challenge hints and solutions 499

Challenge 279, page 158: Use Figure 97 on page 138 for the second half of the trajectory, and
think carefully about the first half. The body falls down slightly to the west of the starting point.
Challenge 280, page 158: Hint: starting rockets at the Equator saves a lot of energy, thus of fuel
and of weight.
Challenge 283, page 161: The flame leans towards the inside.
Challenge 284, page 161: Yes. There is no absolute position and no absolute direction. Equival-
ently, there is no preferred position, and no preferred direction. For time, only the big bang seems
to provide an exception, at first; but when quantum effects are included, the lack of a preferred
time scale is confirmed.
Challenge 285, page 161: For your exam it is better to say that centrifugal force does not exist.
But since in each stationary system there is a force balance, the discussion is somewhat a red
herring.
Challenge 288, page 162: Place the tea in cups on a board and attach the board to four long ropes
that you keep in your hand.
Challenge 289, page 162: The ball leans in the direction it is accelerated to. As a result, one could
imagine that the ball in a glass at rest pulls upwards because the floor is accelerated upwards. We

Motion Mountain – The Adventure of Physics
will come back to this issue in the section of general relativity.
Challenge 290, page 162: The friction of the tides on Earth are the main cause.
Challenge 291, page 163: An earthquake with Richter magnitude of 12 is 1000 times the energy
of the 1960 Chile quake with magnitude 10; the latter was due to a crack throughout the full 40 km
of the Earth’s crust along a length of 1000 km in which both sides slipped by 10 m with respect
to each other. Only the impact of a large asteroid could lead to larger values than 12.
Challenge 293, page 163: Yes; it happens twice a year. To minimize the damage, dishes should
be dark in colour.
Challenge 294, page 163: A rocket fired from the back would be a perfect defence against planes

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
attacking from behind. However, when released, the rocket is effectively flying backwards with
respect to the air, thus turns around and then becomes a danger to the plane that launched it.
Engineers who did not think about this effect almost killed a pilot during the first such tests.
Challenge 295, page 163: Whatever the ape does, whether it climbs up or down or even lets him-
self fall, it remains at the same height as the mass. Now, what happens if there is friction at the
wheel?
Challenge 296, page 163: Yes, if he moves at a large enough angle to the direction of the boat’s
motion.
Challenge 297, page 163: See the article by C. Ucke & H. -J. Schlichting, Faszinierendes
Dynabee, Physik in unserer Zeit 33, pp. 230–231, 2002.
Challenge 298, page 164: See the article by C. Ucke & H. -J. Schlichting, Die kreisende
Büroklammer, Physik in unserer Zeit 36, pp. 33–35, 2005.
Challenge 299, page 164: If a wedding ring rotates on an axis that is not a principal one, angular
momentum and velocity are not parallel.
2
Challenge 300, page 164: The moment of inertia for a homogeneous sphere is Θ = 5 𝑚𝑟2 .
Challenge 301, page 164: The three moments of inertia for the cube are equal, as in the case of
1
the sphere, but the values are Θ = 6
𝑚𝑙2 . The efforts required to put a sphere and a cube into
rotationthe are thus different.
Challenge 304, page 165: Yes, the moon differs in this way. Can you imagine what happens for
an observer on the Equator?
500 challenge hints and solutions

Challenge 305, page 167: A straight line at the zenith, and circles getting smaller at both sides.
See an example on the website apod.nasa.gov/apod/ap021115.html.
Challenge 307, page 167: The plane is described in the websites cited; for a standing human the
plane is the vertical plane containing the two eyes.
Challenge 308, page 168: If you managed, please send the author the video!
Challenge 309, page 169: As said before, legs are simpler than wheels to grow, to maintain and
to repair; in addition, legs do not require flat surfaces (so-called ‘streets’) to work.
Challenge 310, page 170: The staircase formula is an empirical result found by experiment, used
by engineers world-wide. Its origin and explanation seems to be lost in history.
Challenge 311, page 170: Classical or everyday nature is right-left symmetric and thus requires
an even number of legs. Walking on two-dimensional surfaces naturally leads to a minimum of
four legs. Starfish, snails, slugs, clams, eels and snakes are among the most important exceptions
for which the arguments are not valid.
Challenge 313, page 173: The length of the day changes with latitude. So does the length of a
shadow or the elevation of stars at night, facts that are easily checked by telephoning a friend.
Ships appear at the horizon by showing their masts first. These arguments, together with the

Motion Mountain – The Adventure of Physics
round shadow of the earth during a lunar eclipse and the observation that everything falls down-
wards everywhere, were all given already by Aristotle, in his text On the Heavens. It is now known
that everybody in the last 2500 years knew that the Earth is a sphere. The myth that many people
used to believe in a flat Earth was put into the world – as rhetorical polemic – by Copernicus. The
story then continued to be exaggerated more and more during the following centuries, because
Vol. III, page 314 a new device for spreading lies had just been invented: book printing. Fact is that for 2500 years
the vast majority of people knew that the Earth is a sphere.
Challenge 314, page 177: The vector 𝑆𝐹 can be calculated by using 𝑆𝐶 = −(𝐺𝑚𝑀/𝐸) 𝑆𝑃/𝑆𝑃 and
then translating the construction given in the figure into formulae. This exercise yields

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
𝐾
𝑆𝐹 = (146)
𝑚𝐸
where
𝐾 = 𝑝 × 𝐿 − 𝐺𝑀𝑚2 𝑥/𝑥 (147)
is the so-called Runge–Lenz vector. The Runge-Lenz vector is directed along the line that connects
the second focus to the first focus of the ellipse (the Sun). We have used 𝑥 = 𝑆𝑃 for the position of
the orbiting body, 𝑝 for its momentum and 𝐿 for its angular momentum. The Runge–Lenz vector
𝐾 is constant along the orbit of a body, thus has the same value for any position 𝑥 on the orbit.
(Prove it by starting from 𝑥𝐾 = 𝑥𝐾 cos 𝜃.) The Runge–Lenz vector is thus a conserved quantity
in universal gravity. As a result, the vector 𝑆𝐹 is also constant in time.
The Runge–Lenz vector is also often used in quantum mechanics, when calculating the energy
levels of a hydrogen atom, as it appears in all problems with a 1/𝑟 potential. (In fact, the incorrect
name ‘Runge–Lenz vector’ is due to Wolfgang Pauli; the discoverer of the vector was, in 1710,
Jakob Hermann.)
Challenge 316, page 178: On orbits, see page 192.
Challenge 317, page 178: The low gravitational acceleration of the Moon, 1.6 m/s2 , implies that
gas molecules at usual temperatures can escape its attraction.
Challenge 318, page 179: The tip of the velocity arrow, when drawn over time, produces a circle
around the centre of motion.
Challenge 319, page 179: Draw a figure of the situation.
challenge hints and solutions 501

Challenge 320, page 179: Again, draw a figure of the situation.
Challenge 321, page 180: The value of the product 𝐺𝑀 for the Earth is 4.0 ⋅ 1014 m3 /s2 .
Challenge 322, page 180: All points can be reached for general inclinations; but when shooting
horizontally in one given direction, only points on the first half of the circumference can be
reached.
Challenge 323, page 182: On the moon, the gravitational acceleration is 1.6 m/s2 , about one
sixth of the value on Earth. The surface values for the gravitational acceleration for the planets
can be found on many internet sites.
Challenge 324, page 182: The Atwood machine is the answer: two almost equal masses 𝑚1 and
𝑚2 connected by a string hanging from a well-oiled wheel of negligible mass. The heavier one
falls very slowly. Can show that the acceleration 𝑎 of this ‘unfree’ fall is given by 𝑎 = 𝑔(𝑚1 −
𝑚2 )/(𝑚1 + 𝑚2 )? In other words, the smaller the mass difference is, the slower the fall is.
Challenge 325, page 183: You should absolutely try to understand the origin of this expression.
It allows understanding many important concepts of mechanics. The idea is that for small amp-
litudes, the acceleration of a pendulum of length 𝑙 is due to gravity. Drawing a force diagram for
a pendulum at a general angle 𝛼 shows that

Motion Mountain – The Adventure of Physics
𝑚𝑎 = −𝑚𝑔 sin 𝛼
d2 𝛼
𝑚𝑙 = −𝑚𝑔 sin 𝛼
d𝑡2
d2 𝛼
𝑙 2 = −𝑔 sin 𝛼 . (148)
d𝑡
For the mentioned small amplitudes (below 15°) we can approximate this to

d2 𝛼
𝑙 = −𝑔𝛼 . (149)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
d𝑡2
This is the equation for a harmonic oscillation (i.e., a sinusoidal oscillation). The resulting motion
is:
𝛼(𝑡) = 𝐴 sin(𝜔𝑡 + 𝜑) . (150)
The amplitude 𝐴 and the phase 𝜑 depend on the initial conditions; however, the oscillation fre-
quency is given by the length of the pendulum and the acceleration of gravity (check it!):

𝑙
𝜔=√ . (151)
𝑔

(For arbitrary amplitudes, the formula is much more complex; see the internet or special mech-
anics books for more details.)
Challenge 326, page 183: Walking speed is proportional to 𝑙/𝑇, which makes it proportional to
𝑙1/2 . The relation is also true for animals in general. Indeed, measurements show that the max-
imum walking speed (thus not the running speed) across all animals is given by

𝑣maxwalking = (2.2 ± 0.2) m1/2 /s √𝑙 . (152)

Challenge 330, page 186: There is no obvious candidate formula. Can you find one?
Challenge 331, page 186: The acceleration due to gravity is 𝑎 = 𝐺𝑚/𝑟2 ≈ 5 nm/s2 for a mass
of 75 kg. For a fly with mass 𝑚fly = 0.1 g landing on a person with a speed of 𝑣fly = 1 cm/s
502 challenge hints and solutions

F I G U R E 320 The Lagrangian points
and the effective potential that
produces them (NASA).

Motion Mountain – The Adventure of Physics
and deforming the skin (without energy loss) by 𝑑 = 0.3 mm, a person would be accelerated by
𝑎 = (𝑣2 /𝑑)(𝑚fly /𝑚) = 0.4 μm/s2 . The energy loss of the inelastic collision reduces this value at
least by a factor of ten.
Challenge 332, page 187: The calculation shows that a surprisingly high energy value is stored
in thermal motion.
Challenge 333, page 188: Yes, the effect has been measured for skyscrapers. Can you estimate
the values?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 336, page 188: The easiest way to see this is to picture gravity as a flux emanating
from a sphere. This gives a 1/𝑟𝑑−1 dependence for the force and thus a 1/𝑟𝑑−2 dependence of the
potential.
Challenge 338, page 190: Since the paths of free fall are ellipses, which are curves lying in a
plane, this is obvious.
Challenge 340, page 191: A flash of light is sent to the Moon, where several Cat’s-eyes have been
deposited by the Lunokhod and Apollo missions. The measurement precision of the time a flash
take to go and come back is sufficient to measure the Moon’s distance change. For more details,
see challenge 8.
Challenge 342, page 193: A body having zero momentum at spatial infinity is on a parabolic
path. A body with a lower momentum is on an elliptic path and one with a higher momentum is
on a hyperbolic path.
Challenge 345, page 194: The Lagrangian points L4 and L5 are on the orbit, 60° before and be-
hind the orbiting body. They are stable if the mass ratio of the central and the orbiting body is
sufficiently large (above 24.9).
Challenge 346, page 194: The Lagrangian point L3 is located on the orbit, but precisely on the
other side of the central body. The Lagrangian point L1 is located on the line connecting the planet
with the central body, whereas L2 lies outside the orbit, on the same line. If 𝑅 is the radius of the
orbit, the distance between the orbiting body and the L1 and L2 point is √3 𝑚/3𝑀 𝑅, giving around
4 times the distance of the Moon for the Sun-Earth system. L1, L2 and L3 are saddle points, but
effectively stable orbits exist around them. Many satellites make use of these properties, including
challenge hints and solutions 503

F I G U R E 321 The famous ‘vomit comet’, a KC-135,
performing a parabolic ﬂight (NASA).

Motion Mountain – The Adventure of Physics
the famous WMAP satellite that measured the ripples of the big bang, which is located at the ‘quiet’
point L2, where the Sun, the Earth and the Moon are easily shielded and satellite temperature
remains constant.
Challenge 347, page 197: This is a resonance effect, in the same way that a small vibration of a
string can lead to large oscillation of the air and sound box in a guitar.
Challenge 349, page 199: The expression for the strength of tides, namely 2𝐺𝑀/𝑑3 , can be re-
written as (8/3)π𝐺𝜌(𝑅/𝑑)3 . Now, 𝑅/𝑑 is roughly the same for Sun and Moon, as every eclipse
shows. So the density 𝜌 must be much larger for the Moon. In fact, the ratio of the strengths
(height) of the tides of Moon and Sun is roughly 7 : 3. This is also the ratio between the mass

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
densities of the two bodies.
Challenge 350, page 200: The total angular momentum of the Earth and the Moon must remain
constant.
Challenge 352, page 201: Wait for a solar eclipse.
Challenge 354, page 203: Unfortunately, the myth of ‘passive gravitational mass’ is spread by
many books. Careful investigation shows that it is measured in exactly the same way as inertial
mass.
Both masses are measured with the same machines and set-ups. And all these experiments
mix and require both inertial and passive gravitational mass effects. For example, a balance or
bathroom scale has to dampen out any oscillation, which requires inertial mass. Generally speak-
ing, it seems impossible to distinguish inertial mass from the passive gravitational mass due to
all the masses in the rest of the universe. In short, the two concepts are in fact identical.
Challenge 356, page 203: These problems occur because gravitational mass determines poten-
tial energy and inertial mass determines kinetic energy.
Challenge 358, page 205: Either they fell on inclined snowy mountain sides, or they fell into
high trees, or other soft structures. The record was over 7 km of survived free fall. A recent
case made the news in 2007 and is told in www.bbc.co.uk/jersey/content/articles/2006/12/20/
michael_holmes_fall_feature.shtml.
Challenge 360, page 207: For a few thousand Euros, you can experience zero-gravity in a para-
bolic flight, such as the one shown in Figure 321. (Many ‘photographs’ of parabolic flights found
on the internet are in fact computer graphics. What about this one?)
504 challenge hints and solutions

How does zero-gravity feel? It feels similar to floating underwater, but without the resistance
of the water. It also feels like the time in the air when one is diving into the water. However, for
cosmonauts, there is an additional feeling; when they rotate their head rapidly, the sensors for
orientation in our ear are not reset by gravity. Therefore, for the first day or two, most cosmonauts
have feelings of vertigo and of nausea, the so-called space sickness. After that time, the body adapts
and the cosmonaut can enjoy the situation thoroughly.
Challenge 361, page 207: The centre of mass of a broom falls with the usual acceleration; the
end thus falls faster.
Challenge 362, page 207: Just use energy conservation for the two masses of the jumper and the
string. For more details, including the comparison of experimental measurements and theory, see
N. Dubelaar & R. Brantjes, De valversnelling bij bungee-jumping, Nederlands tijdschrift
voor natuurkunde 69, pp. 316–318, October 2003.
Challenge 363, page 207: About 1 ton.
Challenge 364, page 207: About 5 g.
Challenge 365, page 208: Your weight is roughly constant; thus the Earth must be round. On a
flat Earth, the weight would change from place to place, depending on your distance from the

Motion Mountain – The Adventure of Physics
border.
Challenge 366, page 208: Nobody ever claimed that the centre of mass is the same as the centre
of gravity! The attraction of the Moon is negligible on the surface of the Earth.
Challenge 368, page 209: That is the mass of the Earth. Just turn the table on its head.
Challenge 370, page 209: The Moon will be about 1.25 times as far as it is now. The Sun then
will slow down the Earth–Moon system rotation, this time due to the much smaller tidal friction
from the Sun’s deformation. As a result, the Moon will return to smaller and smaller distances
to Earth. However, the Sun will have become a red giant by then, after having swallowed both
the Earth and the Moon.
Challenge 372, page 210: As Galileo determined, for a swing (half a period) the ratio is √2 /π.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
(See challenge 325). But not more than two, maybe three decimals of π can be determined in this
way.
Challenge 373, page 210: Momentum conservation is not a hindrance, as any tennis racket has
the same effect on the tennis ball.
Challenge 374, page 210: In fact, in velocity space, elliptic, parabolic and hyperbolic motions
Ref. 163 are all described by circles. In all cases, the hodograph is a circle.
Challenge 375, page 211: This question is old (it was already asked in Newton’s times) and deep.
One reason is that stars are kept apart by rotation around the galaxy. The other is that galaxies are
kept apart by the momentum they got in the big bang. Without the big bang, all stars would have
collapsed together. In this sense, the big bang can be deduced from the attraction of gravitation
and the immobile sky at night. We shall find out later that the darkness of the night sky gives a
second argument for the big bang.
Challenge 376, page 211: The choice is clear once you notice that there is no section of the orbit
which is concave towards the Sun. Can you show this?
Challenge 378, page 212: The escape velocity, from Earth, to leave the Solar System – without
help of the other planets – is 42 km/s. However, if help by the other planets is allowed, it can be
less than half that value (why?).
If the escape velocity from a body were the speed of light, the body would be a black hole; not
Vol. II, page 262 even light could escape. Black holes are discussed in detail in the volume on relativity.
Challenge 379, page 212: Using a maximal jumping height of ℎ =0.5 m on Earth and an estim-
ated asteroid density of 𝜌 =3 Mg/m3 , we get a maximum radius of 𝑅2 = 3𝑔ℎ/4π𝐺𝜌, or 𝑅 ≈ 2.4 km.
challenge hints and solutions 505

F I G U R E 322 The analemma
photographed, at local noon, from
January to December 2002, at the
Parthenon on Athen’s Acropolis,
and a precision sundial (© Anthony
Ayiomamitis, Stefan Pietrzik).

Challenge 380, page 212: A handle of two bodies.
Challenge 382, page 212: In what does this argument differ from the more common argument

Motion Mountain – The Adventure of Physics
that in the expression 𝑚𝑎 = 𝑔𝑀𝑚/𝑅2 , the left 𝑚 is inertial and the right 𝑚 is gravitational?
Challenge 384, page 212: What counts is local verticality; with respect to it, the river always
flows downhill.
Challenge 385, page 213: The shape of an analemma at local noon is shown in Figure 322. The
shape is known since over 2000 years! The shape of the analemma also illustrates why the earliest
sunrise is not at the longest day of the year.
The vertical extension of the analemma in the figure is due to the obliquity, i.e., the tilt of
the Earth’s axis (it is twice 23.45°). The horizontal extension is due to the combination of the
obliquity and of the ellipticity of the orbit around the Sun. Both effects lead to roughly equal
changes of the position of the Sun at local noon during the course of the year. The asymmetrical

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
position of the central crossing point is purely due to the ellipticity of the orbit. The shape of the
analemma, sometimes shown on globes, is built into the shadow pole or the reading curve of
precision sundials. Examples are the one shown above and the one shown on page 45. For more
details, see B. M. Oliver, The shape of the analemma, Sky & Telescope 44, pp. 20–22, 1972, and
the correction of the figures at 44, p. 303, 1972,
Challenge 386, page 215: Capture of a fluid body is possible if it is split by tidal forces.
Challenge 387, page 216: The tunnel would be an elongated ellipse in the plane of the Equator,
reaching from one point of the Equator to the point at the antipodes. The time of revolution
would not change, compared to a non-rotating Earth. See A. J. Simonson, Falling down a hole
through the Earth, Mathematics Magazine 77, pp. 171–188, June 2004.
Challenge 389, page 216: The centre of mass of the Solar System can be as far as twice the radius
from the centre of the Sun; it thus can be outside the Sun.
Challenge 390, page 216: First, during northern summer time the Earth moves faster around
the Sun than during northern winter time. Second, shallow Sun’s orbits on the sky give longer
days because of light from when the Sun is below the horizon.
Challenge 391, page 216: Apart from the visibility of the Moon, no effect of the Moon on hu-
mans has ever been detected. Gravitational effects – including tidal effects – electrical effects,
magnetic effects and changes in cosmic rays are all swamped by other effects. Indeed the gravity
of passing trucks, factory electromagnetic fields, the weather and solar activity changes have lar-
ger influences on humans than the Moon. The locking of the menstrual cycle to the moon phase
is a visual effect.
506 challenge hints and solutions

Challenge 392, page 216: Distances were difficult to measure. It is easy to observe a planet that
is before the Sun, but it is hard to check whether a planet is behind the Sun. Phases of Venus are
also predicted by the geocentric system; but the phases it predicts do not match the ones that
are observed. Only the phases deduced from the heliocentric system match the observed ones.
Venus orbits the Sun.
Challenge 393, page 216: See the mentioned reference.
Challenge 394, page 217: True.
Challenge 395, page 217: For each pair of opposite shell elements (drawn in yellow), the two
attractions compensate.
Challenge 396, page 218: There is no practical way; if the masses on the shell could move, along
the surface (in the same way that charges can move in a metal) this might be possible, provided
that enough mass is available.
Challenge 400, page 219: Yes, one could, and this has been thought of many times, including by
Jules Verne. The necessary speed depends on the direction of the shot with respect of the rotation
of the Earth.
Challenge 401, page 219: Never. The Moon points always towards the Earth. The Earth changes

Motion Mountain – The Adventure of Physics
position a bit, due to the ellipticity of the Moon’s orbit. Obviously, the Earth shows phases.
Challenge 403, page 219: There are no such bodies, as the chapter of general relativity will show.
Challenge 405, page 221: The oscillation is a purely sinusoidal, or harmonic oscillation, as the
restoring force increases linearly with distance from the centre of the Earth. The period 𝑇 for a
homogeneous Earth is 𝑇 = 2π√𝑅3 /𝐺𝑀 = 84 min.
Challenge 406, page 223: The period is the same for all such tunnels and thus in particular it
is the same as the 84 min period that is valid also for the pole to pole tunnel. See for example,
R. H. Romer, The answer is forty-two – many mechanics problems, only one answer, Physics
Teacher 41, pp. 286–290, May 2003.
Challenge 407, page 223: If the Earth were not rotating, the most general path of a falling stone

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
would be an ellipse whose centre is the centre of the Earth. For a rotating Earth, the ellipse
precesses. Simoson speculates that the spirographics swirls in the Spirograph Nebula, found at
antwrp.gsfc.nasa.gov/apod/ap021214.html, might be due to such an effect. A special case is a path
starting vertically at the equator; in this case, the path is similar to the path of the Foucault
Ref. 178 pendulum, a pointed star with about 16 points at which the stone resurfaces around the Equator.
Challenge 408, page 223: There is no simple answer: the speed depends on the latitude and on
other parameters. The internet also provides videos of solar eclipses seen from space, showing
how the shadow moves over the surface of the Earth.
Challenge 409, page 224: The centrifugal force must be equal to the gravitational force. Call
𝑅+𝑙
the constant linear mass density 𝑑 and the unknown length 𝑙. Then we have 𝐺𝑀𝑑 ∫𝑅 d𝑟/𝑟2 =
𝑅+𝑙
𝜔2 𝑑 ∫𝑅 𝑟 d𝑟. This gives 𝐺𝑀𝑑𝑙/(𝑅2 + 𝑅𝑙) = (2𝑅𝑙 + 𝑙2 )𝜔2 𝑑/2, yielding 𝑙 = 0.14 Gm. For more on
space elevators or lifts, see challenge 574.
Challenge 411, page 224: The inner rings must rotate faster than the outer rings. If the rings
were solid, they would be torn apart. But this reasoning is true only if the rings are inside a
certain limit, the so-called Roche limit. The Roche limit is that radius at which gravitational force
𝐹g and tidal force 𝐹t cancel on the surface of the satellite. For a satellite with mass 𝑚 and radius
𝑟, orbiting a central mass 𝑀 at distance 𝑑, we look at the forces on a small mass 𝜇 on its surface.
We get the condition 𝐺𝑚𝜇/𝑟2 = 2𝐺𝑀𝜇𝑟/𝑑3 . A bit of algebra yields the approximate Roche limit
value
𝜌 1/3
𝑑Roche = 𝑅 (2 𝑀 ) . (153)
𝜌𝑚
challenge hints and solutions 507

Below that distance from a central mass 𝑀, fluid satellites cannot exist. The calculation shown
here is only an approximation; the actual Roche limit is about two times that value.
Challenge 414, page 228: The load is 5 times the load while standing. This explains why race
horses regularly break their legs.
Challenge 415, page 228: At school, you are expected to answer that the weight is the same.
This is a good approximation. But in fact the scale shows a slightly larger weight for the stead-
ily running hourglass compared to the situation where the all the sand is at rest. Looking at
the momentum flow explains the result in a simple way: the only issue that counts is the mo-
Ref. 87 mentum of the sand in the upper chamber, all other effects being unimportant. That momentum
slowly decreases during running. This requires a momentum flow from the scale: the effective
weight increases. See also the experimental confirmation and its explanation by F. Tuinstra
& B. F. Tuinstra, The weight of an hourglass, Europhysics News 41, pp. 25–28, March 2010,
also available online.
If we imagine a photon bouncing up and down in a box made of perfect mirrors, the ideas
from the hourglass puzzle imply that the scale shows an increased weight compared to the situ-
ation without a photon. The weight increase is 𝐸𝑔/𝑐2 , where 𝐸 is the energy of the photon,
𝑔 = 9.81 m/s2 and 𝑐 is the speed of light. This story is told by E. Huggins, Weighing photons

Motion Mountain – The Adventure of Physics
using bathroom scales: a thought experiment, The Physics Teacher 48, pp. 287–288, May 2010,
Challenge 416, page 229: The electricity consumption of a rising escalator indeed increases
when the person on it walks upwards. By how much?
Challenge 417, page 229: Knowledge is power. Time is money. Now, power is defined as work
per time. Inserting the previous equations and transforming them yields
work
money = , (154)
knowledge
which shows that the less you know, the more money you make. That is why scientists have low

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
salaries.
Challenge 418, page 229: In reality muscles keep an object above ground by continuously lifting
and dropping it; that requires energy and work.
Challenge 421, page 234: Yes, because side wind increases the effective speed 𝑣 in air due to vec-
tor addition, and because air resistance is (roughly) proportional to 𝑣2 .
Challenge 422, page 234: The lack of static friction would avoid that the fluid stays attached to
the body; the so-called boundary layer would not exist. One then would have no wing effect.
Challenge 424, page 235: True?
Challenge 426, page 236: From d𝑣/d𝑡 = 𝑔 − 𝑣2 (1/2𝑐𝑤 𝐴𝜌/𝑚) and using the abbreviation 𝑐 =
1/2𝑐𝑤 𝐴𝜌, we can solve for 𝑣(𝑡) by putting all terms containing the variable 𝑣 on one side, all
terms with 𝑡 on the other, and integrating on both sides. We get 𝑣(𝑡) = √𝑔𝑚/𝑐 tanh √𝑐𝑔/𝑚 𝑡.
Challenge 427, page 237: For extended deformable bodies, the intrinsic properties are given by
the mass density – thus a function of space and time – and the state is described by the density
of kinetic energy, local linear and angular momentum, as well as by its stress and strain distribu-
tions.
Challenge 428, page 237: Electric charge.
Challenge 429, page 238: The phase space has 3𝑁 position coordinates and 3𝑁 momentum co-
ordinates.
Challenge 430, page 238: We recall that when a stone is thrown, the initial conditions summar-
ize the effects of the thrower, his history, the way he got there etc.; in other words, initial con-
ditions summarize the past of a system, i.e., the effects that the environment had during the
508 challenge hints and solutions

The hidden geometry of the
Peaucellier-Lipkin linkage

F M C P

The Peaucellier-Lipkin linkage leads to a
straight motion of point P if point C
moves on a circle.

Motion Mountain – The Adventure of Physics
F I G U R E 323 How to draw a straight line with a compass (drawn by Zach Joseph Espiritu).

history of a system. Therefore, the universe has no initial conditions and no phase space. If you
have found reasons to answer yes, you overlooked something. Just go into more detail and check
whether the concepts you used apply to the universe. Also define carefully what you mean by
‘universe’.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Vol. III, page 122 Challenge 431, page 238: The light mill is an example.
Challenge 433, page 240: A system showing energy or matter motion faster than light would
imply that for such systems there are observers for which the order between cause and effect
are reversed. A space-time diagram (and a bit of exercise from the section on special relativity)
shows this.
Challenge 434, page 240: If reproducibility would not exist, we would have difficulties in check-
ing observations; also reading the clock is an observation. The connection between reproducib-
ility and time shall become important in the final part of our adventure.
Challenge 435, page 241: Even if surprises were only rare, each surprise would make it im-
possible to define time just before and just after it.
Challenge 438, page 242: Of course; moral laws are summaries of what others think or will do
about personal actions.
Challenge 439, page 243: The fastest glide path between two points, the brachistochrone, turns
out to be the cycloid, the curve generated by a point on a wheel that is rolling along a horizontal
plane.
The proof can be found in many ways. The simplest is by Johann Bernoulli and is given on
en.wikipedia.org/wiki/Brachistochrone_problem.
Challenge 441, page 244: When F, C and P are aligned, this circle has a radius given by 𝑅 =
√FCFP ; F is its centre. In other words, the Peaucellier-Lipkin linkage realizes an inversion at a
circle.
challenge hints and solutions 509

Motion Mountain – The Adventure of Physics
F I G U R E 324 The mechanism inside the south-pointing carriage.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 325 Falling brick chimneys – thus with limited stiffness – fall with a V shape (© John Glaser,
Frank Siebner).

Challenge 442, page 244: When F, C and P are aligned, the circle to be followed has a radius
given by half the distance FC; its centre lies midway between F and C. Figure 323 illustrates the
situation.
Challenge 443, page 244: Figure 324 shows the most credible reconstruction of a south-pointing
carriage.
Challenge 445, page 245: The water is drawn up along the sides of the spinning egg. The fastest
way to empty a bottle of water is to spin the water while emptying it.
Challenge 446, page 246: The right way is the one where the chimney falls like a V, not like an
inverted V. See challenge 361 on falling brooms for inspiration on how to deduce the answer.
510 challenge hints and solutions

Two examples are shown in Figure 325. It turns out that the chimney breaks (if it is not fastened
to the base) at a height between half or two thirds of the total, depending at the angle at which
this happens. For a complete solution of the problem, see the excellent paper G. Vareschi &
K. Kamiya, Toy models for the falling chimney, American Journal of Physics 71, pp. 1025–1031,
2003.
Challenge 448, page 252: The definition of the integral given in the text is a simplified version
of the so-called Riemann integral. It is sufficient for all uses in nature. Have a look at its exact
definition in a mathematics text if you want more details.
Challenge 454, page 255: In one dimension, the expression 𝐹 = 𝑚𝑎 can be written as −d𝑉/d𝑥 =
𝑚d2 𝑥/d𝑡2 . This can be rewritten as d(−𝑉)/d𝑥 − d/d𝑡[d/d𝑥(̇ 21 𝑚𝑥̇2 )] = 0. This can be expanded to
∂/∂𝑥( 12 𝑚𝑥̇2 − 𝑉(𝑥)) − d/[∂/∂𝑥(̇ 12 𝑚𝑥̇2 − 𝑉(𝑥))] = 0, which is Lagrange’s equation for this case.
Challenge 456, page 256: Do not despair. Up to now, nobody has been able to imagine a uni-
verse (that is not necessarily the same as a ‘world’) different from the one we know. So far, such
attempts have always led to logical inconsistencies.
Challenge 458, page 256: The two are equivalent since the equations of motion follow from the
principle of minimum action and at the same time the principle of minimum action follows from

Motion Mountain – The Adventure of Physics
the equations of motion.
Challenge 460, page 258: For gravity, all three systems exist: rotation in galaxies, pressure in
Vol. IV, page 135 planets and the Pauli pressure in stars that is due to Pauli’s exclusion principle. Against the strong
interaction, the exclusion principle acts in nuclei and neutron stars; in neutron stars maybe also
rotation and pressure complement the Pauli pressure. But for the electromagnetic interaction
there are no composites other than our everyday matter, which is organized by the Pauli’s exclu-
sion principle alone, acting among electrons.
Challenge 461, page 259: Aggregates often form by matter converging to a centre. If there is only
a small asymmetry in this convergence – due to some external influence – the result is a final
aggregate that rotates.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 462, page 262: Angular momentum is the change with respect to angle, whereas ro-
tational energy is again the change with respect to time, as all energy is.
Challenge 463, page 262: Not in this way. A small change can have a large effect, as every switch
shows. But a small change in the brain must be communicated outside, and that will happen
roughly with a 1/𝑟2 dependence. That makes the effects so small, that even with the most sensitive
switches – which for thoughts do not exist anyway – no effects can be realized.
Challenge 465, page 262: This is a wrong question. 𝑇 − 𝑈 is not minimal, only its average is.
Challenge 466, page 263: No. A system tends to a minimum potential only if it is dissipative.
One could, however, deduce that conservative systems oscillate around potential minima.
Challenge 467, page 263: The relation is

𝑐1 sin 𝛼1
= . (155)
𝑐2 sin 𝛼2
The particular speed ratio between air (or vacuum, which is almost the same) and a material
gives the index of refraction 𝑛:
𝑐 sin 𝛼1
𝑛= 1 = (156)
𝑐0 sin 𝛼0
Challenge 468, page 263: The principle for the growth of trees is simply the minimum of poten-
tial energy, since the kinetic energy is negligible. The growth of vessels inside animal bodies is
minimized for transport energy; that is again a minimum principle. The refraction of light is the
challenge hints and solutions 511

path of shortest time; thus it minimizes change as well, if we imagine light as moving entities
moving without any potential energy involved.
Challenge 469, page 264: Special relativity requires that an invariant measure of the action exist.
It is presented later in the walk.
Challenge 470, page 264: The universe is not a physical system. This issue will be discussed in
Vol. VI, page 106 detail later on.
Challenge 471, page 264: Use either the substitution 𝑢 = tan 𝑡/2 or use the historical trick

1 cos 𝜑 cos 𝜑
sec 𝜑 = 2
( + ) . (157)
1 + sin 𝜑 1 − sin 𝜑

Challenge 472, page 264: A skateboarder in a cycloid has the same oscillation time independ-
ently of the oscillation amplitude. But a half-pipe needs to have vertical ends, in order to avoid
jumping outside it. A cycloid never has a vertical end.
Challenge 475, page 267: We talk to a person because we know that somebody understands us.
Thus we assume that she somehow sees the same things we do. That means that observation is

Motion Mountain – The Adventure of Physics
partly viewpoint-independent. Thus nature is symmetric.
Challenge 476, page 270: Memory works because we recognize situations. This is possible be-
cause situations over time are similar. Memory would not have evolved without this reproducib-
ility.
Challenge 477, page 271: Taste differences are not fundamental, but due to different viewpoints
and – mainly – to different experiences of the observers. The same holds for feelings and judge-
ments, as every psychologist will confirm.
Challenge 478, page 273: The integers under addition form a group. Does a painter’s set of oil
colours with the operation of mixing form a group?

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 480, page 273: There is only one symmetry operation: a rotation about π around the
central point. That is the reason that later on the group D4 is only called the approximate sym-
metry group of Figure 202.
Challenge 486, page 278: Scalar is the magnitude of any vector; thus the speed, defined as 𝑣 =
|𝑣|, is a scalar, whereas the velocity 𝑣 is not. Thus the length of any vector (or pseudo-vector),
such as force, acceleration, magnetic field, or electric field, is a scalar, whereas the vector itself is
not a scalar.
Challenge 489, page 278: The charge distribution of an extended body can be seen as a sum of
a point charge, a charge dipole, a charge quadrupole, a charge octupole, etc. The quadrupole is
described by a tensor.
Compare: The inertia against motion of an extended body can be seen as sum of a point mass,
a mass dipole, a mass quadrupole, a mass octupole, etc. The mass quadrupole is described by the
moment of inertia.
Challenge 493, page 281: The conserved charge for rotation invariance is angular momentum.
Challenge 497, page 285: The graph is a logarithmic spiral (can you show this?); it is illustrated
in Figure 326. The travelled distance has a simple answer.
Challenge 498, page 285: An oscillation has a period in time, i.e., a discrete time translation
symmetry. A wave has both discrete time and discrete space translation symmetry.
Challenge 499, page 285: Motion reversal is a symmetry for any closed system; despite the ob-
servations of daily life, the statements of thermodynamics and the opinion of several famous
physicists (who form a minority though) all ideally closed systems are reversible.
512 challenge hints and solutions

First Second
Turtle Turtle

Motion Mountain – The Adventure of Physics
Fourth Third
Turtle Turtle

F I G U R E 326 The motion of four turtles chasing each other (drawn by Zach Joseph Espiritu).

Challenge 500, page 285: The symmetry group is a Lie group and called U(1), for ‘unitary group
in 1 dimension’.
Challenge 501, page 285: See challenge 301

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 502, page 285: There is no such thing as a ‘perfect’ symmetry.
Challenge 504, page 286: The rotating telephone dial had the digits 1 to 0 on the corners of a
regular 14-gon. The even and the odd numbers were on the angles of regular heptagons.
Challenge 508, page 289: Just insert 𝑥(𝑡) into the Lagrangian 𝐿 = 0, the minimum possible value
for a system that transforms all kinetic energy into potential energy and vice versa.
Challenge 517, page 300: The potential energy is due to the ‘bending’ of the medium; a simple
displacement produces no bending and thus contains no energy. Only the gradient captures the
bending idea.
Challenge 519, page 300: The phase changes by π.
Challenge 520, page 301: A wave that carries angular momentum has to be transversal and has
to propagate in three dimensions.
Challenge 521, page 301: Waves can be damped to extremely low intensities. If this is not pos-
sible, the observation is not a wave.
Challenge 522, page 301: The way to observe diffraction and interference with your naked fin-
gers is told on page 101 in volume III.
Challenge 533, page 315: Interference can make radio signals unintelligible. Due to diffraction,
radio signals are weakened behind a wall; this is valid especially for short wavelengths, such
as those used in mobile phones. Refraction makes radio communication with submarines im-
possible for usual radio frequencies. Dispersion in glass fibres makes it necessary to add repeat-
ers in sea cables roughly every 100 km. Damping makes it impossible to hear somebody speaking
challenge hints and solutions 513

at larger distances. Radio signals can lose their polarisation and thus become hard to detect by
usual Yagi antennas that have a fixed polarisation.
Challenge 535, page 320: Skiers scrape snow from the lower side of each bump towards the up-
per side of the next bump. This leads to an upward motion of ski bumps.
Challenge 536, page 320: If the distances to the loudspeaker is a few metres, and the distance to
the orchestra is 20 m, as for people with enough money, the listener at home hears it first.
Challenge 537, page 320: As long as the amplitude is small compared to the length 𝑙, the period
𝑇 is given by
𝑙
𝑇 = 2π√ . (158)
𝑔
The formula does not contain the mass 𝑚 at all. Independently of the mass 𝑚 at its end, the
pendulum has always the same period. In particular, for a length of 1 m, the period is about 2 s.
Half a period, or one swing thus takes about 1 s. (This is the original reason for choosing the unit
of metre.)
For an extremely long pendulum, the answer is a finite value though, and corresponds to the
situation of challenge 26.

Motion Mountain – The Adventure of Physics
Challenge 538, page 320: In general, the body moves along an ellipse (as for planets around the
Sun) but with the fixed point as centre. In contrast to planets, where the Sun is in a focus of the
ellipse and there is a perihelion and an apohelion, such a body moves symmetrically around the
centre of the ellipse. In special cases, the body moves back and forward along a straight segment.
Challenge 540, page 320: This follows from the formula that the frequency of a string is given by
𝑓 = √𝑇/𝜇 /(2𝑙), where 𝑇 is the tension, 𝜇 is the linear mass density, and 𝑙 is the length of a string.
This is discussed in the beautiful paper by G. Barnes, Physics and size in biological systems, The
Physics Teacher 27, pp. 234–253, 1989.
Challenge 542, page 321: The sound of thunder or of car traffic gets lower and lower in fre-
quency with increasing distance.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 545, page 321: Neither; both possibilities are against the properties of water: in sur-
face waves, the water molecules move in circles.
Challenge 546, page 322: Swimmers are able to cover 100 m in 48 s, or slightly better than 2 m/s.
(Swimmer with fins achieve just over 3 m/s.) With a body length of about 1.9 m, the critical speed
is 1.7 m/s. That is why short-distance swimming depends mainly on training; for longer distances
the technique plays a larger role, as the critical speed has not been attained yet. The formula also
predicts that on the 1500 m distance, a 2 m tall swimmer has a potential advantage of over 45 s
on one with a body height of 1.8 m. In addition, longer swimmers have an additional advantage:
they swim shorter distances in pools (why?). It is thus predicted that successful long-distance
swimmers will get taller and taller over time. This is a pity for a sport that so far could claim to
have had champions of all sizes and body shapes, in contrast to many other sports.
Challenge 549, page 324: To reduce noise reflection and thus hall effects. They effectively diffuse
the arriving wavefronts.
Challenge 551, page 324: Waves in a river are never elliptical; they remain circular.
Challenge 552, page 324: The lens is a cushion of material that is ‘transparent’ to sound. The
speed of sound is faster in the cushion than in the air, in contrast to a glass lens, where the speed of
light is slower in the glass. The shape is thus different: the cushion must look like a large biconcave
optical lens.
Challenge 553, page 324: Experiments show that the sound does not depend on air flows (find
out how), but does depend on external sound being present. The sound is due to the selective
amplification by the resonances resulting from the geometry of the shell shape.
514 challenge hints and solutions

Challenge 554, page 324: The Sun is always at a different position than the one we observe it
to be. What is the difference, measured in angular diameters of the Sun? Despite this position
difference, the timing of the sunrise is determoned by the position of the horizon, not by the
position of the Sun. (Imagine the it would not: in that case a room would not get dark when the
window is closed, but eight minutes later ...) In short, there is no measurable effect of the speed
of light on the sunrise.
Challenge 557, page 326: An overview of systems being tested at present can be found in K. -
U. Graw, Energiereservoir Ozean, Physik in unserer Zeit 33, pp. 82–88, Februar 2002. See also
Oceans of electricity – new technologies convert the motion of waves into watts, Science News 159,
pp. 234–236, April 2001.
Challenge 558, page 326: In everyday life, the assumption is usually justified, since each spot
can be approximately represented by an atom, and atoms can be followed. The assumption is
questionable in situations such as turbulence, where not all spots can be assigned to atoms, and
most of all, in the case of motion of the vacuum itself. In other words, for gravity waves, and in
particular for the quantum theory of gravity waves, the assumption is not justified.
Challenge 564, page 333: There are many. One would be that the transmission and thus reflec-
tion coefficient for waves would almost be independent of wavelength.

Motion Mountain – The Adventure of Physics
Challenge 565, page 334: A drop with a diameter of 3 mm would cover a surface of 7.1 m2 with
a 2 nm film.
Challenge 566, page 338: The wind will break tall trees that are too thin. For small and thus thin
trees, the wind does not damage.
𝛽
Challenge 567, page 338: The critical height for a column of material is given by ℎ4crit = 𝑚𝐸,
4π𝑔 𝜌2
where 𝛽 ≈ 1.9 is the constant determined by the calculation when a column buckles under its
own weight.
Challenge 569, page 339: One possibility is to describe particles as clouds; another is given in
the last part of the text.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 258 Challenge 570, page 341: The results gives a range between 1 and 8 ⋅ 1023 .
Challenge 572, page 345: Check your answers with the delightful text by P. Goldrich,
S. Mahajan & S. Phinney, Order-of-Magnitude Physics: Understanding the World with
Dimensional Analysis, Educated Guesswork, and White Lies, available on the internet.
Challenge 573, page 345: Glass shatters, glass is elastic, glass shows transverse sound waves,
glass does not flow (in contrast to what many books state), not even on scale of centuries, glass
molecules are fixed in space, glass is crystalline at small distances, a glass pane supported at the
ends does not hang through.
Challenge 574, page 345: No metal wire allows building such a long wire or rope. Only the idea
of carbon nanotubes has raised hope again; some dream of wire material based on them, stronger
than any material known so far. However, no such material is known yet. The system faces many
dangers, such as fabrication defects, lightning, storms, meteoroids and space debris. All would
lead to the breaking of the wires – if such wires will ever exist. But the biggest of all dangers is
the lack of cash to build it. Nevertheless, numerous people are working towards the goal.
Challenge 575, page 346: The 3 × 3 × 3 cube has a rigid system of three perpendicular axes, on
which a square can rotate at each of the 6 ends. The other squares are attached to pieces moving
around these axes. The 4 × 4 × 4 cube is different though; just find out. From 7 × 7 × 7 onwards,
the parts do not all have the same size or shape. The present limit on the segment number in
commercially available ‘cubes’ is 17 × 17 × 17! It can be found at www.shapeways.com/shops/
oskarpuzzles. The website www.oinkleburger.com/Cube/applet allows playing with virtual cubes
up to 100 × 100 × 100, and more.
challenge hints and solutions 515

Challenge 578, page 347: A medium-large earthquake would be generated.
Challenge 579, page 347: A stalactite contains a thin channel along its axis through which the
water flows, whereas a stalagmite is massive throughout.
Challenge 580, page 347: About 1 part in a thousand.
Challenge 582, page 348: Even though the iron core of the Earth formed by collecting the iron
from colliding asteroids which then sunk into the centre of the Earth, the scheme will not work
today: in its youth, the Earth was much more liquid than today. The iron will most probably not
sink. In addition, there is no known way to build a measurement probe that can send strong
enough sound waves for this scheme. The temperature resistance is also an issue, but this may be
solvable.
Challenge 584, page 351: Atoms are not infinitely hard, as quantum theory shows. Atoms are
Vol. IV, page 80 more similar to deformable clouds.
Challenge 588, page 360: If there is no friction, all three methods work equally fast – including
the rightmost one.
Challenge 591, page 362: The constant 𝑘 follows from the conservation of energy and that of
mass:

Motion Mountain – The Adventure of Physics
2
𝑘=√ . (159)
𝜌(𝐴21 /𝐴22 − 1)
The cross sections are denoted by 𝐴 and the subscript 1 refers to any point far from the constric-
tion, and the subscript 2 to the constriction.
Challenge 594, page 368: The pressure destroys the lung. Snorkeling is only possible at the water
surface, not below the water! This experiment is even dangerous when tried in your own bathtub!
Breathing with a long tube is only possible if a pump at the surface pumps air down the tube at
the correct pressure.
Challenge 596, page 370: Some people notice that in some cases friction is too high, and start

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
sucking at one end of the tube to get the flow started; while doing so, they can inhale or swallow
gasoline, which is poisonous.
Challenge 601, page 372: Calculation yields 𝑁 = 𝐽/𝑗 = (0.0001 m3 /s)/(7 μm2 0.0005 m/s), or
about 6⋅109 ; in reality, the number is much larger, as most capillaries are closed at a given instant.
The reddening of the face shows what happens when all small blood vessels are opened at the
same time.
Challenge 602, page 373: Throwing the stone makes the level fall, throwing the water or the
piece of wood leaves it unchanged.
Challenge 603, page 373: The ship rises higher into the sky. (Why?)
Challenge 605, page 373: The motion of a helium-filled balloon is opposite to that of an air-filled
balloon or of people: the helium balloon moves towards the front when the car accelerates and
to the back when the car decelerates. It also behaves differently in bends. Several films on the
internet show the details.
Challenge 608, page 373: The pumps worked in suction; but air pressure only allows 10 m of
height difference for such systems.
Challenge 609, page 373: This argument is comprehensible only when we remember that ‘twice
the amount’ means ‘twice as many molecules’.
Challenge 610, page 373: The alcohol is frozen and the chocolate is put around it.
Challenge 611, page 374: The author suggested in an old edition of this text that a machine
should be based on the same machines that throw the clay pigeons used in the sports of trap
516 challenge hints and solutions

TA B L E 62 Gaseous composition of dry air, at present time𝑎 (sources: NASA, IPCC).

Gas Symbol Vo l u m e
pa rt𝑏
Nitrogen N2 78.084 %
Oxygen (pollution dependent) O2 20.946 %
Argon Ar 0.934 %
Carbon dioxide (in large part due to human pollution) CO2 403 ppm
Neon Ne 18.18 ppm
Helium He 5.24 ppm
Methane (mostly due to human pollution) CH4 1.79 ppm
Krypton Kr 1.14 ppm
Hydrogen H2 0.55 ppm
Nitrous oxide (mostly due to human pollution) N2 O 0.3 ppm
Carbon monoxide (partly due to human pollution) CO 0.1 ppm

Motion Mountain – The Adventure of Physics
Xenon Xe 0.087 ppm
Ozone (strongly influenced by human pollution) O3 0 to 0.07 ppm
Nitrogen dioxide (mostly due to human pollution) NO2 0.02 ppm
Iodine I2 0.01 ppm
Ammonia (mostly due to human pollution) NH3 traces
Radon Ra traces
Halocarbons and other fluorine compounds (all being 20 types 0.0012 ppm
humans pollutants)
Mercury, other metals, sulfur compounds, other organic numerous concentration

𝑎. Wet air can contain up to 4 % water vapour, depending on the weather. Apart from gases, air can con-
tain water droplets, ice, sand, dust, pollen, spores, volcanic ash, forest fire ash, fuel ash, smoke particles,
pollutants of all kinds, meteoroids and cosmic ray particles. During the history of the Earth, the gaseous
composition varied strongly. In particular, oxygen is part of the atmosphere only in the second half of the
Earth’s lifetime.
𝑏. The abbreviation ppm means ‘parts per million’.

shooting and skeet. In the meantime, Lydéric Bocquet and Christophe Clanet have built such
a stone-skipping machine, but using a different design; a picture can be found on the website
ilm-perso.univ-lyon1.fr/~lbocquet.
Challenge 612, page 374: The third component of air is the noble gas argon, making up about
1 %. A longer list of components is given in Table 62.
Challenge 613, page 374: The pleural cavity between the lungs and the thorax is permanently be-
low atmospheric pressure, usually 5 mbar, but even 10 mbar at inspiration. A hole in it, formed
for example by a bullet, a sword or an accident, leads to the collapse of the lung – the so-
called pneumothorax – and often to death. Open chest operations on people have become pos-
sible only after the surgeon Ferdinand Sauerbruch learned in 1904 how to cope with the prob-
lem. Nowadays however, surgeons keep the lung under higher than atmospheric pressure until
everything is sealed again.
Challenge 614, page 374: The fountain shown in the figure is started by pouring water into the
challenge hints and solutions 517

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 327 A way to ride head-on against the wind using wind power (© Tobias Klaus).

uppermost container. The fountain then uses the air pressure created by the water flowing down-
wards.
Challenge 615, page 375: Yes. The bulb will not resist two such cars though.
Challenge 616, page 375: Radon is about 8 times as heavy as air; it is the densest gas known. In
comparison, Ni(CO) is 6 times, SiCl4 4 times heavier than air. Mercury vapour (obviously also a
gas) is 7 times heavier than air. In comparison, bromine vapour is 5.5 times heavier than air.
Challenge 618, page 375: Yes, as the ventomobil shown in Figure 327 proves. It achieves the feat
already for low wind speeds.
Challenge 619, page 375: None.
Challenge 621, page 375: He brought the ropes into the cabin by passing them through liquid
mercury.
518 challenge hints and solutions

Challenge 623, page 376: There are no official solutions for these questions; just check your as-
sumptions and calculations carefully. The internet is full of such calculations.
Challenge 624, page 377: The soap flows down the bulb, making it thicker at the bottom and
thinner at the top, until it reaches the thickness of two molecular layers. Later, it bursts.
Challenge 625, page 377: The temperature leads to evaporation of the involved liquid, and the
vapour prevents direct contact between the two non-gaseous bodies.
Challenge 626, page 378: For this to happen, friction would have to exist on the microscopic
scale and energy would have to disappear.
Challenge 627, page 378: The longer funnel is empty before the short one. (If you do not believe
it, try it out.) In the case that the amount of water in the funnel outlet can be neglected, one can
use energy conservation for the fluid motion. This yields the famous Bernoulli equation 𝑝/𝜌 +
𝑔ℎ + 𝑣2 /2 = const, where 𝑝 is pressure, 𝜌 the density of water, and 𝑔 is 9.81 m/s2 . Therefore, the
speed 𝑣 is higher for greater lengths ℎ of the thin, straight part of the funnel: the longer funnel
empties first.
But this is strange: the formula gives a simple free fall relation, as the air pressure is the same
above and below and disappears from the calculation. The expression for the speed is thus inde-

Motion Mountain – The Adventure of Physics
pendent of whether a tube is present or not. The real reason for the faster emptying of the tube is
thus that a tube forces more water to flow out than the lack of a tube. Without tube, the diameter
of the water flow diminishes during fall. With tube, it stays constant. This difference leads to the
faster emptying for longer tubes.
Alternatively, you can look at the water pressure value inside the funnel. You will discover that
the water pressure is lowest at the start of the exit tube. This internal water pressure is lower for
longer tubes and sucks out the water faster in those cases.
Challenge 628, page 378: The eyes of fish are positioned in such a way that the pressure reduc-
tion by the flow is compensated by the pressure increase of the stall. By the way, their heart is
positioned in such a way that it is helped by the underpressure.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 630, page 378: This feat has been achieved for lower mountains, such as the Monte
Bianco in the Alps. At present however, there is no way to safely hover at the high altitudes of the
Himalayas.
Challenge 632, page 378: Press the handkerchief into the glass, and lower the glass into the water
with the opening first, while keeping the opening horizontal. This method is also used to lower
people below the sea. The paper ball in the bottle will fly towards you. Blowing into a funnel
will keep the ping-pong ball tightly into place, and the more so the stronger you blow. Blowing
through a funnel towards a candle will make it lean towards you.
Challenge 639, page 388: In 5000 million years, the present method will stop, and the Sun will
become a red giant. But it will burn for many more years after that.
Challenge 640, page 390: Bernoulli argued that the temperature describes the average kinetic
energy of the constituents of the gas. From the kinetic energy he deduced the average momentum
of the constituents. An average momentum leads to a pressure. Adding the details leads to the
ideal gas relation.
Challenge 641, page 390: The answer depends on the size of the balloons, as the pressure is not
a monotonous function of the size. If the smaller balloon is not too small, the smaller balloon
wins.
Challenge 644, page 391: Measure the area of contact between tires and street (all four) and then
multiply by 200 kPa, the usual tire pressure. You get the weight of the car.
Challenge 648, page 394: If the average square displacement is proportional to time, the liquid
is made of smallest particles. This was confirmed by the experiments of Jean Perrin. The next
challenge hints and solutions 519

step is to deduce the number of these particles from the proportionality constant. This constant,
defined by ⟨𝑑2 ⟩ = 4𝐷𝑡, is called the diffusion constant (the factor 4 is valid for random motion in
two dimensions). The diffusion constant can be determined by watching the motion of a particle
under the microscope.
We study a Brownian particle of radius 𝑎. In two dimensions, its square displacement is given
by
4𝑘𝑇
⟨𝑑2 =⟩ 𝑡, (160)
𝜇
where 𝑘 is the Boltzmann constant and 𝑇 the temperature. The relation is deduced by studying the
motion of a particle with drag force −𝜇𝑣 that is subject to random hits. The linear drag coefficient
𝜇 of a sphere of radius 𝑎 is given by
𝜇 = 6π𝜂𝑎 , (161)
where 𝜂 is the kinematic viscosity. In other words, one has

6π𝜂𝑎 ⟨𝑑2 ⟩
𝑘= . (162)
4𝑇 𝑡

Motion Mountain – The Adventure of Physics
All quantities on the right can be measured, thus allowing us to determine the Boltzmann con-
stant 𝑘. Since the ideal gas relation shows that the ideal gas constant 𝑅 is related to the Boltzmann
constant by 𝑅 = 𝑁A 𝑘, the Avogadro constant 𝑁A that gives the number of molecules in a mole
is also found in this way.
Challenge 653, page 402: The possibility of motion inversion for all observed phenomena is in-
deed a fundamental property of nature. It has been confirmed for all interactions and all exper-
iments every performed. Independent of this is the fact that realizing the inversion might be
extremely hard, because inverting the motion of many atoms is usually not feasible.
Challenge 654, page 403: This is a trick question. To a good approximation, any tight box is an
example. However, if we ask for complete precision, all systems radiate some energy, lose some

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
atoms or particles and bend space; ideal closed systems do not exist.
Challenge 659, page 405: We will find out later that the universe is not a physical system; thus
Vol. VI, page 106 the concept of entropy does not apply to it. Thus the universe is neither isolated nor closed.
Challenge 661, page 406: Egg white starts to harden at lower temperature than yolk, but for
complete hardening, the opposite is true. White hardens completely at 80°C, egg yolk hardens
considerably at 66 to 68°C. Cook an egg at the latter temperature, and the feat is possible; the
white remains runny, but does not remain transparent, though. Note again that the cooking time
plays no role, only the precise temperature value.
Challenge 663, page 407: Yes, the effect is easily noticeable.
Challenge 666, page 407: Hot air is less dense and thus wants to rise.
Challenge 667, page 407: Keep the paper wet.
Challenge 668, page 407: Melting ice at 0°C to water at 0°C takes 334 kJ/kg. Cooling water by
1°C or 1 K yields 4.186 kJ/kgK. So the hot water needs to cool down to 20.2°C to melt the ice, so
that the final mixing temperature will be 10.1°C.
Challenge 669, page 408: The air had to be dry.
Challenge 670, page 408: In general, it is impossible to draw a line through three points. Since
absolute zero and the triple point of water are fixed in magnitude, it was practically a sure bet
that the boiling point would not be at precisely 100°C.
Challenge 671, page 408: No, as a water molecule is heavier than that. However, if the water is
allowed to be dirty, it is possible. What happens if the quantum of action is taken into account?
520 challenge hints and solutions

F I G U R E 328 A candle on Earth and in microgravity (NASA).

Challenge 672, page 408: The danger is not due to the amount of energy, but due to the time in
which it is available.
Challenge 673, page 409: The internet is full of solutions.
Challenge 674, page 409: There are 2𝑛 possible sequences of 𝑛 coin throws. Of those, 𝑛!/( 𝑛2 !)2

Motion Mountain – The Adventure of Physics
contain 𝑛/2 heads and 𝑛/2 tails. For a fair coin, the probability 𝑝 of getting 𝑛/2 heads in 𝑛 throws
is thus
𝑛!
𝑝= 2
. (163)
2 ( 𝑛2 !)
𝑛

We approximate this result with the help of Gosper’s formula 𝑛! ≈ √(2𝑛 + 13 )π ( 𝑛𝑒 )𝑛 and get

√(2𝑛 + 13 )π ( 𝑛𝑒 )𝑛 √2𝑛 + 1
3
𝑝≈ 2
= . (164)
𝑛 (𝑛 + 13 )√π
2𝑛 (√(𝑛 + 13 )π ( 2𝑒𝑛 ) 2 )

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
1
For 𝑛 = 1 000 000, we get a probability 𝑝 ≈ 0.0007979, thus a rather small value between 1254 and
1
1253
.
Challenge 675, page 409: The entropy can be defined for the universe as a whole only if the uni-
verse is a closed system. But is the universe closed? Is it a system? This issue is discussed in the
final part of our adventure.
Challenge 678, page 410: For such small animals the body temperature would fall too low. They
could not eat fast enough to get the energy needed to keep themselves warm.
Challenge 681, page 410: The answer depends on the volume, of course. But several families
have died overnight because they had modified their mobile homes to be airtight.
Challenge 682, page 410: The metal salts in the ash act as catalysts, and the sugar burns instead
of just melting. Watch the video of the experiment at www.youtube.com/watch?v=BfBgAaeaVgk.
Challenge 687, page 411: It is about 10−9 that of the Earth.
Challenge 689, page 411: The thickness of the folds in the brain, the bubbles in the lung, the
density of blood vessels and the size of biological cells.
Challenge 690, page 411: The mercury vapour above the liquid gets saturated.
Challenge 691, page 411: A dedicated NASA project studied this question. Figure 328 gives an
example comparison. You can find more details on their website.
Challenge 692, page 411: The risks due to storms and the financial risks are too high.
challenge hints and solutions 521

Challenge 693, page 412: The vortex inside the tube is cold near its axis and hot in the regions
away from the axis. Through the membrane in the middle of the tube (shown in Figure 285 on
page 412) the air from the axis region is sent to one end and the air from the outside region to
the other end. The heating of the outside region is due to the work that the air rotating inside
has to do on the air outside to get a rotation that consumes angular momentum. For a detailed
explanation, see the beautiful text by Mark P. Silverman, And Yet it Moves: Strange Systems
and Subtle Questions in Physics, Cambridge University Press, 1993, p. 221.
Challenge 694, page 412: No.
Challenge 695, page 412: At the highest possible mass concentration, entropy is naturally the
highest possible.
Challenge 696, page 412: The units do not match.
Challenge 697, page 412: In the case of water, a few turns mixes the ink, and turning backwards
increases the mixing. In the case of glycerine, a few turns seems to mix the ink, and turning
backwards undoes the mixing.
Challenge 698, page 413: Put them in clothes.
Challenge 702, page 413: Negative temperatures are a conceptual crutch definable only for sys-

Motion Mountain – The Adventure of Physics
tems with a few discrete states; they are not real temperatures, because they do not describe
equilibrium states, and indeed never apply to systems with a continuum of states.
Challenge 703, page 415: This is also true for the shape of human bodies, the brain control of
human motion, the growth of flowers, the waves of the sea, the formation of clouds, the processes
leading to volcano eruptions, etc.
Page 319 Challenge 706, page 422: See the puzzle about the motion of ski moguls.
Challenge 711, page 425: First, there are many more butterflies than tornadoes. Second, tor-
nadoes do not rely on small initial disturbances for their appearance. Third, the belief in the
butterfly ‘effect’ completely neglects an aspect of nature that is essential for self-organization:
friction and dissipation. The butterfly ‘effect’, assuming that it existed, would require that dis-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
sipation in the air should have completely unrealistic properties. This is not the case in the at-
mosphere. But most important of all, there is no experimental basis for the ‘effect’: it has never
been observed. Thus it does not exist.
Challenge 721, page 437: No. Nature does not allow more than about 20 digits of precision, as
we will discover later in our walk. That is not sufficient for a standard book. The question whether
such a number can be part of its own book thus disappears.
Challenge 722, page 437: All three statements are hogwash. A drag coefficient implies that the
cross area of the car is known to the same precision. This is actually extremely difficult to measure
and to keep constant. In fact, the value 0.375 for the Ford Escort was a cheat, as many other meas-
urements showed. The fuel consumption is even more ridiculous, as it implies that fuel volumes
and distances can be measured to that same precision. Opinion polls are taken by phoning at
most 2000 people; due to the difficulties in selecting the right representative sample, that gives a
precision of at most 3 % for typical countries.
Challenge 724, page 438: Space-time is defined using matter; matter is defined using space-
time.
Challenge 725, page 438: Fact is that physics has been based on a circular definition for hun-
dreds of years. Thus it is possible to build even an exact science on sand. Nevertheless, the elim-
ination of the circularity is an important aim.
Challenge 726, page 439: Every measurement is a comparison with a standard; every compar-
ison requires light or some other electromagnetic field. This is also the case for time measure-
ments.
522 challenge hints and solutions

Challenge 727, page 439: Every mass measurement is a comparison with a standard; every com-
parison requires light or some other electromagnetic field.
Challenge 728, page 439: Angle measurements have the same properties as length or time meas-
urements.
Challenge 730, page 455: Mass is a measure of the amount of energy. The ‘square of mass’ makes
no sense.
Challenge 733, page 457: About 10 μg.
Challenge 734, page 458: Probably the quantity with the biggest variation is mass, where a prefix
for 1 eV/c2 would be useful, as would be one for the total mass in the universe, which is about
1090 times larger.
Challenge 735, page 459: The formula with 𝑛 − 1 is a better fit. Why?
Challenge 738, page 460: No! They are much too precise to make sense. They are only given as
an illustration of the behaviour of the Gaussian distribution. Real measurement distributions are
not Gaussian to the precision implied in these numbers.
Challenge 739, page 460: About 0.3 m/s. It is not 0.33 m/s, it is not 0.333 m/s and it is not any

Motion Mountain – The Adventure of Physics
longer strings of threes!
Challenge 741, page 466: The slowdown goes quadratically with time, because every new slow-
down adds to the old one!
Challenge 742, page 466: No, only properties of parts of the universe are listed. The universe
Vol. VI, page 112 itself has no properties, as shown in the last volume.
Challenge 743, page 527: For example, speed inside materials is slowed, but between atoms,
light still travels with vacuum speed.

“ ”
Aiunt enim multum legendum esse, non multa.
Plinius, Epistulae.*

1 For a history of science in antiquity, see Lucio Russo, La rivoluzione dimenticata, Fel-
trinelli, 1996, also available in several other languages. Cited on page 15.
2 If you want to catch up secondary school physics, the clearest and shortest introduction

Motion Mountain – The Adventure of Physics
worldwide is a free school text, available in English and several other languages, written by
a researcher who has dedicated all his life to the teaching of physics in secondary school, to-
gether with his university team: Friedrich Herrmann, The Karlsruhe Physics Course,
free to download in English, Spanish, Russian, Italian and Chinese at www.physikdidaktik.
uni-karlsruhe.de/index_en.html. It is one of the few secondary school texts that captivates
and surprises even professional physicists. (The 2013 paper on this book by C. Strunk &
K. Rincke, Zum Gutachten der Deutschen Physikalischen Gesellschaft über den Karlsruher
Physikkurs, available on the internet, makes many interesting points and is enlightening
for every physicist.) This can be said even more of the wonderfully daring companion text
Friedrich Herrmann & Georg Job, Historical Burdens on Physics, whose content is

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
also freely available on the Karlsruhe site, in English and in several other languages.
A beautiful book explaining physics and its many applications in nature and technology
vividly and thoroughly is Paul G. Hewitt, John Suchocki & Leslie A. Hewitt,
Conceptual Physical Science, Bejamin/Cummings, 1999.
A great introduction is Klaus Dransfeld, Paul Kienle & Georg Kalvius,
Physik 1: Mechanik und Wärme, Oldenburg, 2005.
A book series famous for its passion for curiosity is Richard P. Feynman,
Robert B. Leighton & Matthew Sands, The Feynman Lectures on Physics, Ad-
dison Wesley, 1977. The volumes can now be read online for free at www.feynmanlectures.
info.
A lot can be learned about motion from quiz books. One of the best is the well-
structured collection of beautiful problems that require no mathematics, written by
Jean-Marc Lév y-Leblond, La physique en questions – mécanique, Vuibert, 1998.
Another excellent quiz collection is Yakov Perelman, Oh, la physique, Dunod,
2000, a translation from the Russian original. (One of Perelman’s books, in English, is
found at archive.org/details/physicsforentert035428mbp/page/n5/mode/2up.)
A good problem book is W. G. Rees, Physics by Example: 200 Problems and Solutions,
Cambridge University Press, 1994.

* ‘Read much, but not anything.’ Ep. 7, 9, 15. Gaius Plinius Secundus (b. 23/4 Novum Comum,
d. 79 Vesuvius eruption), Roman writer, especially famous for his large, mainly scientific work Historia nat-
uralis, which has been translated and read for almost 2000 years.
524 bibliography

A good history of physical ideas is given in the excellent text by David Park, The How
and the Why, Princeton University Press, 1988.
An excellent introduction into physics is Robert Pohl, Pohl’s Einführung in die
Physik, Klaus Lüders & Robert O. Pohl editors, Springer, 2004, in two volumes with CDs.
It is a new edition of a book that is over 70 years old; but the didactic quality, in particular
of the experimental side of physics, is unsurpassed.
Another excellent Russian physics problem book, the so-called Saraeva, seems to exist
only as Spanish translation: B.B. Bújovtsev, V.D. Krívchenkov, G.Ya. Miák-
ishev & I.M. Saráeva Problemas seleccionados de física elemental, Mir, 1979.
Another good physics problem book is Giovanni Tonzig, Cento errori di fisica
pronti per l’uso, Sansoni, third edition, 2006. See also his www.giovannitonzig.it website.
Cited on pages 15, 120, 219, 324, and 532.
3 An overview of motion illusions can be found on the excellent website www.michaelbach.
de/ot. The complex motion illusion figure is found on www.michaelbach.de/ot/
mot_rotsnake/index.html; it is a slight variation of the original by Kitaoka Akiyoshi at
www.ritsumei.ac.jp/~akitaoka/rotsnake.gif, published as A. Kitaoka & H. Ashida,
Phenomenal characteristics of the peripheral drift illusion, Vision 15, pp. 261–262, 2003. A

Motion Mountain – The Adventure of Physics
common scam is to claim that the illusion is due to or depends on stress. Cited on page 16.
4 These and other fantastic illusions are also found in Akiyoshi Kitaoka, Trick Eyes,
Barnes & Noble, 2005. Cited on page 16.
5 A well-known principle in the social sciences states that, given a question, for every possible
answer, however weird it may seem, there is somebody – and often a whole group – who
holds it as his opinion. One just has to go through literature (or the internet) to confirm
this.
About group behaviour in general, see R. Axelrod, The Evolution of Cooperation,
Harper Collins, 1984. The propagation and acceptance of ideas, such as those of physics,
are also an example of human cooperation, with all its potential dangers and weaknesses.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Cited on page 16.
6 All the known texts by Parmenides and Heraclitus can be found in Jean-Paul Dumont,
Les écoles présocratiques, Folio-Gallimard, 1988. Views about the non-existence of motion
have also been put forward by much more modern and much more contemptible authors,
Vol. III, page 325 such as in 1710 by Berkeley. Cited on page 17.
7 An example of people worried by Zeno is given by William McLaughlin, Resolving
Zeno’s paradoxes, Scientific American pp. 66–71, November 1994. The actual argument was
not about a hand slapping a face, but about an arrow hitting the target. See also Ref. 66.
Cited on page 17.
8 The full text of La Beauté and the other poems from Les fleurs du mal, one of the finest
books of poetry ever written, can be found at the hypermedia.univ-paris8.fr/bibliotheque/
Baudelaire/Spleen.html website. Cited on page 18.
9 A famous collection of interesting examples of motion in everyday life is the excel-
lent book by Jearl Walker, The Flying Circus of Physics, Wiley, 1975. Its website
is at www.flyingcircusofphysics.com. Another beautiful book is Christian Ucke &
H. Joachim Schlichting, Spiel, Physik und Spaß – Physik zum Mitdenken und Mit-
machen, Wiley-VCH, 2011. For more interesting physical effects in everyday life, see Er-
wein Flachsel, Hundertfünfzig Physikrätsel, Ernst Klett Verlag, 1985. The book also cov-
ers several clock puzzles, in puzzle numbers 126 to 128. Cited on page 19.
10 A concise and informative introduction into the history of classical physics is given in the
first chapter of the book by Floyd Karker Richtmyer, Earle Hesse Kennard
bibliography 525

& John N. Cooper, Introduction to Modern Physics, McGraw–Hill, 1969. Cited on page
19.
11 An introduction into perception research is E. Bruce Goldstein, Perception,
Books/Cole, 5th edition, 1998. Cited on pages 21 and 25.
12 A good overview over the arguments used to prove the existence of god from motion is
given by Michael Buckley, Motion and Motion’s God, Princeton University Press, 1971.
The intensity of the battles waged around these failed attempts is one of the tragicomic
chapters of history. Cited on page 21.
13 Thomas Aquinas, Summa Theologiae or Summa Theologica, 1265–1273, online in Latin
at www.newadvent.org/summa, in English on several other servers. Cited on page 21.
14 For an exploration of ‘inner’ motions, see the beautiful text by Richard Schwartz,
Internal Family Systems Therapy, The Guilford Press, 1995. Cited on page 21.
15 For an authoritative description of proper motion development in babies and about how it
leads to a healthy character see Emmi Pikler, Laßt mir Zeit - Die selbstständige Bewegung-
sentwicklung des Kindes bis zum freien Gehen, Pflaum Verlag, 2001, and her other books. See
also the website www.pikler.org. Cited on page 21.

Motion Mountain – The Adventure of Physics
16 See e.g. the fascinating text by David G. Chandler, The Campaigns of Napoleon – The
Mind and Method of History’s Greatest Soldier, Macmillan, 1966. Cited on page 21.
17 Richard Marcus, American Roulette, St Martin’s Press, 2003, a thriller and a true story.
Cited on page 21.
18 A good and funny book on behaviour change is the well-known text Richard Bandler,
Using Your Brain for a Change, Real People Press, 1985. See also Richard Bandler &
John Grinder, Frogs into princes – Neuro Linguistic Programming, Eden Grove Editions,
1990. Cited on pages 21 and 32.
19 A beautiful book about the mechanisms of human growth from the original cell to full size

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
is Lewis Wolpert, The Triumph of the Embryo, Oxford University Press, 1991. Cited on
page 21.
20 On the topic of grace and poise, see e.g. the numerous books on the Alexander tech-
nique, such as M. Gelb, Body Learning – An Introduction to the Alexander Technique,
Aurum Press, 1981, and Richard Brennan, Introduction to the Alexander Technique,
Little Brown and Company, 1996. Among others, the idea of the Alexander technique is
to return to the situation that the muscle groups for sustainment and those for motion are
used only for their respective function, and not vice versa. Any unnecessary muscle ten-
sion, such as neck stiffness, is a waste of energy due to the use of sustainment muscles for
movement and of motion muscles for sustainment. The technique teaches the way to return
to the natural use of muscles.
Motion of animals was discussed extensively already in the seventeenth century by
G. B orelli, De motu animalium, 1680. An example of a more modern approach is
J. J. Collins & I. Stewart, Hexapodal gaits and coupled nonlinear oscillator models,
Biological Cybernetics 68, pp. 287–298, 1993. See also I. Stewart & M. Golubitsky,
Fearful Symmetry, Blackwell, 1992. Cited on pages 23 and 122.
21 The results on the development of children mentioned here and in the following have been
drawn mainly from the studies initiated by Jean Piaget; for more details on child devel-
Vol. III, page 256 opment, see later on. At www.piaget.org you can find the website maintained by the Jean
Piaget Society. Cited on pages 24, 40, and 42.
22 The reptilian brain (eat? flee? ignore?), also called the R-complex, includes the brain stem,
the cerebellum, the basal ganglia and the thalamus; the old mammalian (emotions) brain,
526 bibliography

also called the limbic system, contains the amygdala, the hypothalamus and the hippocam-
pus; the human (and primate) (rational) brain, called the neocortex, consists of the famous
grey matter. For images of the brain, see the atlas by John Nolte, The Human Brain: An
Introduction to its Functional Anatomy, Mosby, fourth edition, 1999. Cited on page 25.
23 The lower left corner film can be reproduced on a computer after typing the following lines
in the Mathematica software package: Cited on page 25.

« Graphics‘Animation‘
Nxpixels=72; Nypixels=54; Nframes=Nxpixels 4/3;
Nxwind=Round[Nxpixels/4]; Nywind=Round[Nypixels/3];
front=Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}];
back =Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}];
frame=Table[front,{nf,1,Nframes}];
Do[ If[ x>n-Nxwind && x<n && y>Nywind && y<2Nywind,
frame[[n,y,x]]=back[[y,x-n]] ],
{x,1,Nxpixels}, {y,1,Nypixels}, {n,1,Nframes}];
ﬁlm=Table[ListDensityPlot[frame[[nf ]], Mesh-> False,
Frame-> False, AspectRatio-> N[Nypixels/Nxpixels],

Motion Mountain – The Adventure of Physics
DisplayFunction-> Identity], {nf,1,Nframes}]
ShowAnimation[ﬁlm]

But our motion detection system is much more powerful than the example shown in the
lower left corners. The following, different film makes the point.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
frame=Table[front,{nf,1,Nframes}];
Do[ If[ x>n-Nxwind && x<n && y>Nywind && y<2Nywind,
frame[[n,y,x]]=back[[y,x]] ],
{x,1,Nxpixels}, {y,1,Nypixels}, {n,1,Nframes}];
ﬁlm=Table[ListDensityPlot[frame[[nf ]], Mesh-> False,
Frame-> False, AspectRatio-> N[Nypixels/Nxpixels],
DisplayFunction-> Identity], {nf,1,Nframes}]
ShowAnimation[ﬁlm]

Similar experiments, e.g. using randomly changing random patterns, show that the eye
perceives motion even in cases where all Fourier components of the image are practic-
ally zero; such image motion is called drift-balanced or non-Fourier motion. Several ex-
amples are presented in J. Zanker, Modelling human motion perception I: Classical stim-
uli, Naturwissenschaften 81, pp. 156–163, 1994, and J. Zanker, Modelling human motion
perception II: Beyond Fourier motion stimuli, Naturwissenschaften 81, pp. 200–209, 1994.
Modern research has helped to find the corresponding neuronal structures, as shown in
S. A. Baccus, B. P. Olveczky, M. Manu & M. Meister, A retinal circuit that com-
putes object motion, Journal of Neuroscience 28, pp. 6807–6817, 2008.
24 All fragments from Heraclitus are from John Mansley Robinson, An Introduction to
Early Greek Philosophy, Houghton Muffin 1968, chapter 5. Cited on page 27.
25 On the block and tackle, see the explanations by Donald Simanek at http://www.lhup.edu/
~dsimanek/TTT-fool/fool.htm. Cited on page 31.
bibliography 527

26 An overview over these pretty puzzles is found in E. D. Demaine, M. L. Demaine,
Y. N. Minski, J. S. B. Mitchell, R. L. Rivest & M. Patrascu, Picture-hanging
puzzles, preprint at arxiv.org/abs/1203.3602. Cited on page 33.
27 An introduction to Newton the alchemist are the books by Betty Jo Teeter Dobbs,
The Foundations of Newton’s Alchemy, Cambridge University Press, 1983, and The Janus
Face of Genius, Cambridge University Press, 1992. Newton is found to be a sort of highly
intellectual magician, desperately looking for examples of processes where gods inter-
act with the material world. An intense but tragic tale. A good overview is provided by
R. G. Keesing, Essay Review: Newton’s Alchemy, Contemporary Physics 36, pp. 117–119,
1995.
Newton’s infantile theology, typical for god seekers who grew up without a father, is de-
scribed in the many books summarizing the letter exchanges between Clarke, his secretary,
and Leibniz, Newton’s rival for fame. Cited on page 34.
28 An introduction to the story of classical mechanics, which also destroys a few of the myths
surrounding it – such as the idea that Newton could solve differential equations or that he
introduced the expression 𝐹 = 𝑚𝑎 – is given by Clifford A. Truesdell, Essays in the
History of Mechanics, Springer, 1968. Cited on pages 34, 178, and 228.

Motion Mountain – The Adventure of Physics
29 The slowness of the effective speed of light inside the Sun is due to the frequent scattering
of photons by solar matter. The best estimate of its value is by R. Mitalas & K. R. Sills,
On the photon diffusion time scale for the Sun, The Astrophysical Journal 401, pp. 759–760,
1992. They give an average speed of 0.97 cm/s over the whole Sun and a value about 10 times
smaller at its centre. Cited on page 36.
30 C. Liu, Z. Dutton, C. H. Behroozi & L. Vestergaard Hau, Observation of co-
herent optical information storage in an atomic medium using halted light pulses, Nature 409,
pp. 490–493, 2001. There is also a comment on the paper by E. A. Cornell, Stopping light
in its track, 409, pp. 461–462, 2001. However, despite the claim, the light pulses of course

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
have not been halted. Can you give at least two reasons without even reading the paper, and
Challenge 743 s maybe a third after reading it?
The work was an improvement on the previous experiment where a group velocity of
light of 17 m/s had been achieved, in an ultracold gas of sodium atoms, at nanokelvin tem-
peratures. This was reported by L. Vestergaard Hau, S. E. Harris, Z. Dutton &
C. H. Behroozi, Light speed reduction to 17 meters per second in an ultracold atomic gas,
Nature 397, pp. 594–598, 1999. Cited on page 36.
31 Rainer Flindt, Biologie in Zahlen – Eine Datensammlung in Tabellen mit über 10.000
Einzelwerten, Spektrum Akademischer Verlag, 2000. Cited on page 36.
32 Two jets with that speed have been observed by I. F. Mirabel & L. F. Rodríguez, A
superluminal source in the Galaxy, Nature 371, pp. 46–48, 1994, as well as the comments on
p. 18. Cited on page 36.
33 A beautiful introduction to the slowest motions in nature, the changes in landscapes, is
Detlev Busche, Jürgen Kempf & Ingrid Stengel, Landschaftsformen der Erde –
Bildatlas der Geomorphologie, Primus Verlag, 2005. Cited on page 37.
34 To build your own sundial, see the pretty and short Arnold Zenkert, Faszination
Sonnenuhr, VEB Verlag Technik, 1984. See also the excellent and complete introduction
into this somewhat strange world at the www.sundials.co.uk website. Cited on page 42.
35 An introduction to the sense of time as a result of clocks in the brain is found in R. B. Ivry
& R. Spencer, The neural representation of time, Current Opinion in Neurobiology 14,
pp. 225–232, 2004. The chemical clocks in our body are described in John D. Palmer,
528 bibliography

The Living Clock, Oxford University Press, 2002, or in A. Ahlgren & F. Halberg,
Cycles of Nature: An Introduction to Biological Rhythms, National Science Teachers Asso-
ciation, 1990. See also the www.msi.umn.edu/~halberg/introd website. Cited on page 43.
36 This has been shown among others by the work of Anna Wierzbicka that is discussed in
Vol. III, page 280 more detail in one of the subsequent volumes. The passionate bestseller by the Chomskian
author Steven Pinker, The Language Instinct – How the Mind Creates Language, Harper
Perennial, 1994, also discusses issues related to this matter, refuting amongst others on page
63 the often repeated false statement that the Hopi language is an exception. Cited on page
43.
37 For more information, see the excellent and freely downloadable books on biological clocks
by Wolfgang Engelmann on the website www.uni-tuebingen.de/plantphys/bioclox. Cited
on page 44.
38 B. Günther & E. Morgado, Allometric scaling of biological rhythms in mammals, Bio-
logical Research 38, pp. 207–212, 2005. Cited on page 44.
39 Aristotle rejects the idea of the flow of time in chapter IV of his Physics. See the full text on
the classics.mit.edu/Aristotle/physics.4.iv.html website. Cited on page 48.

Motion Mountain – The Adventure of Physics
40 Perhaps the most informative of the books about the ‘arrow of time’ is Heinz-
Dieter Zeh, The Physical Basis of the Direction of Time, Springer Verlag, 4th edition,
2001. It is still the best book on the topic. Most other texts exist – have a look on the internet
– but lack clarity of ideas.
A typical conference proceeding is J. J. Halliwell, J. Pérez-Mercader & Wo-
jciech H. Zurek, Physical Origins of Time Asymmetry, Cambridge University Press,
1994. Cited on page 49.
41 On the issue of absolute and relative motion there are many books about few issues. Ex-
amples are Julian Barbour, Absolute or Relative Motion? Vol. 1: A Study from the Ma-
chian Point of View of the Discovery and the Structure of Spacetime Theories, Cambridge

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
University Press, 1989, Julian Barbour, Absolute or Relative Motion? Vol. 2: The Deep
Structure of General Relativity, Oxford University Press, 2005, or John Earman, World
Enough and Spacetime: Absolute vs Relational Theories of Spacetime, MIT Press, 1989. A
speculative solution on the alternative between absolute and relative motion is presented in
Vol. VI, page 65 volume VI. Cited on page 52.
42 Coastlines and other fractals are beautifully presented in Heinz-Otto Peitgen,
Hartmut Jürgens & Dietmar Saupe, Fractals for the Classroom, Springer Verlag,
1992, pp. 232–245. It is also available in several other languages. Cited on page 54.
43 R. Dougherty & M. Foreman, Banach–Tarski decompositions using sets with the
property of Baire, Journal of the American Mathematical Society 7, pp. 75–124, 1994.
See also Alan L.T. Paterson, Amenability, American Mathematical Society, 1998,
and Robert M. French, The Banach–Tarski theorem, The Mathematical Intelligen-
cer 10, pp. 21–28, 1998. Finally, there are the books by Bernard R. Gelbaum &
John M. H. Olmsted, counter-examples in Analysis, Holden–Day, 1964, and their The-
orems and counter-examples in Mathematics, Springer, 1993. Cited on page 57.
44 The beautiful but not easy text is Stan Wagon, The Banach Tarski Paradox, Cambridge
University Press, 1993. Cited on pages 58 and 480.
45 About the shapes of saltwater bacteria, see the corresponding section in the interesting book
by Bernard Dixon, Power Unseen – How Microbes Rule the World, W.H. Freeman, 1994.
The book has about 80 sections, in which as many microorganisms are vividly presented.
Cited on page 59.
bibliography 529

46 Olaf Medenbach & Harry Wilk, Zauberwelt der Mineralien, Sigloch Edition, 1977.
It combines beautiful photographs with an introduction into the science of crystals, min-
erals and stones. About the largest crystals, see P. C. Rickwood, The largest crystals,
66, pp. 885–908, 1981, also available on www.minsocam.org/MSA/collectors_corner/arc/
large_crystals.htm. For an impressive example, the Naica cave in Mexico, see www.naica.
com.mx/ingles/index.htm Cited on page 59.
47 See the websites www.weltbildfrage.de/3frame.htm and www.lhup.edu/~dsimanek/hollow/
morrow.htm. Cited on page 60.
48 R. Tan, V. Osman & G. Tan, Ear size as a predictor of chronological age, Archives of
Gerontology and Geriatrics 25, pp. 187–191, 1997. Cited on page 61.
49 The smallest distances are probed in particle accelerators; the distance can be determined
from the energy of the particle beam. In 1996, the value of 10−19 m (for the upper limit of
the size of quarks) was taken from the experiments described in F. Abe & al., Measurement
of dijet angular distributions by the collider detector at Fermilab, Physical Review Letters 77,
pp. 5336–5341, 1996. Cited on page 66.
50 More on the Moon illusion can be found at the website science.nasa.gov/science-news/

Motion Mountain – The Adventure of Physics
science-at-nasa/2008/16jun_moonillusion/. All the works of Ptolemy are found online at
www.ptolemaeus.badw.de. Cited on page 69.
51 These puzzles are taken from the puzzle collection at www.mathematische-basteleien.de.
Cited on page 70.
52 Alexander K. Dewdney, The Planiverse – Computer Contact with a Two-dimensional
World, Poseidon Books/Simon & Schuster, 1984. See also Edwin A. Abbott, Flatland: A
romance of many dimensions, 1884. Several other fiction authors had explored the option of
a two-dimensional universe before, always answering, incorrectly, in the affirmative. Cited
on page 71.
J. B ohr & K. Olsen, The ancient art of laying rope, preprint at arxiv.org/abs/1004.0814

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
53
Cited on page 72.
54 For an overview and references see www.pbrc.hawaii.edu/~petra/animal_olympians.html.
Cited on page 72.
55 P. Pieranski, S. Przybyl & A. Stasiak, Tight open knots, European Physical Journal
E 6, pp. 123–128, 2001, preprint at arxiv.org/abs/physics/0103016. Cited on page 73.
56 On the world of fireworks, see the frequently asked questions list of the usenet
group rec.pyrotechnics, or search the web. A simple introduction is the article by
J. A. Conkling, Pyrotechnics, Scientific American pp. 66–73, July 1990. Cited on page
75.
57 There is a whole story behind the variations of 𝑔. It can be discovered in Chuji Tsuboi,
Gravity, Allen & Unwin, 1979, or in Wolf gang Torge, Gravimetry, de Gruyter, 1989,
or in Milan Burša & Karel Pěč, The Gravity Field and the Dynamics of the Earth,
Springer, 1993. The variation of the height of the soil by up to 0.3 m due to the Moon is one
of the interesting effects found by these investigations. Cited on pages 76 and 197.
58 Stillman Drake, Galileo: A Very Short Introduction, Oxford University Press, 2001.
Cited on page 76.
59 Andrea Frova, La fisica sotto il naso – 44 pezzi facili, Biblioteca Universale Rizzoli, Mil-
ano, 2001. Cited on page 77.
60 On the other hand, other sciences enjoy studying usual paths in all detail. See, for example,
Heini Hediger, editor, Die Straßen der Tiere, Vieweg & Sohn, 1967. Cited on page 77.
530 bibliography

61 H. K. Eriksen, J. R. Kristiansen, Ø. Langangen & I. K. Wehus, How fast could
Usain Bolt have run? A dynamical study, American Journal of Physics 77, pp. 224–228, 2009.
See also the references at en.wikipedia.org/wiki/Footspeed. Cited on page 78.
62 This was discussed in the Frankfurter Allgemeine Zeitung, 2nd of August, 1997, at the time
of the world athletics championship. The values are for the fastest part of the race of a
100 m sprinter; the exact values cited were called the running speed world records in 1997,
and were given as 12.048 m/s = 43.372 km/h by Ben Johnson for men, and 10.99 m/s =
39.56 km/h for women. Cited on page 78.
63 Long jump data and literature can be found in three articles all entitled Is a good long
jumper a good high jumper?, in the American Journal of Physics 69, pp. 104–105, 2001. In
particular, world-class long jumpers run at 9.35 ± 0.15 m/s, with vertical take-off speeds of
3.35 ± 0.15 m/s, giving take-off angles of about (only) 20°. A new technique for achieving
higher take-off angles would allow the world long jump record to increase dramatically.
Cited on page 78.
64 The study of shooting faeces (i.e., shit) and its mechanisms is a part of modern biology. The
reason that caterpillars do this was determined by M. Weiss, Good housekeeping: why do
shelter-dwelling caterpillars fling their frass?, Ecology Letters 6, pp. 361–370, 2003, who also

Motion Mountain – The Adventure of Physics
gives the present record of 1.5 m for the 24 mg pellets of Epargyreus clarus. The picture of
the flying frass is from S. Caveney, H. McLean & D. Surry, Faecal firing in a skipper
caterpillar is pressure-driven, The Journal of Experimental Biology 201, pp. 121–133, 1998.
Cited on page 79.
65 H. C. Bennet-Clark, Scale effects in jumping animals, pp. 185–201, in T. J. Pedley,
editor, Scale Effects in Animal Locomotion, Academic Press, 1977. Cited on page 80.
66 The arguments of Zeno can be found in Aristotle, Physics, VI, 9. It can be found
translated in almost any language. The classics.mit.edu/Aristotle/physics.6.vi.html website
provides an online version in English. Cited on pages 83 and 524.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
67 See, for example, K. V. Kumar & W. T. Norfleet, Issues of human acceleration toler-
ance after long-duration space flights, NASA Technical Memorandum 104753, pp. 1–55, 1992,
available at ntrs.nasa.gov. Cited on page 85.
68 Etymology can be a fascinating topic, e.g. when research discovers the origin of the Ger-
man word ‘Weib’ (‘woman’, related to English ‘wife’). It was discovered, via a few texts in
Tocharian – an extinct Indo-European language from a region inside modern China – to
mean originally ‘shame’. It was used for the female genital region in an expression mean-
ing ‘place of shame’. With time, this expression became to mean ‘woman’ in general, while
being shortened to the second term only. This connection was discovered by the linguist
Klaus T. Schmidt; it explains in particular why the word is not feminine but neutral, i.e.,
why it uses the article ‘das’ instead of ‘die’. Julia Simon, private communication.
Etymology can also be simple and plain fun, for example when one discovers in the Ox-
ford English Dictionary that ‘testimony’ and ‘testicle’ have the same origin; indeed in Latin
the same word ‘testis’ was used for both concepts. Cited on pages 86 and 101.
69 An overview of the latest developments is given by J. T. Armstrong, D. J. Hunter,
K. J. Johnston & D. Mozurkewich, Stellar optical interferometry in the 1990s, Phys-
ics Today pp. 42–49, May 1995. More than 100 stellar diameters were known already in 1995.
Several dedicated powerful instruments are being planned. Cited on page 87.
70 A good biology textbook on growth is Arthur F. Hopper & Nathan H. Hart,
Foundations of Animal Development, Oxford University Press, 2006. Cited on page 89.
71 This is discussed for example in C. L. Stong, The amateur scientist – how to supply electric
power to something which is turning, Scientific American pp. 120–125, December 1975. It also
bibliography 531

discusses how to make a still picture of something rotating simply by using a few prisms, the
so-called Dove prisms. Other examples of attaching something to a rotating body are given
by E. Rieflin, Some mechanisms related to Dirac’s strings, American Journal of Physics
47, pp. 379–381, 1979. Cited on page 89.
72 James A. Young, Tumbleweed, Scientific American 264, pp. 82–87, March 1991. The
tumbleweed is in fact quite rare, except in Hollywood westerns, where all directors feel
obliged to give it a special appearance. Cited on page 90.
73 The classic book on the topic is James Gray, Animal Locomotion, Weidenfeld & Nicolson,
1968. Cited on page 90.
74 About N. decemspinosa, see R. L. Caldwell, A unique form of locomotion in a stoma-
topod – backward somersaulting, Nature 282, pp. 71–73, 1979, and R. Full, K. Earls,
M. Wong & R. Caldwell, Locomotion like a wheel?, Nature 365, p. 495, 1993. About
rolling caterpillars, see J. Brackenbury, Caterpillar kinematics, Nature 330, p. 453, 1997,
and J. Brackenbury, Fast locomotion in caterpillars, Journal of Insect Physiology 45,
pp. 525–533, 1999. More images around legs can be found on rjf9.biol.berkeley.edu/twiki/
bin/view/PolyPEDAL/LabPhotographs. Cited on page 90.

Motion Mountain – The Adventure of Physics
75 The locomotion of the spiders of the species Cebrennus villosus has been described by
Ingo Rechenberg from Berlin. See the video at www.youtube.com/watch?v=Aayb_h31RyQ.
Cited on page 90.
76 The first experiments to prove the rotation of the flagella were by M. Silverman &
M. I. Simon, Flagellar rotation and the mechanism of bacterial motility, Nature 249, pp. 73–
74, 1974. For some pretty pictures of the molecules involved, see K. Namba, A biological
molecular machine: bacterial flagellar motor and filament, Wear 168, pp. 189–193, 1993, or
the website www.nanonet.go.jp/english/mailmag/2004/011a.html. The present record speed
of rotation, 1700 rotations per second, is reported by Y. Magariyama, S. Sugiyama,
K. Muramoto, Y. Maekawa, I. Kawagishi, Y. Imae & S. Kudo, Very fast flagellar

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
rotation, Nature 371, p. 752, 1994.
More on bacteria can be learned from David Dusenbery, Life at a Small Scale, Sci-
entific American Library, 1996. Cited on page 92.
77 S. Chen & al., Structural diversity of bacterial flagellar motors, EMBO Journal 30,
pp. 2972–2981, 2011, also online at emboj.embopress.org/content/30/14/2972. Cited on page
92.
78 M. P. Brenner, S. Hilgenfeldt & D. Lohse, Single bubble sonoluminescence, Re-
views of Modern Physics 74, pp. 425–484, 2002. Cited on page 96.
79 K. R. Weninger, B. P. Barber & S. J. Putterman, Pulsed Mie scattering measure-
ments of the collapse of a sonoluminescing bubble, Physical Review Letters 78, pp. 1799–
1802, 1997. Cited on page 96.
80 On shadows, see the agreeable popular text by Roberto Casati, Alla scoperta dell’ombra
– Da Platone a Galileo la storia di un enigma che ha affascinato le grandi menti dell’umanità,
Oscar Mondadori, 2000, and his website located at www.shadowes.org. Cited on page 98.
81 There is also the beautiful book by Penelope Farrant, Colour in Nature, Blandford,
1997. Cited on page 98.
82 The ‘laws’ of cartoon physics can easily be found using any search engine on the internet.
Cited on page 98.
83 For the curious, an overview of the illusions used in the cinema and in television, which
lead to some of the strange behaviour of images mentioned above, is given in Bern-
ard Wilkie, The Technique of Special Effects in Television, Focal Press, 1993, and his other
532 bibliography

books, or in the Cinefex magazine. On digital cinema techniques, see Peter C. Slansky,
editor, Digitaler film – digitales Kino, UVK Verlag, 2004. Cited on page 99.
84 Aetius, Opinions, I, XXIII, 3. See Jean-Paul Dumont, Les écoles présocratiques, Folio
Essais, Gallimard, p. 426, 1991. Cited on page 99.
85 Giuseppe Fumagalli, Chi l’ha detto?, Hoepli, 1983. It is from Pappus of Alexandria’s
opus Synagoge, book VIII, 19. Cited on pages 100 and 237.
86 See www.straightdope.com/classics/a5_262.html and the more dubious en.wikipedia.org/
wiki/Guillotine. Cited on page 102.
87 See the path-breaking paper by A. DiSessa, Momentum flow as an alternative perspective
in elementary mechanics, 48, p. 365, 1980, and A. DiSessa, Erratum: “Momentum flow as
an alternative perspective in elementary mechanics” [Am. J. Phys. 48, 365 (1980)], 48, p. 784,
1980. Also the wonderful free textbook by Friedrich Herrmann, The Karlsruhe Physics
Course, makes this point extensively; see Ref. 2. Cited on pages 109, 228, 231, and 507.
88 For the role and chemistry of adenosine triphosphate (ATP) in cells and in living beings, see
any chemistry book, or search the internet. The uncovering of the mechanisms around ATP
has led to Nobel Prizes in Chemistry in 1978 and in 1997. Cited on page 109.

Motion Mountain – The Adventure of Physics
89 A picture of this unique clock can be found in the article by A. Garrett, Perpetual motion
– a delicious delirium, Physics World pp. 23–26, December 1990. Cited on page 110.
90 Esger Brunner, Het ongelijk van Newton – het kleibakexperiment van ’s Gravesande
nagespeld, Nederland tijdschrift voor natuurkunde pp. 95–96, Maart 2012. The paper con-
tains photographs of the mud imprints. Cited on page 111.
91 A Shell study estimated the world’s total energy consumption in 2000 to be 500 EJ. The
US Department of Energy estimated it to be around 416 EJ. We took the lower value here.
A discussion and a breakdown into electricity usage (14 EJ) and other energy forms, with
variations per country, can be found in S. Benka, The energy challenge, Physics Today

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
55, pp. 38–39, April 2002, and in E. J. Monitz & M. A. Kenderdine, Meeting energy
challenges: technology and policy, Physics Today 55, pp. 40–46, April 2002. Cited on pages
113 and 114.
92 L. M. Miller, F. Gans & A. Kleidon, Estimating maximum global land surface wind
power extractability and associated climatic consequences, Earth System Dynamics 2, pp. 1–
12, 2011. Cited on page 114.
93 For an overview, see the paper by J. F. Mulligan & H. G. Hertz, An unpublished lecture
by Heinrich Hertz: ‘On the energy balance of the Earth’, American Journal of Physics 65,
pp. 36–45, 1997. Cited on page 114.
94 For a beautiful photograph of this feline feat, see the cover of the journal and the article of
J. Darius, A tale of a falling cat, Nature 308, p. 109, 1984. Cited on page 121.
95 Natthi L. Sharma, A new observation about rolling motion, European Journal of Phys-
ics 17, pp. 353–356, 1996. Cited on page 121.
96 C. Singh, When physical intuition fails, American Journal of Physics 70, pp. 1103–1109,
2002. Cited on page 121.
97 There is a vast literature on walking. Among the books on the topic, two well-known intro-
ductions are Robert McNeill Alexander, Exploring Biomechanics: Animals in Mo-
tion, Scientific American Library, 1992, and Steven Vogel, Comparative Biomechanics -
Life’s Physical World, Princeton University Press, 2003. Cited on page 122.
98 Serge Gracovetsky, The Spinal Engine, Springer Verlag, 1990. It is now also known
that human gait is chaotic. This is explained by M. Perc, The dynamics of human gait,
bibliography 533

European Journal of Physics 26, pp. 525–534, 2005. On the physics of walking and running,
see also the respective chapters in the delightful book by Werner Gruber, Unglaublich
einfach, einfach unglaublich: Physik für jeden Tag, Heyne, 2006. Cited on page 123.
99 M. Llobera & T. J. Sluckin, Zigzagging: theoretical insights on climbing strategies,
Journal of Theoretical Biology 249, pp. 206–217, 2007. Cited on page 125.
100 This description of life and death is called the concept of maximal metabolic scope. Look up
details in your favourite library. A different phrasing is the one by M. Ya. Azbel, Universal
biological scaling and mortality, Proceedings of the National Academy of Sciences of the
USA 91, pp. 12453–12457, 1994. He explains that every atom in an organism consumes, on
average, 20 oxygen molecules per lifespan. Cited on page 125.
101 Duncan MacDougall, Hypothesis concerning soul substance together with experi-
mental evidence of the existence of such substance, American Medicine 2, pp. 240–243,
April 1907, and Duncan MacDougall, Hypothesis concerning soul substance, Amer-
ican Medicine 2, pp. 395–397, July 1907. Reading the papers shows that the author has little
practice in performing reliable weight and time measurements. Cited on page 126.
102 A good roulette prediction story from the 1970s is told by Thomas A. Bass, The Eudae-

Motion Mountain – The Adventure of Physics
monic Pie also published under the title The Newtonian Casino, Backinprint, 2000. An over-
view up to 1998 is given in the paper Edward O. Thorp, The invention of the first wear-
able computer, Proceedings of the Second International Symposium on Wearable Computers
(ISWC 1998), 19-20 October 1998, Pittsburgh, Pennsylvania, USA (IEEE Computer Society),
pp. 4–8, 1998, downloadable at csdl.computer.org/comp/proceedings/iswc/1998/9074/00/
9074toc.htm. Cited on page 127.
103 This and many other physics surprises are described in the beautiful lecture script by
Josef Zweck, Physik im Alltag, the notes of his lectures held in 1999/2000 at the Uni-
versität Regensburg. Cited on pages 128 and 133.
104 The equilibrium of ships, so important in car ferries, is an interesting part of shipbuilding;

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
an introduction was already given by Leonhard Euler, Scientia navalis, 1749. Cited on
page 129.
105 Thomas Heath, Aristarchus of Samos – the Ancient Copernicus, Dover, 1981, reprinted
from the original 1913 edition. Aristarchus’ treaty is given in Greek and English. Aristarchus
was the first proposer of the heliocentric system. Aristarchus had measured the length of
the day (in fact, by determining the number of days per year) to the astonishing precision of
less than one second. This excellent book also gives an overview of Greek astronomy before
Aristarchus, explained in detail for each Greek thinker. Aristarchus’ text is also reprinted
in Aristarchus, On the sizes and the distances of the Sun and the Moon, c. 280 bce in
Michael J. Crowe, Theories of the World From Antiquity to the Copernican Revolution,
Dover, 1990, especially on pp. 27–29. No citations.
106 T. Gerkema & L. Gostiaux, A brief history of the Coriolis force, Europhysics News 43,
pp. 14–17, 2012. Cited on page 138.
107 See for example the videos on the Coriolis effect at techtv.mit.edu/videos/3722 and techtv.
mit.edu/videos/3714, or search for videos on youtube.com. Cited on page 139.
108 The influence of the Coriolis effect on icebergs was studied most thoroughly by the physicist
turned oceanographer Walfrid Ekman (b. 1874 Stockholm, d. 1954 Gostad); the topic was
suggested by the great explorer Fridtjof Nansen, who also made the first observations. In
his honour, one speaks of the Ekman layer, Ekman transport and Ekman spirals. Any text
on oceanography or physical geography will give more details about them. Cited on page
139.
534 bibliography

109 An overview of the effects of the Coriolis acceleration 𝑎 = −2𝜔 × 𝑣 in the rotating frame is
given by Edward A. Desloge, Classical Mechanics, Volume 1, John Wiley & Sons, 1982.
Even the so-called Gulf Stream, the current of warm water flowing from the Caribbean to
the North Sea is influenced by it. Cited on page 139.
110 The original publication is by A. H. Shapiro, Bath-tub vortex, Nature 196, pp. 1080–1081,
1962. He also produced two films of the experiment. The experiment has been repeated
many times in the northern and in the southern hemisphere, where the water drains clock-
wise; the first southern hemisphere test was L.M. Trefethen & al., The bath-tub vortex
in the southern hemisphere, Nature 201, pp. 1084–1085, 1965. A complete literature list is
found in the letters to the editor of the American Journal of Physics 62, p. 1063, 1994. Cited
on page 140.
111 The tricks are explained by H. Richard Crane, Short Foucault pendulum: a way to
eliminate the precession due to ellipticity, American Journal of Physics 49, pp. 1004–1006,
1981, and particularly in H. Richard Crane, Foucault pendulum wall clock, American
Journal of Physics 63, pp. 33–39, 1993. The Foucault pendulum was also the topic of the
thesis of Heike Kamerling Onnes, Nieuwe bewijzen der aswenteling der aarde, Uni-
versiteit Groningen, 1879. Cited on page 140.

Motion Mountain – The Adventure of Physics
112 The reference is J. G. Hagen, La rotation de la terre : ses preuves mécaniques anciennes et
nouvelles, Sp. Astr. Vaticana Second. App. Rome, 1910. His other experiment is published as
J. G. Hagen, How Atwood’s machine shows the rotation of the Earth even quantitatively,
International Congress of Mathematics, Aug. 1912. Cited on page 141.
113 The original papers are A. H. Compton, A laboratory method of demonstrating the Earth’s
rotation, Science 37, pp. 803–806, 1913, A. H. Compton, Watching the Earth revolve, Sci-
entific American Supplement no. 2047, pp. 196–197, 1915, and A. H. Compton, A determ-
ination of latitude, azimuth and the length of the day independent of astronomical observa-
tions, Physical Review (second series) 5, pp. 109–117, 1915. Cited on page 141.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
114 The G-ring in Wettzell is so precise, with a resolution of less than 10−8 , that it has de-
tected the motion of the poles. For details, see K. U. Schreiber, A. Velikoseltsev,
M. Rothacher, T. Kluegel, G. E. Stedman & D. L. Wiltshire, Direct measure-
ment of diurnal polar motion by ring laser gyroscopes, Journal of Geophysical Research 109
B, p. 06405, 2004, an a review article at T. Klügel, W. Schlüter, U. Schreiber &
M. Schneider, Großringlaser zur kontinuierlichen Beobachtung der Erdrotation, Zeits-
chrift für Vermessungswesen 130, pp. 99–108, February 2005. Cited on page 142.
115 R. Anderson, H. R. Bilger & G. E. Stedman, The Sagnac-effect: a century of Earth-
rotated interferometers, American Journal of Physics 62, pp. 975–985, 1994.
See also the clear and extensive paper by G. E. Stedman, Ring laser tests of funda-
mental physics and geophysics, Reports on Progress in Physics 60, pp. 615–688, 1997. Cited
on page 144.
116 About the length of the day, see the maia.usno.navy.mil website, or the books by
K. Lambeck, The Earth’s Variable Rotation: Geophysical Causes and Consequences,
Cambridge University Press, 1980, and by W. H. Munk & G. J. F. MacDonald, The
Rotation of the Earth, Cambridge University Press, 1960. For a modern ring laser set-up,
see www.wettzell.ifag.de. Cited on pages 145 and 200.
117 H. Bucka, Zwei einfache Vorlesungsversuche zum Nachweis der Erddrehung, Zeitschrift für
Physik 126, pp. 98–105, 1949, and H. Bucka, Zwei einfache Vorlesungsversuche zum Nach-
weis der Erddrehung. II. Teil, Zeitschrift für Physik 128, pp. 104–107, 1950. Cited on page
145.
bibliography 535

118 One example of data is by C. P. Sonett, E. P. Kvale, A. Zakharian, M. A. Chan &
T. M. Demko, Late proterozoic and paleozoic tides, retreat of the moon, and rotation of the
Earth, Science 273, pp. 100–104, 5 July 1996. They deduce from tidal sediment analysis that
days were only 18 to 19 hours long in the Proterozoic, i.e., 900 million years ago; they as-
sume that the year was 31 million seconds long from then to today. See also C. P. Sonett
& M. A. Chan, Neoproterozoic Earth-Moon dynamics – rework of the 900 MA Big Cotton-
wood canyon tidal laminae, Geophysical Research Letters 25, pp. 539–542, 1998. Another
determination was by G. E. Williams, Precambrian tidal and glacial clastic deposits: im-
plications for precambrian Earth–Moon dynamics and palaeoclimate, Sedimentary Geology
120, pp. 55–74, 1998. Using a geological formation called tidal rhythmites, he deduced that
about 600 million years ago there were 13 months per year and a day had 22 hours. Cited
on page 145.
119 For the story of this combination of history and astronomy see Richard Stephenson,
Historical Eclispes and Earth’s Rotation, Cambridge University Press, 1996. Cited on page
146.
120 B. F. Chao, Earth Rotational Variations excited by geophysical fluids, IVS 2004 General
Meeting proceedings/ pages 38-46. Cited on page 146.

Motion Mountain – The Adventure of Physics
121 On the rotation and history of the Solar System, see S. Brush, Theories of the origin of the
solar system 1956–1985, Reviews of Modern Physics 62, pp. 43–112, 1990. Cited on page
146.
122 The website hpiers.obspm.fr/eop-pc shows the motion of the Earth’s axis over the last ten
years. The International Latitude Service founded by Küstner is now part of the Interna-
tional Earth Rotation Service; more information can be found on the www.iers.org web-
site. The latest idea is that two-thirds of the circular component of the polar motion, which
in the USA is called ‘Chandler wobble’ after the person who attributed to himself the dis-
covery by Küstner, is due to fluctuations of the ocean pressure at the bottom of the oceans

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
and one-third is due to pressure changes in the atmosphere of the Earth. This is explained
by R. S. Gross, The excitation of the Chandler wobble, Geophysical Physics Letters 27,
pp. 2329–2332, 2000. Cited on page 147.
123 S. B. Lambert, C. Bizouard & V. Dehant, Rapid variations in polar motion during
the 2005-2006 winter season, Geophysical Research Letters 33, p. L13303, 2006. Cited on
page 147.
124 For more information about Alfred Wegener, see the (simple) text by Klaus Rohrbach,
Alfred Wegener – Erforscher der wandernden Kontinente, Verlag Freies Geistesleben, 1993;
about plate tectonics, see the www.scotese.com website. About earthquakes, see the www.
geo.ed.ac.uk/quakexe/quakes and the www.iris.edu/seismon website. See the vulcan.wr.
usgs.gov and the www.dartmouth.edu/~volcano websites for information about volcanoes.
Cited on page 150.
125 J. Jouzel & al., Orbital and millennial Antarctic climate variability over the past 800,000
years, Science 317, pp. 793–796, 2007, takes the data from isotope concentrations in ice
cores. In contrast, J. D. Hays, J. Imbrie & N. J. Shackleton, Variations in the Earth’s
orbit: pacemaker of the ice ages, Science 194, pp. 1121–1132, 1976, confirmed the connection
with orbital parameters by literally digging in the mud that covers the ocean floor in certain
places. Note that the web is full of information on the ice ages. Just look up ‘Milankovitch’
in a search engine. Cited on pages 153 and 154.
126 R. Humphreys & J. Larsen, The sun’s distance above the galactic plane, Astronomical
Journal 110, pp. 2183–2188, November 1995. Cited on page 153.
536 bibliography

127 C. L. Bennet, M. S. Turner & M. White, The cosmic rosetta stone, Physics Today 50,
pp. 32–38, November 1997. Cited on page 155.
128 The website www.geoffreylandis.com/vacuum.html gives a description of what happened.
See also the www.geoffreylandis.com/ebullism.html and imagine.gsfc.nasa.gov/docs/
ask_astro/answers/970603.html websites. They all give details on the effects of vacuum on
humans. Cited on page 161.
129 R. McN. Alexander, Leg design and jumping technique for humans, other vertebrates
and insects, Philosophical Transactions of the Royal Society in London B 347, pp. 235–249,
1995. Cited on page 169.
130 J. W. Glasheen & T. A. McMahon, A hydrodynamic model of locomotion in the basilisk
lizard, Nature 380, pp. 340–342, For pictures, see also New Scientist, p. 18, 30 March 1996,
or Scientific American, pp. 48–49, September 1997, or the website by the author at rjf2.biol.
berkeley.edu/Full_Lab/FL_Personnel/J_Glasheen/J_Glasheen.html.
Several shore birds also have the ability to run over water, using the same mechanism.
Cited on page 169.
131 A. Fernandez-Nieves & F. J. de las Nieves, About the propulsion system of a kayak

Motion Mountain – The Adventure of Physics
and of Basiliscus basiliscus, European Journal of Physics 19, pp. 425–429, 1998. Cited on
page 170.
132 Y. S. Song, S. H. Suhr & M. Sitti, Modeling of the supporting legs for designing biomi-
metic water strider robot, Proceedings of the IEEE International Conference on Robotics
and Automation, Orlando, USA, 2006. S. H. Suhr, Y. S. Song, S. J. Lee & M. Sitti,
Biologically inspired miniature water strider robot, Proceedings of the Robotics: Science and
Systems I, Boston, USA, 2005. See also the website www.me.cmu.edu/faculty1/sitti/nano/
projects/waterstrider. Cited on page 170.
133 J. Iriarte-Díaz, Differential scaling of locomotor performance in small and large ter-
restrial mammals, The Journal of Experimental Biology 205, pp. 2897–2908, 2002. Cited

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
on pages 171 and 560.
134 M. Wittlinger, R. Wehner & H. Wolf, The ant odometer: stepping on stilts and
stumps, Science 312, pp. 1965–1967, 2006. Cited on page 171.
135 P. G. Weyand, D. B. Sternlight, M. J. Bellizzi & S. Wright, Faster top running
speeds are achieved with greater ground forces not more rapid leg movements, Journal of Ap-
plied Physiology 89, pp. 1991–1999, 2000. Cited on page 171.
136 The material on the shadow discussion is from the book by Robert M. Pryce, Cook and
Peary, Stackpole Books, 1997. See also the details of Peary’s forgeries in Wally Herbert,
The Noose of Laurels, Doubleday 1989. The sad story of Robert Peary is also told in the
centenary number of National Geographic, September 1988. Since the National Geographic
Society had financed Peary in his attempt and had supported him until the US Congress had
declared him the first man at the Pole, the (partial) retraction is noteworthy. (The magazine
then changed its mind again later on, to sell more copies, and now again claims that Peary
reached the North Pole.) By the way, the photographs of Cook, who claimed to have been
at the North Pole even before Peary, have the same problem with the shadow length. Both
men have a history of cheating about their ‘exploits’. As a result, the first man at the North
Pole was probably Roald Amundsen, who arrived there a few years later, and who was also
the first man at the South Pole. Cited on page 135.
137 The story is told in M. Nauenberg, Hooke, orbital motion, and Newton’s Principia, Amer-
ican Journal of Physics 62, 1994, pp. 331–350. Cited on page 178.
138 More details are given by D. Rawlins, in Doubling your sunsets or how anyone can meas-
bibliography 537

ure the Earth’s size with wristwatch and meter stick, American Journal of Physics 47, 1979,
pp. 126–128. Another simple measurement of the Earth radius, using only a sextant, is given
by R. O ’ Keefe & B. Ghavimi-Alagha, in The World Trade Center and the distance to
the world’s center, American Journal of Physics 60, pp. 183–185, 1992. Cited on page 179.
139 More details on astronomical distance measurements can be found in the beautiful little
book by A. van Helden, Measuring the Universe, University of Chicago Press, 1985, and
in Nigel Henbest & Heather Cooper, The Guide to the Galaxy, Cambridge Univer-
sity Press, 1994. Cited on page 179.
140 A lot of details can be found in M. Jammer, Concepts of Mass in Classical and Modern
Physics, reprinted by Dover, 1997, and in Concepts of Force, a Study in the Foundations of
Mechanics, Harvard University Press, 1957. These eclectic and thoroughly researched texts
provide numerous details and explain various philosophical viewpoints, but lack clear state-
ments and conclusions on the accurate description of nature; thus are not of help on fun-
damental issues.
Jean Buridan (c. 1295 to c. 1366) criticizes the distinction of sublunar and translunar mo-
tion in his book De Caelo, one of his numerous works. Cited on page 180.

Motion Mountain – The Adventure of Physics
141 D. Topper & D. E. Vincent, An analysis of Newton’s projectile diagram, European
Journal of Physics 20, pp. 59–66, 1999. Cited on page 180.
142 The absurd story of the metre is told in the historical novel by Ken Alder, The Measure
of All Things : The Seven-Year Odyssey and Hidden Error that Transformed the World, The
Free Press, 2003. Cited on page 183.
143 H. Cavendish, Experiments to determine the density of the Earth, Philosophical Trans-
actions of the Royal Society 88, pp. 469–526, 1798. In fact, the first value of the gravita-
tional constant 𝐺 found in the literature is only from 1873, by Marie-Alfred Cornu and
Jean-Baptistin Baille, who used an improved version of Cavendish’s method. Cited on page
185.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
144 About the measurement of spatial dimensions via gravity – and the failure to find any hint
for a number different from three – see the review by E. G. Adelberger, B. R. Heckel
& A. E. Nelson, Tests of the gravitational inverse-square law, Annual Review of Nuclear
and Particle Science 53, pp. 77–121, 2003, also arxiv.org/abs/hep-ph/0307284, or the re-
view by J. A. Hewett & M. Spiropulu, Particle physics probes of extra spacetime di-
mensions, Annual Review of Nuclear and Particle Science 52, pp. 397–424, 2002, arxiv.org/
abs/hep-ph/0205106. Cited on page 188.
145 There are many books explaining the origin of the precise shape of the Earth, such as the
pocket book S. Anders, Weil die Erde rotiert, Verlag Harri Deutsch, 1985. Cited on page
188.
146 The shape of the Earth is described most precisely with the World Geodetic System. For
a presentation, see the en.wikipedia.org/wiki/World_Geodetic_System and www.dqts.net/
wgs84.htm websites. See also the website of the International Earth Rotation Service at
hpiers.obspm.fr. Cited on page 188.
147 G. Heckman & M. van Haandel, De vele beweijzen van Kepler’s wet over ellipsen-
banen: een nieuwe voor ‘het Boek’?, Nederlands tijdschrift voor natuurkunde 73, pp. 366–
368, November 2007. Cited on page 177.
148 W. K. Hartman, R. J. Phillips & G. J. Taylor, editors, Origin of the Moon, Lunar
and Planetary Institute, 1986. Cited on page 191.
149 If you want to read about the motion of the Moon in all its fascinating details, have a look
at Martin C. Gutzwiller, Moon–Earth–Sun: the oldest three body problem, Reviews
538 bibliography

of Modern Physics 70, pp. 589–639, 1998. Cited on page 191.
150 Dietrich Neumann, Physiologische Uhren von Insekten – Zur Ökophysiologie lunarperi-
odisch kontrollierter Fortpflanzungszeiten, Naturwissenschaften 82, pp. 310–320, 1995. Cited
on page 192.
151 The origin of the duration of the menstrual cycle is not yet settled; however, there are ex-
planations on how it becomes synchronized with other cycles. For a general explanation see
Arkady Pikovsky, Michael Rosenblum & Jürgen Kurths, Synchronization: A
Universal Concept in Nonlinear Science, Cambridge University Press, 2002. Cited on page
192.
152 J. Laskar, F. Joutel & P. Robutel, Stability of the Earth’s obliquity by the moon,
Nature 361, pp. 615–617, 1993. However, the question is not completely settled, and other
opinions exist. Cited on page 192.
153 Neil F. Comins, What if the Moon Did not Exist? – Voyages to Earths that Might Have
Been, Harper Collins, 1993. Cited on page 192.
154 A recent proposal is M. Ćuk, D. P. Hamilton, S. J. Lock & S. T. Stewart, Tidal
evolution of the Moon from a high-obliquity, high-angular-momentum Earth, Nature 539,

Motion Mountain – The Adventure of Physics
pp. 402–406, 2016. Cited on page 192.
155 M. Connors, C. Veillet, R. Brasser, P. A. Wiegert, P. W. Chodas,
S. Mikkola & K. A. Innanen, Discovery of Earth’s quasi-satellite, Meteoritics
& Planetary Science 39, pp. 1251–1255, 2004, and R. Brasser, K. A. Innanen,
M. Connors, C. Veillet, P. A. Wiegert, S. Mikkola & P. W. Chodas, Tran-
sient co-orbital asteroids, Icarus 171, pp. 102–109, 2004. See also the orbits drawn
in M. Connors, C. Veillet, R. Brasser, P. A. Wiegert, P. W. Chodas,
S. Mikkola & K. A. Innanen, Horseshoe asteroids and quasi-satellites in Earth-like
orbits, Lunar and Planetary Science 35, p. 1562, 2004,, preprint at www.lpi.usra.edu/
meetings/lpsc2004/pdf/1565.pdf. Cited on page 195.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
156 P. A. Wiegert, K. A. Innanen & S. Mikkola, An asteroidal companion to the Earth,
Nature 387, pp. 685–686, 12 June 1997, together with the comment on pp. 651–652. Details
on the orbit and on the fact that Lagrangian points do not always form equilateral triangles
can be found in F. Namouni, A. A. Christou & C. D. Murray, Coorbital dynamics
at large eccentricity and inclination, Physical Review Letters 83, pp. 2506–2509, 1999. Cited
on page 194.
157 Simon Newcomb, Astronomical Papers of the American Ephemeris 1, p. 472, 1882. Cited
on page 197.
158 For an animation of the tides, have a look at www.jason.oceanobs.com/html/applications/
marees/m2_atlantique_fr.html. Cited on page 197.
159 A beautiful introduction is the classic G. Falk & W. Ruppel, Mechanik, Relativität, Grav-
itation – ein Lehrbuch, Springer Verlag, Dritte Auflage, 1983. Cited on page 197.
160 J. Soldner, Berliner Astronomisches Jahrbuch auf das Jahr 1804, 1801, p. 161. Cited on
page 201.
161 The equality was first tested with precision by R. von Eötvös, Annalen der Physik &
Chemie 59, p. 354, 1896, and by R. von Eötvös, V. Pekár, E. Fekete, Beiträge zum
Gesetz der Proportionalität von Trägheit und Gravität, Annalen der Physik 4, Leipzig 68,
pp. 11–66, 1922. He found agreement to 5 parts in 109 . More experiments were performed by
P. G. Roll, R. Krotkow & R. H. Dicke, The equivalence of inertial and passive grav-
itational mass, Annals of Physics (NY) 26, pp. 442–517, 1964, one of the most interesting
and entertaining research articles in experimental physics, and by V. B. Braginsky &
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V. I. Panov, Soviet Physics – JETP 34, pp. 463–466, 1971. Modern results, with errors less
than one part in 1012 , are by Y. Su & al., New tests of the universality of free fall, Phys-
ical Review D50, pp. 3614–3636, 1994. Several experiments have been proposed to test the
equality in space to less than one part in 1016 . Cited on page 202.
162 H. Edelmann, R. Napiwotzki, U. Heber, N. Christlieb & D. Reimers, HE
0437-5439: an unbound hyper-velocity B-type star, The Astrophysical Journal 634, pp. L181–
L184, 2005. Cited on page 210.
163 This is explained for example by D.K. Firpić & I.V. Aniçin, The planets, after all, may
run only in perfect circles – but in the velocity space!, European Journal of Physics 14, pp. 255–
258, 1993. Cited on pages 210 and 504.
164 See L. Hodges, Gravitational field strength inside the Earth, American Journal of Physics
59, pp. 954–956, 1991. Cited on page 211.
165 The controversial argument is proposed in A. E. Chubykalo & S. J. Vlaev, Theorem
on the proportionality of inertial and gravitational masses in classical mechanics, European
Journal of Physics 19, pp. 1–6, 1998, preprint at arXiv.org/abs/physics/9703031. This paper
might be wrong; see the tough comment by B. Jancovici, European Journal of Physics

Motion Mountain – The Adventure of Physics
19, p. 399, 1998, and the reply in arxiv.org/abs/physics/9805003. Cited on page 212.
166 P. Mohazzabi & M. C. James, Plumb line and the shape of the Earth, American Journal
of Physics 68, pp. 1038–1041, 2000. Cited on page 212.
167 From Neil de Gasse Tyson, The Universe Down to Earth, Columbia University Press,
1994. Cited on page 213.
168 G. D. Quinlan, Planet X: a myth exposed, Nature 363, pp. 18–19, 1993. Cited on page 213.
169 See en.wikipedia.org/wiki/90377_Sedna. Cited on page 214.
170 See R. Matthews, Not a snowball’s chance ..., New Scientist 12 July 1997, pp. 24–27. The
original claim is by Louis A. Frank, J. B. Sigwarth & J. D. Craven, On the influx of

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
small comets into the Earth’s upper atmosphere, parts I and II, Geophysical Research Letters
13, pp. 303–306, pp. 307–310, 1986. The latest observations have disproved the claim. Cited
on page 215.
171 The ray form is beautifully explained by J. Evans, The ray form of Newton’s law of motion,
American Journal of Physics 61, pp. 347–350, 1993. Cited on page 216.
172 This is a small example from the beautiful text by Mark P. Silverman, And Yet It Moves:
Strange Systems and Subtle Questions in Physics, Cambridge University Press, 1993. It is a
treasure chest for anybody interested in the details of physics. Cited on page 217.
173 G. -L. Lesage, Lucrèce Newtonien, Nouveaux mémoires de l’Académie Royale des
Sciences et Belles Lettres pp. 404–431, 1747, or www3.bbaw.de/bibliothek/digital/
struktur/03-nouv/1782/jpg-0600/00000495.htm. See also en.wikipedia.org/wiki/
Le_Sage’s_theory_of_gravitation. In fact, the first to propose the idea of gravitation as
a result of small particles pushing masses around was Nicolas Fatio de Duillier in 1688.
Cited on page 218.
174 J. Laskar, A numerical experiment on the chaotic behaviour of the solar system, Nature
338, pp. 237–238, 1989, and J. Laskar, The chaotic motion of the solar system - A numerical
estimate of the size of the chaotic zones, Icarus 88, pp. 266–291, 1990. The work by Laskar was
later expanded by Jack Wisdom, using specially built computers, following only the planets,
without taking into account the smaller objects. For more details, see G. J. Sussman &
J. Wisdom, Chaotic Evolution of the Solar System, Science 257, pp. 56–62, 1992. Today,
such calculations can be performed on your home PC with computer code freely available
on the internet. Cited on page 219.
540 bibliography

175 B. Dubrulle & F. Graner, Titius-Bode laws in the solar system. 1: Scale invariance ex-
plains everything, Astronomy and Astrophysics 282, pp. 262–268, 1994, and Titius-Bode
laws in the solar system. 2: Build your own law from disk models, Astronomy and Astro-
physics 282, pp. 269–-276, 1994. Cited on page 221.
176 M. Lecar, Bode’s Law, Nature 242, pp. 318–319, 1973, and M. Henon, A comment on
“The resonant structure of the solar system” by A.M. Molchanov, Icarus 11, pp. 93–94, 1969.
Cited on page 221.
177 Cassius Dio, Historia Romana, c. 220, book 37, 18. For an English translation, see the site
penelope.uchicago.edu/Thayer/E/Roman/Texts/Cassius_Dio/37*.html. Cited on page 221.
178 See the beautiful paper A. J. Simoson, Falling down a hole through the Earth, Mathematics
Magazine 77, pp. 171–188, 2004. See also A. J. Simoson, The gravity of Hades, 75, pp. 335–
350, 2002. Cited on pages 223 and 506.
179 M. Bevis, D. Alsdorf, E. Kendrick, L. P. Fortes, B. Forsberg, R. Malley &
J. Becker, Seasonal fluctuations in the mass of the Amazon River system and Earth’s elastic
response, Geophysical Research Letters 32, p. L16308, 2005. Cited on page 223.
180 D. Hestenes, M. Wells & G. Swackhamer, Force concept inventory, Physics

Motion Mountain – The Adventure of Physics
Teacher 30, pp. 141–158, 1982. The authors developed tests to check the understanding
of the concept of physical force in students; the work has attracted a lot of attention in the
field of physics teaching. Cited on page 229.
181 For a general overview on friction, from physics to economics, architecture and organiz-
ational theory, see N. Åkerman, editor, The Necessity of Friction – Nineteen Essays on a
Vital Force, Springer Verlag, 1993. Cited on page 234.
182 See M. Hirano, K. Shinjo, R. Kanecko & Y. Murata, Observation of superlubricity
by scanning tunneling microscopy, Physical Review Letters 78, pp. 1448–1451, 1997. See also
the discussion of their results by Serge Fayeulle, Superlubricity: when friction stops,
Physics World pp. 29–30, May 1997. Cited on page 234.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
183 C. Donald Ahrens, Meteorology Today: An Introduction to the Weather, Climate, and
the Environment, West Publishing Company, 1991. Cited on page 235.
184 This topic is discussed with lucidity by J. R. Mureika, What really are the best 100 m per-
formances?, Athletics: Canada’s National Track and Field Running Magazine, July 1997. It
can also be found as arxiv.org/abs/physics/9705004, together with other papers on similar
topics by the same author. Cited on page 235.
185 F. P. B owden & D. Tabor, The Friction and Lubrication of Solids, Oxford University
Press, Part I, revised edition, 1954, and part II, 1964. Cited on page 236.
186 A powerful book on human violence is James Gilligan, Violence – Our Deadly Epidemic
and its Causes, Grosset/Putnam, 1992. Cited on page 236.
187 The main tests of randomness of number series – among them the gorilla test – can be found
in the authoritative paper by G. Marsaglia & W. W. Tsang, Some difficult-to-pass tests
of randomness, Journal of Statistical Software 7, p. 8, 2002. It can also be downloaded from
www.jstatsoft.org/v07/i03. Cited on page 239.
188 See the interesting book on the topic by Jaroslaw Strzalko, Juliusz Grabski,
Przemyslaw Perlikowski, Andrzeij Stefanski & Tomasz Kapitaniak, Dy-
namics of Gambling: Origin of Randomness in Mechanical Systems, Springer, 2009, as well
as the more recent publications by Kapitaniak. Cited on page 239.
189 For one aspect of free will, see the captivating book by Bert Hellinger, Zweierlei Glück,
Carl Auer Systeme Verlag, 1997. The author explains how to live serenely and with the
bibliography 541

highest possible responsibility for one’s actions, by reducing entanglements with the des-
tiny of others. He describes a powerful technique to realise this goal.
A completely different viewpoint is given by Aung San Suu Kyi, Freedom from Fear,
Penguin, 1991. One of the bravest women on Earth, she won the Nobel Peace Price in 1991.
An effective personal technique is presented by Phil Stutz & Barry Michels, The
Tools, Spiegel & Grau, 2012. Cited on page 242.
190 Henrik Walter, Neurophilosophie der Willensfreiheit, Mentis Verlag, Paderborn 1999.
Also available in English translation. Cited on page 242.
191 Giuseppe Fumagalli, Chi l’ha detto?, Hoepli, 1983. Cited on page 242.
192 See the tutorial on the Peaucellier-Lipkin linkage by D.W. Henderson and D. Taimina
found on kmoddl.library.cornell.edu/tutorials/11/index.php. The internet contains many
other pages on the topic. Cited on page 243.
193 The beautiful story of the south-pointing carriage is told in Appendix B of James Foster
& J. D. Nightingale, A Short Course in General Relativity, Springer Verlag, 2nd edition,
1998. Such carriages have existed in China, as told by the great sinologist Joseph Needham,
but their construction is unknown. The carriage described by Foster and Nightingale is the

Motion Mountain – The Adventure of Physics
one reconstructed in 1947 by George Lancaster, a British engineer. Cited on page 244.
194 T. Van de Kamp, P. Vagovic, T. Baumbach & A. Riedel, A biological screw in a
beetle’s leg, Science 333, p. 52, 2011. Cited on page 244.
195 M. Burrows & G. P. Sutton, Interacting gears synchronise propulsive leg movements in
a jumping insect, Science 341, pp. 1254–1256, 2013. Cited on page 244.
196 See for example Z. Ghahramani, Building blocks of movement, Nature 407, pp. 682–683,
2000. Researchers in robot control are also interested in such topics. Cited on page 244.
197 G. Gutierrez, C. Fehr, A. Calzadilla & D. Figueroa, Fluid flow up the wall of
a spinning egg, American Journal of Physics 66, pp. 442–445, 1998. Cited on page 244.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
198 A historical account is given in Wolf gang Yourgray & Stanley Mandelstam,
Variational Principles in Dynamics and Quantum Theory, Dover, 1968. Cited on pages 248
and 256.
199 C. G. Gray & E. F. Taylor, When action is not least, American Journal of Physics 75,
pp. 434–458, 2007. Cited on page 253.
200 Max Päsler, Prinzipe der Mechanik, Walter de Gruyter & Co., 1968. Cited on page 254.
201 The relations between possible Lagrangians are explained by Herbert Goldstein,
Classical Mechanics, 2nd edition, Addison-Wesley, 1980. Cited on page 255.
202 The Hemingway statement is quoted by Marlene Dietrich in Aaron E. Hotchner, Papa
Hemingway, Random House, 1966, in part 1, chapter 1. Cited on page 256.
203 C. G. Gray, G. Karl & V. A. Novikov, From Maupertius to Schrödinger. Quantization
of classical variational principles, American Journal of Physics 67, pp. 959–961, 1999. Cited
on page 257.
204 J. A. Moore, An innovation in physics instruction for nonscience majors, American Journal
of Physics 46, pp. 607–612, 1978. Cited on page 257.
205 See e.g. Alan P. B oss, Extrasolar planets, Physics Today 49, pp. 32–38. September 1996.
The most recent information can be found at the ‘Extrasolar Planet Encyclopaedia’ main-
tained at www.obspm.fr/planets by Jean Schneider at the Observatoire de Paris. Cited on
page 261.
206 A good review article is by David W. Hughes, Comets and Asteroids, Contemporary
Physics 35, pp. 75–93, 1994. Cited on page 261.
542 bibliography

207 G. B. West, J. H. Brown & B. J. Enquist, A general model for the origin of allometric
scaling laws in biology, Science 276, pp. 122–126, 4 April 1997, with a comment on page
34 of the same issue. The rules governing branching properties of blood vessels, of lymph
systems and of vessel systems in plants are explained. For more about plants, see also the
paper G. B. West, J. H. Brown & B. J. Enquist, A general model for the structure and
allometry of plant vascular systems, Nature 400, pp. 664–667, 1999. Cited on page 263.
208 J. R. Banavar, A. Martin & A. Rinaldo, Size and form in efficient transportation net-
works, Nature 399, pp. 130–132, 1999. Cited on page 263.
209 N. Moreira, New striders - new humanoids with efficient gaits change the robotics land-
scape, Science News Online 6th of August, 2005. Cited on page 264.
210 Werner Heisenberg, Der Teil und das Ganze, Piper, 1969. Cited on page 266.
211 See the clear presenttion by E. H. Lockwood & R. H. Macmillan, Geometric Sym-
metry, Cambridge University Press, 1978. Cited on page 266.
212 John Mansley Robinson, An Introduction to Early Greek Philosophy, Houghton
Muffin 1968, chapter 5. Cited on page 270.
213 See e.g. B. B ower, A child’s theory of mind, Science News 144, pp. 40–41. Cited on page

Motion Mountain – The Adventure of Physics
271.
214 The most beautiful book on this topic is the text by Branko Grünbaum &
G. C. Shephard, Tilings and Patterns, W.H. Freeman and Company, New York, 1987. It
has been translated into several languages and republished several times. Cited on page
273.
215 About tensors and ellipsoids in three-dimensional space, see mysite.du.edu/~jcalvert/phys/
ellipso.htm. In four-dimensional space-time, tensors are more abstract to comprehend.
With emphasis on their applications in relativity, such tensors are explained in R. Frosch,
Four-tensors, the mother tongue of classical physics, vdf Hochschulverlag, 2006, partly avail-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
able on books.google.com. Cited on page 278.
216 U. Niederer, The maximal kinematical invariance group of the free Schrödinger equa-
tion, Helvetica Physica Acta 45, pp. 802–810, 1972. See also the introduction by O. Jahn
& V. V. Sreedhar, The maximal invariance group of Newton’s equations for a free point
particle, arxiv.org/abs/math-ph/0102011. Cited on page 279.
217 The story is told in the interesting biography of Einstein by A. Pais, ‘Subtle is the Lord...’
– The Science and the Life of Albert Einstein, Oxford University Press, 1982. Cited on page
280.
218 W. Zürn & R. Widmer-Schnidrig, Globale Eigenschwingungen der Erde, Physik
Journal 1, pp. 49–55, 2002. Cited on page 290.
219 N. Gauthier, What happens to energy and momentum when two oppositely-moving wave
pulses overlap?, American Journal of Physics 71, pp. 787–790, 2003. Cited on page 300.
220 An informative and modern summary of present research about the ear and the details of
its function is www.physicsweb.org/article/world/15/5/8. Cited on page 303.
221 A renowned expert of the physics of singing is Ingo Titze. Among his many books
and papers is the popular introduction I. R. Titze, The human instrument, Scientific
American pp. 94–101, January 2008. Several of his books, papers and presentation
are free to download on the website www.ncvs.org of the National Center of Voice &
Speech. They are valuable to everybody who has a passion for singing and the human
voice. See also the article and sound clips at www.scientificamerican.com/article.cfm?
id=sound-clips-human-instrument. An interesting paper is also M. Kob & al., Ana-
bibliography 543

lysing and understanding the singing voice: recent progress and open questions, Current
Bioinformatics 6, pp. 362–374, 2011. Cited on page 308.
222 S. Adachi, Principles of sound production in wind instruments, Acoustical Science and
Technology 25, pp. 400–404, 2004. Cited on page 309.
223 The literature on tones and their effects is vast. For example, people have explored the
differences and effects of various intonations in great detail. Several websites, such as
bellsouthpwp.net/j/d/jdelaub/jstudio.htm, allow listening to music played with different in-
tonations. People have even researched whether animals use just or chromatic intonation.
(See, for example, K. Leutw yler, Exploring the musical brain, Scientific American Janu-
ary 2001.) There are also studies of the effects of low frequencies, of beat notes, and of many
other effects on humans. However, many studies mix serious and non-serious arguments.
It is easy to get lost in them. Cited on page 309.
224 M. Fatemi, P. L. Ogburn & J. F. Greenleaf, Fetal stimulation by pulsed diagnostic ul-
trasound, Journal of Ultrasound in Medicine 20, pp. 883–889, 2001. See also M. Fatemi,
A. Alizad & J. F. Greenleaf, Characteristics of the audio sound generated by ultra-
sound imaging systems, Journal of the Acoustical Society of America 117, pp. 1448–1455,

Motion Mountain – The Adventure of Physics
2005. Cited on page 313.
225 I know a female gynecologist who, during her own pregnancy, imaged her child every day
with her ultrasound machine. The baby was born with strong hearing problems that did
not go away. Cited on page 313.
226 R. Mole, Possible hazards of imaging and Doppler ultrasound in obstetrics, Birth Issues in
Perinatal Care 13, pp. 29–37, 2007. Cited on page 313.
227 A. L. Hodgkin & A. F. Huxley, A quantitative description of membrane current and its
application to conduction and excitation in nerve, Journal of Physiology 117, pp. 500–544,
1952. This famous paper of theoretical biology earned the authors the Nobel Prize in Medi-
cine in 1963. Cited on page 315.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
228 T. Filippov, The Versatile Soliton, Springer Verlag, 2000. See also J. S. Russel, Report
of the Fourteenth Meeting of the British Association for the Advancement of Science, Murray,
London, 1844, pp. 311–390. Cited on pages 316 and 318.
229 R. S. Ward, Solitons and other extended field configurations, preprint at arxiv.org/abs/
hep-th/0505135. Cited on page 318.
230 D. B. Bahr, W. T. Pfeffer & R. C. Browning, The surprising motion of ski moguls,
Physics Today 62, pp. 68–69, November 2009. Cited on page 319.
231 N. J. Zabusky & M. D. Kruskal, Interaction of solitons in a collisionless plasma and the
recurrence of initial states, Physical Review Letters 15, pp. 240–243, 1965. Cited on page 317.
232 O. Muskens, De kortste knal ter wereld, Nederlands tijdschrift voor natuurkunde pp. 70–
73, 2005. Cited on page 318.
233 E. Heller, Freak waves: just bad luck, or avoidable?, Europhysics News pp. 159–161,
September/October 2005, downloadable at www.europhysicsnews.org. Cited on page 320.
234 See the beautiful article by D. Aarts, M. Schmidt & H. Lekkerkerker, Dir-
ecte visuele waarneming van thermische capillaire golven, Nederlands tijdschrift voor
natuurkunde 70, pp. 216–218, 2004. Cited on page 321.
235 For more about the ocean sound channel, see the novel by Tom Clancy, The Hunt for Red
October. See also the physics script by R. A. Muller, Government secrets of the oceans,
atmosphere, and UFOs, web.archive.org/web/*/muller.lbl.gov/teaching/Physics10/chapters/
9-SecretsofUFOs.html 2001. Cited on page 322.
544 bibliography

236 B. Wilson, R. S. Batty & L. M. Dill, Pacific and Atlantic herring produce burst pulse
sounds, Biology Letters 271, number S3, 7 February 2004. Cited on page 323.
237 A. Chabchoub & M. Fink, Time-reversal generation of rogue Wwaves, Physical Review
Letters 112, p. 124101, 2014, preprint at arxiv.org/abs/1311.2990. See also the cited references.
Cited on page 324.
238 See for example the article by G. Fritsch, Infraschall, Physik in unserer Zeit 13, pp. 104–
110, 1982. Cited on page 325.
239 Wavelet transformations were developed by the French mathematicians Alex Grossmann,
Jean Morlet and Thierry Paul. The basic paper is A. Grossmann, J. Morlet & T. Paul,
Integral transforms associated to square integrable representations, Journal of Mathematical
Physics 26, pp. 2473–2479, 1985. For a modern introduction, see Stéphane Mallat, A
Wavelet Tour of Signal Processing, Academic Press, 1999. Cited on page 326.
240 P. Manogg, Knall und Superknall beim Überschallflug, Der mathematische und naturwis-
senschaftliche Unterricht 35, pp. 26–33, 1982, my physics teacher in secondary school. Cited
on page 326.
241 See the excellent introduction by L. Ellerbreok & L. van den Hoorn, In het kielzog

Motion Mountain – The Adventure of Physics
van Kelvin, Nederlands tijdschrift voor natuurkunde 73, pp. 310–313, 2007. About ex-
ceptions to the Kelvin angle, see www.graingerdesigns.net/oshunpro/design-technology/
wave-cancellation. Cited on page 327.
242 Jay Ingram, The Velocity of Honey - And More Science of Everyday Life, Viking, 2003. See
also W. W. L. Au & J. A. Simmons, Echolocation in dolphins and bats, Physics Today 60,
pp. 40–45, 2007. Cited on page 328.
243 M. B oiti, J. -P. Leon, L. Martina & F. Pempinelli, Scattering of localized solitons
in the plane, Physics Letters A 132, pp. 432–439, 1988, A. S. Fokas & P. M. Santini,
Coherent structures in multidimensions, Physics Review Letters 63, pp. 1329–1333, 1989,

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
J. Hietarinta & R. Hirota, Multidromion solutions to the Davey–Stewartson equation,
Physics Letters A 145, pp. 237–244, 1990. Cited on page 328.
244 For some of this fascinating research, see J. L. Hammack, D. M. Henderson &
H. Segur, Progressive waves with persistent two-dimensional surface patterns in deep
water, Journal of Fluid Mechanics 532, pp. 1–52, 2005. For a beautiful photograph of
crossing cnoidal waves, see A. R. Osborne, Hyperfast Modeling of Shallow-Water Waves:
The KdV and KP Equations, International Geophysics 97, pp. 821–856, 2010. See also en.
wikipedia.org/wiki/Waves_and_shallow_water, en.wikipedia.org/wiki/Cnoidal_wave and
en.wikipedia.org/wiki/Tidal_bore for a first impression. Cited on page 329.
245 The sound frequency change with bottle volume is explained on hyperphysics.phy-astr.gsu.
edu/Hbase/Waves/cavity.html. Cited on page 329.
246 A passionate introduction is Neville H. Fletcher & Thomas D. Rossing, The
Physics of Musical Instruments, second edition, Springer 2000. Cited on page 330.
247 M. Ausloos & D. H. Berman, Multivariate Weierstrass–Mandelbrot function, Proceed-
ings of the Royal Society in London A 400, pp. 331–350, 1985. Cited on page 333.
248 Catechism of the Catholic Church, Part Two, Section Two, Chapter One, Article 3, state-
ments 1376, 1377 and 1413, found at www.vatican.va/archive/ENG0015/__P41.HTM or
www.vatican.va/archive/ITA0014/__P40.HTM with their explanations on www.vatican.va/
archive/compendium_ccc/documents/archive_2005_compendium-ccc_en.html and www.
vatican.va/archive/compendium_ccc/documents/archive_2005_compendium-ccc_it.
html. Cited on page 336.
bibliography 545

249 The original text of the 1633 conviction of Galileo can be found on it.wikisource.org/wiki/
Sentenza_di_condanna_di_Galileo_Galilei. Cited on page 336.
250 The retraction that Galileo was forced to sign in 1633 can be found on it.wikisource.org/
wiki/Abiura_di_Galileo_Galilei. Cited on page 336.
251 M. Artigas, Un nuovo documento sul caso Galileo: EE 291, Acta Philosophica 10, pp. 199–
214, 2001. Cited on page 337.
252 Most of these points are made, directly or indirectly, in the book by Annibale Fantoli,
Galileo: For Copernicanism and for the Church, Vatican Observatory Publications, second
edition, 1996, and by George Coyne, director of the Vatican observatory, in his speeches
and publications, for example in G. Coyne, Galileo: for Copernicanism and for the church,
Zwoje 3/36, 2003, found at www.zwoje-scrolls.com/zwoje36/text05p.htm. Cited on page
337.
253 Thomas A. McMahon & John T. B onner, On Size and Life, Scientific Amer-
ican/Freeman, 1983. Another book by John Bonner, who won the Nobel Prize in Biology, is
John T. B onner, Why Size Matters: From Bacteria to Blue Whales, Princeton University
Press, 2011. Cited on page 337.

Motion Mountain – The Adventure of Physics
254 G. W. Koch, S. C. Sillett, G. M. Jennings & S. D. Davis, The limits to tree height,
Nature 428, pp. 851–854, 2004. Cited on page 338.
255 A simple article explaining the tallness of trees is A. Mineyev, Trees worthy of Paul
Bunyan, Quantum pp. 4–10, January–February 1994. (Paul Bunyan is a mythical giant lum-
berjack who is the hero of the early frontier pioneers in the United States.) Note that
the transport of liquids in trees sets no limits on their height, since water is pumped up
along tree stems (except in spring, when it is pumped up from the roots) by evaporation
from the leaves. This works almost without limits because water columns, when nucle-
ation is carefully avoided, can be put under tensile stresses of over 100 bar, corresponding
to 1000 m. See also P. Nobel, Plant Physiology, Academic Press, 2nd Edition, 1999. An

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
artificial tree – though extremely small – using the same mechanism was built and stud-
ied by T. D. Wheeler & A. D. Stroock, The transpiration of water at negative pres-
sures in a synthetic tree, Nature 455, pp. 208–212, 2008. See also N. M. Holbrook &
M. A. Zwieniecki, Transporting water to the top of trees, Physics Today pp. 76–77, 2008.
Cited on pages 338 and 359.
256 Such information can be taken from the excellent overview article by M. F. Ashby, On the
engineering properties of materials, Acta Metallurgica 37, pp. 1273–1293, 1989. The article
explains the various general criteria which determine the selection of materials, and gives
numerous tables to guide the selection. Cited on page 338.
257 See the beautiful paper by S. E. Virgo, Loschmidt’s number, Science Progress 27, pp. 634–
649, 1933. It is also freely available in HTML format on the internet. Cited on pages 340
and 342.
258 See the delightful paper by Peter Pesic, Estimating Avogadro’s number from skylight and
airlight, European Journal of Physics 26, pp. 183–187, 2005. The mistaken statement that the
blue colour is due to density fluctuations is dispelled in C. F. B ohren & A. B. Fraser,
Color of the Sky, Physics Teacher 238, pp. 267–272, 1985. It also explains that the variations
of the sky colour, like the colour of milk, are due to multiple scattering. Cited on pages 341
and 514.
259 For a photograph of a single barium atom – named Astrid – see Hans Dehmelt,
Experiments with an isolated subatomic particle at rest, Reviews of Modern Physics
62, pp. 525–530, 1990. For another photograph of a barium ion, see W. Neuhauser,
546 bibliography

M. Hohenstatt, P. E. Toschek & H. Dehmelt, Localized visible Ba+ mono-ion
oscillator, Physical Review A 22, pp. 1137–1140, 1980. See also the photograph on page 344.
Cited on page 344.
260 Holograms of atoms were first produced by Hans-Werner Fink & al., Atomic resolution
in lens-less low-energy electron holography, Physical Review Letters 67, pp. 1543–1546, 1991.
Cited on page 344.
261 A single–atom laser was built in 1994 by K. An, J. J. Childs, R. R. Dasari &
M. S. Feld, Microlaser: a laser with one atom in an optical resonator, Physical Review
Letters 73, p. 3375, 1994. Cited on page 344.
262 The photograph on the left of Figure 240 on page 344 is the first image that showed sub-
atomic structures (visible as shadows on the atoms). It was published by F. J. Giessibl,
S. Hembacher, H. Bielefeldt & J. Mannhart, Subatomic features on the silicon
(111)-(7x7) surface observed by atomic force microscopy, Science 289, pp. 422 – 425, 2000.
Cited on page 344.
263 See for example C. Schiller, A. A. Koomans, T. L. van Rooy, C. Schönenberger
& H. B. Elswijk, Decapitation of tungsten field emitter tips during sputter sharpening, Sur-

Motion Mountain – The Adventure of Physics
face Science Letters 339, pp. L925–L930, 1996. Cited on page 344.
264 U. Weierstall & J. C. H. Spence, An STM with time-of-flight analyzer for atomic spe-
cies identification, MSA 2000, Philadelphia, Microscopy and Microanalysis 6, Supplement
2, p. 718, 2000. Cited on page 345.
265 P. Krehl, S. Engemann & D. Schwenkel, The puzzle of whip cracking – uncovered by
a correlation of whip-tip kinematics with shock wave emission, Shock Waves 8, pp. 1–9, 1998.
The authors used high-speed cameras to study the motion of the whip. A new aspect has
been added by A. Goriely & T. McMillen, Shape of a cracking whip, Physical Review
Letters 88, p. 244301, 2002. This article focuses on the tapered shape of the whip. However,
the neglection of the tuft – a piece at the end of the whip which is required to make it crack

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
– in the latter paper shows that there is more to be discovered still. Cited on page 349.
266 Z. Sheng & K. Yamafuji, Realization of a Human Riding a Unicycle by a Robot, Proceed-
ings of the 1995 IEEE International Conference on Robotics and Automation, Vol. 2, pp. 1319–
1326, 1995. Cited on page 350.
267 On human unicycling, see Jack Wiley, The Complete Book of Unicycling, Lodi, 1984, and
Sebastian Hoeher, Einradfahren und die Physik, Reinbeck, 1991. Cited on page 351.
268 W. Thomson, Lecture to the Royal Society of Edinburgh, 18 February 1867, Proceedings of
the Royal Society in Edinborough 6, p. 94, 1869. Cited on page 351.
269 S. T. Thoroddsen & A. Q. Shen, Granular jets, Physics of Fluids 13, pp. 4–6, 2001, and
A. Q. Shen & S. T. Thoroddsen, Granular jetting, Physics of Fluids 14, p. S3, 2002,
Cited on page 351.
270 M. J. Hancock & J. W. M. Bush, Fluid pipes, Journal of Fluid Mechanics 466, pp. 285–
304, 2002. A. E. Hosoi & J. W. M. Bush, Evaporative instabilities in climbing films,
Journal of Fluid Mechanics 442, pp. 217–239, 2001. J. W. M. Bush & A. E. Hasha, On the
collision of laminar jets: fluid chains and fishbones, Journal of Fluid Mechanics 511, pp. 285–
310, 2004. Cited on page 354.
271 The present record for negative pressure in water was achieved by Q. Zheng,
D. J. Durben, G. H. Wolf & C. A. Angell, Liquids at large negative pressures: water
at the homogeneous nucleation limit, Science 254, pp. 829–832, 1991. Cited on page 359.
272 H. Maris & S. Balibar, Negative pressures and cavitation in liquid helium, Physics
Today 53, pp. 29–34, 2000. Sebastien Balibar has also written several popular books that
bibliography 547

are presented on his website www.lps.ens.fr/~balibar. Cited on page 359.
273 The present state of our understanding of turbulence is described by G. Falkovich &
K. P. Sreenivasan, Lessons from hydrodynamic turbulence, Physics Today 59, pp. 43–49,
2006. Cited on page 363.
274 K. Weltner, A comparison of explanations of aerodynamical lifting force, American
Journal of Physics 55, pp. 50–54, 1987, K. Weltner, Aerodynamic lifting force, The Physics
Teacher 28, pp. 78–82, 1990. See also the user.uni-frankfurt.de/~weltner/Flight/PHYSIC4.
htm and the www.av8n.com/how/htm/airfoils.html websites. Cited on page 363.
275 L. Lanotte, J. Mauer, S. Mendez, D. A. Fedosov, J. -M. Fromental,
V. Claveria, F. Nicoud, G. Gompper & M. Abkarian, Red cells’ dynamic mor-
phologies govern blood shear thinning under microcirculatory flow conditions, Proceedings
of the National Academy of Sciences of the United States of America 2016, preprint at arxiv.
org/abs/1608.03730. Cited on page 368.
276 S. Gekle, I. R. Peters, J. M. Gordillo, D. van der Meer & D. Lohse, Super-
sonic air flow due to solid-liquid impact, Physical Review Letters 104, p. 024501, 2010. Films
of the effect can be found at physics.aps.org/articles/v3/4. Cited on page 370.

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277 See the beautiful book by Rainer F. Foelix, Biologie der Spinnen, Thieme Verlag, 1996,
also available in an even newer edition in English as Rainer F. Foelix, Biology of Spiders,
Oxford University Press, third edition, 2011. Special fora dedicated only to spiders can be
found on the internet. Cited on page 370.
278 See the website www.esa.int/esaCP/SEMER89U7TG_index_0.html. Cited on page 370.
279 For a fascinating account of the passion and the techniques of apnoea diving, see Um-
berto Pelizzari, L’Homme et la mer, Flammarion, 1994. Palizzari cites and explains
the saying by Enzo Maiorca: ‘The first breath you take when you come back to the surface
is like the first breath with which you enter life.’ Cited on page 371.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
280 Lydéric B ocquet, The physics of stone skipping, American Journal of Physics
17, pp. 150–155, 2003. The present recod holder is Kurt Steiner, with 40 skips. See
pastoneskipping.com/steiner.htm and www.stoneskipping.com. The site www.yeeha.net/
nassa/guin/g2.html is by the a previous world record holder, Jerdome Coleman-McGhee.
Cited on page 373.
281 S. F. Kistler & L. E. Scriven, The teapot effect: sheetforming flows with deflection, wet-
ting, and hysteresis, Journal of Fluid Mechanics 263, pp. 19–62, 1994. Cited on page 377.
282 J. Walker, Boiling and the Leidenfrost effect, a chapter from David Halliday,
Robert Resnick & Jearl Walker, Fundamentals of Physics, Wiley, 2007. The chapter
can also be found on the internet as pdf file. Cited on page 377.
283 E. Hollander, Over trechters en zo ..., Nederlands tijdschrift voor natuurkunde 68,
p. 303, 2002. Cited on page 378.
284 S. Dorbolo, H. Caps & N. Vandewalle, Fluid instabilities in the birth and death of
antibubbles, New Journal of Physics 5, p. 161, 2003. Cited on page 379.
285 T. T. Lim, A note on the leapfrogging between two coaxial vortex rings at low Reynolds num-
bers, Physics of Fluids 9, pp. 239–241, 1997. Cited on page 379.
286 P. Aussillous & D. Quéré, Properties of liquid marbles, Proc. Roy. Soc. London 462,
pp. 973–999, 2006, and references therein. Cited on page 380.
287 Thermostatics and thermodynamics are difficult to learn also because the fields were not
discovered in a systematic way. See C. Truesdell, The Tragicomical History of Thermody-
namics 1822–1854, Springer Verlag, 1980. An excellent advanced textbook on thermostatics
548 bibliography

and thermodynamics is Linda Reichl, A Modern Course in Statistical Physics, Wiley, 2nd
edition, 1998. Cited on page 383.
288 Gas expansion was the main method used for the definition of the official temperature
scale. Only in 1990 were other methods introduced officially, such as total radiation thermo-
metry (in the range 140 K to 373 K), noise thermometry (2 K to 4 K and 900 K to 1235 K),
acoustical thermometry (around 303 K), magnetic thermometry (0.5 K to 2.6 K) and optical
radiation thermometry (above 730 K). Radiation thermometry is still the central method
in the range from about 3 K to about 1000 K. This is explained in detail in R. L. Rusby,
R. P. Hudson, M. Durieux, J. F. Schooley, P. P. M. Steur & C. A. Swenson,
The basis of the ITS-90, Metrologia 28, pp. 9–18, 1991. On the water boiling point see also
Ref. 316. Cited on pages 385, 550, and 555.
289 Other methods to rig lottery draws made use of balls of different mass or of balls that are
more polished. One example of such a scam was uncovered in 1999. Cited on page 384.
290 See for example the captivating text by Gino Segrè, A Matter of Degrees: What Temper-
ature Reveals About the Past and Future of Our Species, Planet and Universe, Viking, New
York, 2002. Cited on page 385.

Motion Mountain – The Adventure of Physics
291 D. Karstädt, F. Pinno, K. -P. Möllmann & M. Vollmer, Anschauliche
Wärmelehre im Unterricht: ein Beitrag zur Visualisierung thermischer Vorgänge, Praxis der
Naturwissenschaften Physik 5-48, pp. 24–31, 1999, K. -P. Möllmann & M. Vollmer,
Eine etwas andere, physikalische Sehweise - Visualisierung von Energieumwandlungen und
Strahlungsphysik für die (Hochschul-)lehre, Physikalische Bllätter 56, pp. 65–69, 2000,
D. Karstädt, K. -P. Möllmann, F. Pinno & M. Vollmer, There is more to see
than eyes can detect: visualization of energy transfer processes and the laws of radiation
for physics education, The Physics Teacher 39, pp. 371–376, 2001, K. -P. Möllmann &
M. Vollmer, Infrared thermal imaging as a tool in university physics education, European
Journal of Physics 28, pp. S37–S50, 2007. Cited on page 387.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
292 See for example the article by H. Preston-Thomas, The international temperature scale
of 1990 (ITS-90), Metrologia 27, pp. 3–10, 1990, and the errata H. Preston-Thomas,
The international temperature scale of 1990 (ITS-90), Metrologia 27, p. 107, 1990, Cited on
page 391.
293 For an overview, see Christian Enss & Siegfried Hunklinger, Low-Temperature
Physics, Springer, 2005. Cited on page 391.
294 The famous paper on Brownian motion which contributed so much to Einstein’s fame is
A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewe-
gung von in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik 17, pp. 549–
560, 1905. In the following years, Einstein wrote a series of further papers elaborating on this
topic. For example, he published his 1905 Ph.D. thesis as A. Einstein, Eine neue Bestim-
mung der Moleküldimensionen, Annalen der Physik 19, pp. 289–306, 1906, and he corrected
a small mistake in A. Einstein, Berichtigung zu meiner Arbeit: ‘Eine neue Bestimmung der
Moleküldimensionen’, Annalen der Physik 34, pp. 591–592, 1911, where, using new data, he
found the value 6.6 ⋅ 1023 for Avogadro’s number. However, five years before Smoluchow-
ski and Einstein, a much more practically-minded man had made the same calculations,
but in a different domain: the mathematician Louis Bachelier did so in his PhD about stock
options; this young financial analyst was thus smarter than Einstein. Cited on page 392.
295 The first experimental confirmation of the prediction was performed by J. Perrin,
Comptes Rendus de l’Académie des Sciences 147, pp. 475–476, and pp. 530–532, 1908. He
masterfully sums up the whole discussion in Jean Perrin, Les atomes, Librarie Félix Al-
can, Paris, 1913. Cited on page 394.
bibliography 549

296 Pierre Gaspard & al., Experimental evidence for microscopic chaos, Nature 394, p. 865,
27 August 1998. Cited on page 394.
297 An excellent introduction into the physics of heat is the book by Linda Reichl, A Mod-
ern Course in Statistical Physics, Wiley, 2nd edition, 1998. Cited on page 395.
298 On the bicycle speed record, see the website fredrompelberg.com. It also shows details of
the bicycle he used. On the drag effect of a motorbike behind a bicycle, see B. Blocken,
Y. Toparlar & T. Andrianne, Aerodynamic benefit for a cyclist by a following mo-
torcycle, Journal of Wind Engineering & Industrial Aerodynamics 2016, available for free
download at www.urbanphysics.net. Cited on page 381.
299 F. Herrmann, Mengenartige Größen im Physikunterricht, Physikalische Blätter 54,
pp. 830–832, September 1998. See also his lecture notes on general introductory phys-
ics on the website www.physikdidaktik.uni-karlsruhe.de/skripten. Cited on pages 232, 354,
and 395.
300 These points are made clearly and forcibly, as is his style, by N.G. van Kampen, Entropie,
Nederlands tijdschrift voor natuurkunde 62, pp. 395–396, 3 December 1996. Cited on page
398.

Motion Mountain – The Adventure of Physics
301 This is a disappointing result of all efforts so far, as Grégoire Nicolis always stresses in his
university courses. Seth Lloyd has compiled a list of 31 proposed definitions of complexity,
containing among others, fractal dimension, grammatical complexity, computational com-
plexity, thermodynamic depth. See, for example, a short summary in Scientific American
p. 77, June 1995. Cited on page 398.
302 Minimal entropy is discussed by L. Szilard, Über die Entropieverminderung in einem
thermodynamischen System bei Eingriffen intelligenter Wesen, Zeitschrift für Physik 53,
pp. 840–856, 1929. This classic paper can also be found in English translation in his col-
lected works. Cited on page 399.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
303 G. Cohen-Tannoudji, Les constantes universelles, Pluriel, Hachette, 1998. See also
L. Brillouin, Science and Information Theory, Academic Press, 1962. Cited on page 399.
304 H. W. Zimmermann, Particle entropies and entropy quanta IV: the ideal gas, the second
law of thermodynamics, and the P-t uncertainty relation, Zeitschrift für physikalische
Chemie 217, pp. 55–78, 2003, and H. W. Zimmermann, Particle entropies and entropy
quanta V: the P-t uncertainty relation, Zeitschrift für physikalische Chemie 217, pp. 1097–
1108, 2003. Cited on pages 399 and 400.
305 See for example A. E. Shalyt-Margolin & A. Ya. Tregubovich, Generalized
uncertainty relation in thermodynamics, arxiv.org/abs/gr-qc/0307018, or J. Uffink &
J. van Lith-van Dis, Thermodynamic uncertainty relations, Foundations of Physics 29,
p. 655, 1999. Cited on page 399.
306 B. Lavenda, Statistical Physics: A Probabilistic Approach, Wiley-Interscience, 1991. Cited
on pages 399 and 400.
307 The quote given is found in the introduction by George Wald to the text by
Lawrence J. Henderson, The Fitness of the Environment, Macmillan, New York,
1913, reprinted 1958. Cited on page 400.
308 A fascinating introduction to chemistry is the text by John Emsley, Molecules at an Ex-
hibition, Oxford University Press, 1998. Cited on page 401.
309 B. Polster, What is the best way to lace your shoes?, Nature 420, p. 476, 5 December 2002.
Cited on page 402.
550 bibliography

310 L. B oltzmann, Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmethe-
orie, Sitzungsberichte der königlichen Akademie der Wissenschaften in Wien 53, pp. 155–
220, 1866. Cited on page 403.
311 See for example, the web page www.snopes.com/science/cricket.asp. Cited on page 406.
312 H. de Lang, Moleculaire gastronomie, Nederlands tijdschrift voor natuurkunde 74,
pp. 431–433, 2008. Cited on page 406.
313 Emile B orel, Introduction géométrique à la physique, Gauthier-Villars, 1912. Cited on
page 406.
314 See V. L. Telegdi, Enrico Fermi in America, Physics Today 55, pp. 38–43, June 2002. Cited
on page 407.
315 K. Schmidt-Nielsen, Desert Animals: Physiological Problems of Heat and Water, Ox-
ford University Press, 1964. Cited on page 408.
316 Following a private communication by Richard Rusby, this is the value of 1997, whereas it
was estimated as 99.975°C in 1989, as reported by Gareth Jones & Richard Rusby,
Official: water boils at 99.975°C, Physics World 2, pp. 23–24, September 1989, and
R. L. Rusby, Ironing out the standard scale, Nature 338, p. 1169, March 1989. For more

Motion Mountain – The Adventure of Physics
details on temperature measurements, see Ref. 288. Cited on pages 408 and 548.
317 Why entropy is created when information is erased, but not when it is acquired, is ex-
plained in C. H. Bennett & R. Landauer, Fundamental Limits of Computation, Sci-
entific American 253:1, pp. 48–56, 1985. The conclusion: we should pay to throw the news-
paper away, not to buy it. Cited on page 409.
318 See, for example, G. Swift, Thermoacoustic engines and refrigerators, Physics Today 48,
pp. 22–28, July 1995. Cited on page 412.
319 Quoted in D. Campbell, J. Crutchfield, J. Farmer & E. Jen, Experimental math-
ematics: the role of computation in nonlinear science, Communications of the Association of

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Computing Machinery 28, pp. 374–384, 1985. Cited on page 415.
320 For more about the shapes of snowflakes, see the famous book by W. A. Bentley &
W. J. Humphreys, Snow Crystals, Dover Publications, New York, 1962. This second print-
ing of the original from 1931 shows a large part of the result of Bentley’s lifelong passion,
namely several thousand photographs of snowflakes. Cited on page 415.
321 K. Schwenk, Why snakes have forked tongues, Science 263, pp. 1573–1577, 1994. Cited on
page 418.
322 Human hands do not have five fingers in around 1 case out of 1000. How does nature en-
sure this constancy? The detailed mechanisms are not completely known yet. It is known,
though, that a combination of spatial and temporal self-organization during cell prolifera-
tion and differentiation in the embryo is the key factor. In this self-regulating system, the
GLI3 transcription factor plays an essential role. Cited on page 419.
323 E. Martínez, C. Pérez-Penichet, O. Sotolongo-Costa, O. Ramos,
K.J. Måløy, S. Douady, E. Altshuler, Uphill solitary waves in granular flows,
Physical Review 75, p. 031303, 2007, and E. Altshuler, O. Ramos, E. Martínez,
A. J. Batista-Ley va, A. Rivera & K. E. Bassler, Sandpile formation by revolving
rivers, Physical Review Letters 91, p. 014501, 2003. Cited on page 420.
324 P. B. Umbanhowar, F. Melo & H. L. Swinney, Localized excitations in a vertically
vibrated granular layer, Nature 382, pp. 793–796, 29 August 1996. Cited on page 421.
325 D. K. Campbell, S. Flach & Y. S. Kivshar, Localizing energy through nonlinearity
and discreteness, Physics Today 57, pp. 43–49, January 2004. Cited on page 421.
bibliography 551

326 B. Andreotti, The song of dunes as a wave-particle mode locking, Physical Review Letters
92, p. 238001, 2004. Cited on page 421.
327 D. C. Mays & B. A. Faybishenko, Washboards in unpaved highways as a complex dy-
namic system, Complexity 5, pp. 51–60, 2000. See also N. Taberlet, S. W. Morris &
J. N. McElwaine, Washboard road: the dynamics of granular ripples formed by rolling
wheels, Physical Review Letters 99, p. 068003, 2007. Cited on pages 422 and 562.
328 K. Kötter, E. Goles & M. Markus, Shell structures with ‘magic numbers’ of spheres
in a swirled disk, Physical Review E 60, pp. 7182–7185, 1999. Cited on page 422.
329 A good introduction is the text by Daniel Walgraef, Spatiotemporal Pattern Forma-
tion, With Examples in Physics, Chemistry and Materials Science, Springer 1996. Cited on
page 422.
330 For an overview, see the Ph.D. thesis by Joceline Lega, Défauts topologiques associés à
la brisure de l’invariance de translation dans le temps, Université de Nice, 1989. Cited on
page 424.
331 An idea of the fascinating mechanisms at the basis of the heart beat is given by
A. Babloyantz & A. Destexhe, Is the normal heart a periodic oscillator?, Biolo-

Motion Mountain – The Adventure of Physics
gical Cybernetics 58, pp. 203–211, 1989. Cited on page 425.
332 For a short, modern overview of turbulence, see L. P. Kadanoff, A model of turbulence,
Physics Today 48, pp. 11–13, September 1995. Cited on page 426.
333 For a clear introduction, see T. Schmidt & M. Mahrl, A simple mathematical model of
a dripping tap, European Journal of Physics 18, pp. 377–383, 1997. Cited on page 426.
334 The mathematics of fur patterns has been studied in great detail. By varying parameters
in reaction–diffusion equations, it is possible to explain the patterns on zebras, leopards,
giraffes and many other animals. The equations can be checked by noting, for example, how
the calculated patterns continue on the tail, which usually looks quite different. In fact, most

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
patterns look differently if the fur is not flat but curved. This is a general phenomenon, valid
also for the spot patterns of ladybugs, as shown by S. S. Liaw, C. C. Yang, R. T. Liu &
J. T. Hong, Turing model for the patterns of lady beetles, Physical Review E 64, p. 041909,
2001. Cited on page 426.
335 An overview of science humour can be found in the famous anthology compiled by
R. L. Weber, edited by E. Mendoza, A Random Walk in Science, Institute of Physics,
1973. It is also available in several expanded translations. Cited on page 426.
336 W. Dreybrodt, Physik von Stalagmiten, Physik in unserer Zeit pp. 25–30, Physik in un-
serer Zeit February 2009. Cited on page 427.
337 K. Mertens, V. Putkaradze & P. Vorobieff, Braiding patterns on an inclined
plane, Nature 430, p. 165, 2004. Cited on page 428.
338 These beautifully simple experiments were published in G. Müller, Starch columns: ana-
log model for basalt columns, Journal of Geophysical Research 103, pp. 15239–15253, 1998,
in G. Müller, Experimental simulation of basalt columns, Journal of Volcanology and
Geothermal Research 86, pp. 93–96, 1998, and in G. Müller, Trocknungsrisse in Stärke,
Physikalische Blätter 55, pp. 35–37, 1999. Cited on page 429.
339 To get a feeling for viscosity, see the fascinating text by Steven Vogel, Life in Moving
Fluids: the Physical Biology of Flow, Princeton University Press, 1994. Cited on page 430.
340 B. Hof, C. W. H. van Doorne, J. Westerweel, F. T. M. Nieuwstadt,
H. Wedin, R. Kerswell, F. Waleffe, H. Faisst & B. Eckhardt, Experimental
observation of nonlinear traveling waves in turbulent pipe flow, Science 305, pp. 1594–1598,
552 bibliography

2004. See also B. Hof & al., Finite lifetime of turbulence in shear flows, Nature 443, p. 59,
2006. Cited on page 430.
341 A fascinating book on the topic is Kenneth Laws & Martha Swope, Physics and
the Art of Dance: Understanding Movement, Oxford University Press 2002. See also Ken-
neth Laws & M. Lott, Resource Letter PoD-1: The Physics of Dance, American Journal
of Physics 81, pp. 7–13, 2013. Cited on page 430.
342 The fascinating variation of snow crystals is presented in C. Magono & C. W. Lee, Met-
eorological classification of natural snow crystals, Journal of the Faculty of Science, Hokkaido
University Ser. VII, II, pp. 321–325, 1966, also online at the eprints.lib.hokudai.ac.jp website.
Cited on page 431.
343 Josef H. Reichholf, Eine kurze Naturgeschichte des letzten Jahrtausends, Fischer Ver-
lag, 2007. Cited on page 431.
344 See for example, E. F. Bunn, Evolution and the second law of thermodynamics, American
Journal of Physics 77, pp. 922–925, 2009. Cited on page 432.
345 See the paper J. Maldacena, S. H. Shenker & D. Stanford, A bound on chaos, free
preprint at www.arxiv.org/abs/1503.01409. The bound has not be put into question yet.

Motion Mountain – The Adventure of Physics
Cited on page 432.
346 A good introduction to the physics of bird swarms is T. Feder, Statistical physics is for the
birds, Physics Today 60, pp. 28–30, October 2007. Cited on page 433.
347 The Nagel-Schreckenberg model for vehicle traffic, for example, explains how simple fluc-
tuations in traffic can lead to congestions. Cited on page 434.
348 J. J. Lissauer, Chaotic motion in the solar system, Reviews of Modern Physics 71, pp. 835–
845, 1999. Cited on page 435.
349 See Jean-Paul Dumont, Les écoles présocratiques, Folio Essais, Gallimard, 1991, p. 426.
Cited on page 436.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
350 For information about the number π, and about some other mathematical constants, the
website www.cecm.sfu.ca/~jborwein/pi_cover.html provides the most extensive informa-
tion and references. It also has a link to the many other sites on the topic, including the
overview at mathworld.wolfram.com/Pi.html. Simple formulae for π are
∞
𝑛 2𝑛
π+3= ∑ 2𝑛
(165)
𝑛=1 ( 𝑛 )

or the beautiful formula discovered in 1996 by Bailey, Borwein and Plouffe
∞
1 4 2 1 1
π=∑ 𝑛
( − − − ) . (166)
𝑛=0 16 8𝑛 + 1 8𝑛 + 4 8𝑛 + 5 8𝑛 + 6

The mentioned site also explains the newly discovered methods for calculating specific bin-
ary digits of π without having to calculate all the preceding ones. The known digits of
π pass all tests of randomness, as the mathworld.wolfram.com/PiDigits.html website ex-
plains. However, this property, called normality, has never been proven; it is the biggest
open question about π. It is possible that the theory of chaotic dynamics will lead to a solu-
tion of this puzzle in the coming years.
Another method to calculate π and other constants was discovered and published by
D. V. Chudnovsky & G. V. Chudnovsky, The computation of classical constants, Pro-
ceedings of the National Academy of Sciences (USA) 86, pp. 8178–8182, 1989. The Chud-
nowsky brothers have built a supercomputer in Gregory’s apartment for about 70 000
bibliography 553

euros, and for many years held the record for calculating the largest number of digits of
π. They have battled for decades with Kanada Yasumasa, who held the record in 2000, cal-
culated on an industrial supercomputer. From 2009 on, the record number of (consecutive)
digits of π has been calculated on a desktop PC. The first was Fabrice Bellard, who needed
123 days and used a Chudnovsky formula. Bellard calculated over 2.7 million million digits,
as told on bellard.org. Only in 2019 did people start to use cloud computers. The present
record is 31.415 million million digits. For the most recent records, see en.wikipedia.org/
wiki/Chronology_of_computation_of_%CF%80. New formulae to calculate π are still oc-
casionally discovered.
For the calculation of Euler’s constant γ see also D. W. DeTemple, A quicker conver-
gence to Euler’s constant, The Mathematical Intelligencer, pp. 468–470, May 1993. Cited on
pages 437 and 468.
351 The Johnson quote is found in William Seward, Biographiana, 1799. For details, see the
story in quoteinvestigator.com/2014/11/08/without-effort/. Cited on page 441.
352 The first written record of the letter U seems to be Leon Battista Alberti, Grammat-
ica della lingua toscana, 1442, the first grammar of a modern (non-latin) language, written
by a genius that was intellectual, architect and the father of cryptology. The first written re-

Motion Mountain – The Adventure of Physics
cord of the letter J seems to be Antonio de Nebrija, Gramática castellana, 1492. Before
writing it, Nebrija lived for ten years in Italy, so that it is possible that the I/J distinction is
of Italian origin as well. Nebrija was one of the most important Spanish scholars. Cited on
page 443.
353 For more information about the letters thorn and eth, have a look at the extensive report to
be found on the website www.everytype.com/standards/wynnyogh/thorn.html. Cited on
page 443.
354 For a modern history of the English language, see David Crystal, The Stories of English,
Allen Lane, 2004. Cited on page 443.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
355 Hans Jensen, Die Schrift, Berlin, 1969, translated into English as Sign, Symbol and Script:
an Account of Man’s Efforts to Write, Putnam’s Sons, 1970. Cited on page 443.
356 David R. Lide, editor, CRC Handbook of Chemistry and Physics, 78th edition, CRC Press,
1997. This classic reference work appears in a new edition every year. The full Hebrew alpha-
bet is given on page 2-90. The list of abbreviations of physical quantities for use in formulae
approved by ISO, IUPAP and IUPAC can also be found there.
However, the ISO 31 standard, which defines these abbreviations, costs around a thou-
sand euro, is not available on the internet, and therefore can safely be ignored, like any
standard that is supposed to be used in teaching but is kept inaccessible to teachers. Cited
on pages 445 and 447.
357 See the mighty text by Peter T. Daniels & William Bright, The World’s Writing
Systems, Oxford University Press, 1996. Cited on page 446.
358 The story of the development of the numbers is told most interestingly by
Georges Ifrah, Histoire universelle des chiffres, Seghers, 1981, which has been translated
into several languages. He sums up the genealogy of the number signs in ten beautiful
tables, one for each digit, at the end of the book. However, the book itself contains fac-
tual errors on every page, as explained for example in the review found at www.ams.org/
notices/200201/rev-dauben.pdf and www.ams.org/notices/200202/rev-dauben.pdf. Cited
on page 446.
359 See the for example the fascinating book by Steven B. Smith, The Great Mental Calcu-
lators – The Psychology, Methods and Lives of the Calculating Prodigies, Columbia University
554 bibliography

Press, 1983. The book also presents the techniques that they use, and that anybody else can
use to emulate them. Cited on page 447.
360 See for example the article ‘Mathematical notation’ in the Encyclopedia of Mathematics, 10
volumes, Kluwer Academic Publishers, 1988–1993. But first all, have a look at the informat-
ive and beautiful jeff560.tripod.com/mathsym.html website. The main source for all these
results is the classic and extensive research by Florian Cajori, A History of Mathemat-
ical Notations, 2 volumes, The Open Court Publishing Co., 1928–1929. The square root sign
is used in Christoff Rudolff, Die Coss, Vuolfius Cephaleus Joanni Jung: Argentorati,
1525. (The full title was Behend vnnd Hubsch Rechnung durch die kunstreichen regeln Algebre
so gemeinlicklich die Coss genent werden. Darinnen alles so treülich an tag gegeben, das auch
allein auss vleissigem lesen on allen mündtlich𝑒 vnterricht mag begriffen werden, etc.) Cited
on page 447.
361 J. Tschichold, Formenwamdlungen der et-Zeichen, Stempel AG, 1953. Cited on page
449.
362 Malcolm B. Parkes, Pause and Effect: An Introduction to the History of Punctuation in
the West, University of California Press, 1993. Cited on page 449.

Motion Mountain – The Adventure of Physics
363 This is explained by Berthold Louis Ullman, Ancient Writing and its Influence, 1932.
Cited on page 449.
364 Paul Lehmann, Erforschung des Mittelalters – Ausgewählte Abhandlungen und Aufsätze,
Anton Hiersemann, 1961, pp. 4–21. Cited on page 449.
365 Bernard Bischoff, Paläographie des römischen Altertums und des abendländischen
Mittelalters, Erich Schmidt Verlag, 1979, pp. 215–219. Cited on page 449.
366 Hutton Webster, Rest Days: A Study in Early Law and Morality, MacMillan, 1916. The
discovery of the unlucky day in Babylonia was made in 1869 by George Smith, who also
rediscovered the famous Epic of Gilgamesh. Cited on page 450.

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
367 The connections between Greek roots and many French words – and thus many English
ones – can be used to rapidly build up a vocabulary of ancient Greek without much study,
as shown by the practical collection by J. Chaineux, Quelques racines grecques, Wetteren
– De Meester, 1929. See also Donald M. Ayers, English Words from Latin and Greek
Elements, University of Arizona Press, 1986. Cited on page 451.
368 In order to write well, read William Strunk & E. B. White, The Elements of Style,
Macmillan, 1935, 1979, or Wolf Schneider, Deutsch für Kenner – Die neue Stilkunde,
Gruner und Jahr, 1987. Cited on page 452.
369 Le Système International d’Unités, Bureau International des Poids et Mesures, Pavillon de
Breteuil, Parc de Saint Cloud, 92310 Sèvres, France. All new developments concerning SI
units are published in the journal Metrologia, edited by the same body. Showing the slow
pace of an old institution, the BIPM launched a website only in 1998; it is now reachable at
www.bipm.fr. See also the www.utc.fr/~tthomass/Themes/Unites/index.html website; this
includes the biographies of people who gave their names to various units. The site of its
British equivalent, www.npl.co.uk/npl/reference, is much better; it provides many details
as well as the English-language version of the SI unit definitions. Cited on page 453.
370 The bible in the field of time measurement is the two-volume work by J. Vanier &
C. Audoin, The Quantum Physics of Atomic Frequency Standards, Adam Hilge, 1989. A
popular account is Tony Jones, Splitting the Second, Institute of Physics Publishing, 2000.
The site opdaf1.obspm.fr/www/lexique.html gives a glossary of terms used in the field.
For precision length measurements, the tools of choice are special lasers, such as mode-
locked lasers and frequency combs. There is a huge literature on these topics. Equally large
bibliography 555

is the literature on precision electric current measurements; there is a race going on for the
best way to do this: counting charges or measuring magnetic forces. The issue is still open.
Vol. II, page 71 On mass and atomic mass measurements, see the volume on relativity. On high-precision
temperature measurements, see Ref. 288. Cited on page 454.
371 The unofficial SI prefixes were first proposed in the 1990s by Jeff K. Aronson of the Uni-
versity of Oxford, and might come into general usage in the future. See New Scientist 144,
p. 81, 3 December 1994. Other, less serious proposals also exist. Cited on page 455.
372 The most precise clock built in 2004, a caesium fountain clock, had a precision of one
part in 1015 . Higher precision has been predicted to be possible soon, among others
by M. Takamoto, F. -L. Hong, R. Higashi & H. Katori, An optical lattice clock,
Nature 435, pp. 321–324, 2005. Cited on page 457.
373 J. Bergquist, ed., Proceedings of the Fifth Symposium on Frequency Standards and Met-
rology, World Scientific, 1997. Cited on page 457.
374 J. Short, Newton’s apples fall from grace, New Scientist 2098, p. 5, 6 September 1997. More
details can be found in R. G. Keesing, The history of Newton’s apple tree, Contemporary
Physics 39, pp. 377–391, 1998. Cited on page 458.

Motion Mountain – The Adventure of Physics
375 The various concepts are even the topic of a separate international standard, ISO 5725, with
the title Accuracy and precision of measurement methods and results. A good introduction is
John R. Taylor, An Introduction to Error Analysis: the Study of Uncertainties in Physical
Measurements, 2nd edition, University Science Books, Sausalito, 1997. Cited on page 459.
376 The most recent (2010) recommended values of the fundamental physical constants are
found only on the website physics.nist.gov/cuu/Constants/index.html. This set of constants
results from an international adjustment and is recommended for international use by the
Committee on Data for Science and Technology (CODATA), a body in the International
Council of Scientific Unions, which brings together the International Union of Pure and
Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC)

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
and other organizations. The website of IUPAC is www.iupac.org. Cited on page 461.
377 Some of the stories can be found in the text by N. W. Wise, The Values of Precision,
Princeton University Press, 1994. The field of high-precision measurements, from which
the results on these pages stem, is a world on its own. A beautiful introduction to it
is J. D. Fairbanks, B. S. Deaver, C. W. Everitt & P. F. Michaelson, eds., Near
Zero: Frontiers of Physics, Freeman, 1988. Cited on page 461.
378 For details see the well-known astronomical reference, P. Kenneth Seidelmann, Ex-
planatory Supplement to the Astronomical Almanac, 1992. Cited on page 466.
379 F.F. Stanaway & al., How fast does the Grim Reaper walk? Receiver operating character-
istic curve analysis in healthy men aged 70 and over, British Medical Journal 343, p. 7679,
2011. This paper by an Australian research team, was based on a study of 1800 older men
that were followed over several years; the paper was part of the 2011 Christmas issue and is
freely downloadable at www.bmj.com. Additional research shows that walking and train-
ing to walk rapidly can indeed push death further away, as summarized by K. Jahn &
T. Brandt, Wie Alter und Krankheit den Gang verändern, Akademie Aktuell 03, pp. 22–
25, 2012, The paper also shows that humans walk upright for at least 3.6 million years and
that walking speed decreases about 1 % per year after the age of 60. Cited on page 480.
C R E DI T S

Acknowled gements
Many people who have kept their gift of curiosity alive have helped to make this project come
true. Most of all, Peter Rudolph and Saverio Pascazio have been – present or not – a constant
reference for this project. Fernand Mayné, Ata Masafumi, Roberto Crespi, Serge Pahaut, Luca
Bombelli, Herman Elswijk, Marcel Krijn, Marc de Jong, Martin van der Mark, Kim Jalink, my
parents Peter and Isabella Schiller, Mike van Wijk, Renate Georgi, Paul Tegelaar, Barbara and

Motion Mountain – The Adventure of Physics
Edgar Augel, M. Jamil, Ron Murdock, Carol Pritchard, Richard Hoffman, Stephan Schiller, Franz
Aichinger and, most of all, my wife Britta have all provided valuable advice and encouragement.
Many people have helped with the project and the collection of material. Most useful was the
help of Mikael Johansson, Bruno Barberi Gnecco, Lothar Beyer, the numerous improvements by
Bert Sierra, the detailed suggestions by Claudio Farinati, the many improvements by Eric Shel-
don, the detailed suggestions by Andrew Young, the continuous help and advice of Jonatan Kelu,
the corrections of Elmar Bartel, and in particular the extensive, passionate and conscientious
help of Adrian Kubala.
Important material was provided by Bert Peeters, Anna Wierzbicka, William Beaty, Jim Carr,
John Merrit, John Baez, Frank DiFilippo, Jonathan Scott, Jon Thaler, Luca Bombelli, Douglas

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Singleton, George McQuarry, Tilman Hausherr, Brian Oberquell, Peer Zalm, Martin van der
Mark, Vladimir Surdin, Julia Simon, Antonio Fermani, Don Page, Stephen Haley, Peter Mayr,
Allan Hayes, Norbert Dragon, Igor Ivanov, Doug Renselle, Wim de Muynck, Steve Carlip, Tom
Bruce, Ryan Budney, Gary Ruben, Chris Hillman, Olivier Glassey, Jochen Greiner, squark, Mar-
tin Hardcastle, Mark Biggar, Pavel Kuzin, Douglas Brebner, Luciano Lombardi, Franco Bagnoli,
Lukas Fabian Moser, Dejan Corovic, Paul Vannoni, John Haber, Saverio Pascazio, Klaus Finken-
zeller, Leo Volin, Jeff Aronson, Roggie Boone, Lawrence Tuppen, Quentin David Jones, Arnaldo
Uguzzoni, Frans van Nieuwpoort, Alan Mahoney, Britta Schiller, Petr Danecek, Ingo Thies, Vi-
taliy Solomatin, Carl Offner, Nuno Proença, Elena Colazingari, Paula Henderson, Daniel Darre,
Wolfgang Rankl, John Heumann, Joseph Kiss, Martha Weiss, Antonio González, Antonio Mar-
tos, André Slabber, Ferdinand Bautista, Zoltán Gácsi, Pat Furrie, Michael Reppisch, Enrico Pasi,
Thomas Köppe, Martin Rivas, Herman Beeksma, Tom Helmond, John Brandes, Vlad Tarko, Na-
dia Murillo, Ciprian Dobra, Romano Perini, Harald van Lintel, Andrea Conti, François Belfort,
Dirk Van de Moortel, Heinrich Neumaier, Jarosław Królikowski, John Dahlman, Fathi Namouni,
Paul Townsend, Sergei Emelin, Freeman Dyson, S.R. Madhu Rao, David Parks, Jürgen Janek,
Daniel Huber, Alfons Buchmann, William Purves, Pietro Redondi, Damoon Saghian, Frank
Sweetser, Markus Zecherle, Zach Joseph Espiritu, Marian Denes, Miles Mutka, plus a number
of people who wanted to remain unnamed.
The software tools were refined with extensive help on fonts and typesetting by Michael Zedler
and Achim Blumensath and with the repeated and valuable support of Donald Arseneau; help
came also from Ulrike Fischer, Piet van Oostrum, Gerben Wierda, Klaus Böhncke, Craig Up-
right, Herbert Voss, Andrew Trevorrow, Danie Els, Heiko Oberdiek, Sebastian Rahtz, Don Story,
credits 557

Vincent Darley, Johan Linde, Joseph Hertzlinger, Rick Zaccone, John Warkentin, Ulrich Diez,
Uwe Siart, Will Robertson, Joseph Wright, Enrico Gregorio, Rolf Niepraschk, Alexander Grahn,
Werner Fabian and Karl Köller.
The typesetting and book design is due to the professional consulting of Ulrich Dirr. The
typography was much improved with the help of Johannes Küster and his Minion Math font.
The design of the book and its website also owe much to the suggestions and support of my wife
Britta.
I also thank the lawmakers and the taxpayers in Germany, who, in contrast to most other
countries in the world, allow residents to use the local university libraries.
From 2007 to 2011, the electronic edition and distribution of the Motion Mountain text was
generously supported by the Klaus Tschira Foundation.

Film credits
The beautiful animations of the rotating attached dodecaedron on page 90 and of the embed-
ded ball on page 168 are copyright and courtesy of Jason Hise; he made them for this text and
for the Wikimedia Commons website. Several of his animations are found on his website www.
entropygames.net. The clear animation of a suspended spinning top, shown on page 148, was

Motion Mountain – The Adventure of Physics
made for this text by Lucas Barbosa. The impressive animation of the Solar System on page 155
was made for this text by Rhys Taylor and is now found at his website www.rhysy.net. The beau-
tiful animation of the lunation on page 190 was calculated from actual astronomical data and
is copyright and courtesy by Martin Elsässer. It can be found on his website www.mondatlas.de/
lunation.html. The beautiful film of geostationary satellites on page 195 is copyright and courtesy
by Michael Kunze and can be found on his beautiful site www.sky-in-motion.de/en. The beau-
tiful animation of the planets and planetoids on page 220 is copyright and courtesy by Hans-
Christian Greier. It can be found on his wonderful website www.parallax.at. The film of an os-
cillating quartz on page 291 is copyrighted and courtesy of Micro Crystal, part of the Swatch
Group, found at www.microcrystal.com. The animation illustrating group and wave velocity on

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
page 299 and the animation illustrating the molecular motion in a sound wave on page 311 are
courtesy and copyright of the ISVR at the University of Southampton. The film of the rogue wave
on page 323 is courtesy and copyright of Amin Chabchoub; details can be found at journals.aps.
org/prx/abstract/10.1103/PhysRevX.2.011015. The films of solitons on page 317 and of dromions
on page 329 are copyright and courtesy by Jarmo Hietarinta. They can be found on his website
users.utu.fi/hietarin. The film of leapfrogging vortex rings on page 380 is copyright and courtesy
by Lim Tee Tai. It can be found via his fluid dynamics website serve.me.nus.edu.sg. The film of
the growing snowflake on page 422 is copyright and courtesy by Kenneth Libbrecht. It can be
found on his website www.its.caltech.edu/~atomic/snowcrystals.

Image credits
The photograph of the east side of the Langtang Lirung peak in the Nepalese Himalayas, shown
on the front cover, is courtesy and copyright by Kevin Hite and found on his blog thegettingthere.
com. The lightning photograph on page 14 is courtesy and copyright by Harald Edens and found
on the www.lightningsafety.noaa.gov/photos.htm and www.weather-photography.com websites.
The motion illusion on page 18 is courtesy and copyright by Michael Bach and found on his web-
site www.michaelbach.de/ot/mot_rotsnake/index.html. It is a variation of the illusion by Kitaoka
Akiyoshi found on www.ritsumei.ac.jp/~akitaoka and used here with his permission. The figures
on pages 20, 57 and 207 were made especially for this text and are copyright by Luca Gastaldi.
The high speed photograph of a bouncing tennis ball on page 20 is courtesy and copyright by the
International Tennis Federation, and were provided by Janet Page. The figure of Etna on pages
558 credits

22 and 150 is copyright and courtesy of Marco Fulle and found on the wonderful website www.
stromboli.net. The famous photograph of the Les Poulains and its lighthouse by Philip Plisson
on page 23 is courtesy and copyright by Pechêurs d’Images; see the websites www.plisson.com
and www.pecheurs-d-images.com. It is also found in Plisson’s magnus opus La Mer, a stun-
ning book of photographs of the sea. The picture on page 23 of Alexander Tsukanov jumping
from one ultimate wheel to another is copyright and courtesy of the Moscow State Circus. The
photograph of a deer on page 26 is copyright and courtesy of Tony Rodgers and taken from
his website www.flickr.com/photos/moonm. The photographs of speed measurement devices
on page 35 are courtesy and copyright of the Fachhochschule Koblenz, of Silva, of Tracer and
of Wikimedia. The graph on page 38 is redrawn and translated from the wonderful book by
Henk Tennekes, De wetten van de vliegkunst - Over stijgen, dalen, vliegen en zweven, Aramith
Uitgevers, 1993. The photographs of the ping-pong ball on page 40 and of the dripping water tap
on page 355 are copyright and courtesy of Andrew Davidhazy and found on his website www.rit.
edu/~andpph. The photograph of the bouncing water droplet on page 40 are copyright and cour-
tesy of Max Groenendijk and found on the website www.lightmotif.nl. The photograph of the
precision sundial on page 45 is copyright and courtesy of Stefan Pietrzik and found at commons.
wikimedia.org/wiki/Image:Präzissions-Sonnenuhr_mit_Sommerwalze.jpg The other clock pho-

Motion Mountain – The Adventure of Physics
tographs in the figure are from public domain sources as indicated. The graph on the scaling of
biological rhythms on page 47 is drawn by the author using data from the European Molecu-
lar Biology Organization found at www.nature.com/embor/journal/v6/n1s/fig_tab/7400425_f3.
html and Enrique Morgado. The drawing of the human ear on page page 50 and on page 325
are copyright of Northwestern University and courtesy of Tim Hain; it is found on his website
www.dizziness-and-balance.com/disorders/bppv/otoliths.html. The illustrations of the vernier
caliper and the micrometer screw on page 54 and 65 are copyright of Medien Werkstatt, cour-
tesy of Stephan Bogusch, and taken from their instruction course found on their website www.
medien-werkstatt.de. The photo of the tiger on page 54 is copyright of Naples zoo (in Florida,
not in Italy), and courtesy of Tim Tetzlaff; see their website at www.napleszoo.com. The other

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
length measurement devices on page 54 are courtesy and copyright of Keyence and Leica Geo-
systems, found at www.leica-geosystems.com. The curvimeter photograph on page 55 is copy-
right and courtesy of Frank Müller and found on the www.wikimedia.org website. The crystal
photograph on the left of page 58 is copyright and courtesy of Stephan Wolfsried and found on
the www.mindat.org website. The crystal photograph on the right of page 58 is courtesy of Tullio
Bernabei, copyright of Arch. Speleoresearch & Films/La Venta and found on the www.laventa.
it and www.naica.com.mx websites. The hollow Earth figure on pages 60 is courtesy of Helmut
Diel and was drawn by Isolde Diel. The wonderful photographs on page 69, page 150, page 179,
page 217, page 213 and page 505 are courtesy and copyright by Anthony Ayiomamitis; the story
of the photographs is told on his beautiful website at www.perseus.gr. The anticrepuscular pho-
tograph on page 71 is courtesy and copyright by Peggy Peterson. The rope images on page 72 are
copyright and courtesy of Jakob Bohr. The image of the tight knot on page 73 is courtesy and
copyright by Piotr Pieranski. The firing caterpillar figure of page 79 is courtesy and copyright of
Stanley Caveney. The photograph of an airbag sensor on page 86 is courtesy and copyright of
Bosch; the accelerometer picture is courtesy and copyright of Rieker Electronics; the three draw-
ings of the human ear are copyright of Northwestern University and courtesy of Tim Hain and
found on his website www.dizziness-and-balance.com/disorders/bppv/otoliths.html. The photo-
graph of Orion on page 87 is courtesy and copyright by Matthew Spinelli; it was also featured on
antwrp.gsfc.nasa.gov/apod/ap030207.html. On page 88, the drawing of star sizes is courtesy and
copyright Dave Jarvis. The photograph of Regulus and Mars on page 88 is courtesy and copy-
right of Jürgen Michelberger and found on www.jmichelberger.de. On page 91, the millipede
photograph is courtesy and copyright of David Parks and found on his website www.mobot.
credits 559

org/mobot/madagascar/image.asp?relation=A71. The photograph of the gecko climbing the bus
window on page 91 is courtesy and copyright of Marcel Berendsen, and found on his website
www.flickr.com/photos/berendm. The photograph of the amoeba is courtesy and copyright of
Antonio Guillén Oterino and is taken from his wonderful website Proyecto Agua at www.flickr.
com/photos/microagua. The photograph of N. decemspinosa on page 91 is courtesy and copy-
right of Robert Full, and found on his website rjf9.biol.berkeley.edu/twiki/bin/view/PolyPEDAL/
LabPhotographs. The photograph of P. ruralis on page 91 is courtesy and copyright of John Brack-
enbury, and part of his wonderful collection on the website www.sciencephoto.co.uk. The pho-
tograph of the rolling spider on page 91 is courtesy and copyright of Ingo Rechenberg and can
be found at www.bionik.tu-berlin.de, while the photo of the child somersaulting is courtesy and
copyright of Karva Javi, and can be found at www.flickr.com/photos/karvajavi. The photographs
of flagellar motors on page 93 are copyright and courtesy by Wiley & Sons and found at emboj.
embopress.org/content/30/14/2972. The two wonderful films about bacterial flagella on page 94
and on page 94 are copyright and courtesy of the Graduate School of Frontier Biosciences at
Osaka University. The beautiful photograph of comet McNaught on page 95 is courtesy and
copyright by its discoveror, Robert McNaught; it is taken from his website at www.mso.anu.
edu.au/~rmn and is found also on antwrp.gsfc.nasa.gov/apod/ap070122.html. The sonolumin-

Motion Mountain – The Adventure of Physics
secence picture on page 96 is courtesy and copyright of Detlef Lohse. The photograph of the
standard kilogram on page 100 is courtesy and copyright by the Bureau International des Poids
et Mesures (BIPM). On page 108, the photograph of Mendeleyev’s balance is copyright of Think-
tank Trust and courtesy of Jack Kirby; it can be found on the www.birminghamstories.co.uk
website. The photograph of the laboratory balance is copyright and courtesy of Mettler-Toledo.
The photograph of the cosmonaut mass measurement device is courtesy of NASA. On page 115,
the photographs of the power meters are courtesy and copyright of SRAM, Laser Components
and Wikimedia. The measured graph of the walking human on page 122 is courtesy and copy-
right of Ray McCoy. On page 130, the photograph of the stacked gyros is cortesy of Wikimedia.
The photograph of the clock that needs no winding up is copyright Jaeger-LeCoultre and cour-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
tesy of Ralph Stieber. Its history and working are described in detail in a brochure available from
the company. The company’s website is www.Jaeger-LeCoultre.com. The photograph of the ship
lift at Strépy-Thieux on page 132 is courtesy and copyright of Jean-Marie Hoornaert and found
on Wikimedia Commons. The photograph of the Celtic wobble stone on page 133 is courtesy and
copyright of Ed Keath and found on Wikimedia Commons. The photograph of the star trails on
page 137 is courtesy and copyright of Robert Schwartz; it was featured on apod.nasa.gov/apod/
ap120802.html. The photograph of Foucault’s gyroscope on page 142 is courtesy and copyright of
the museum of the CNAM, the Conservatoire National des Arts et Métiers in Paris, whose web-
site is at www.arts-et-metiers.net. The photograph of the laser gyroscope on page 142 is courtesy
and copyright of JAXA, the Japan Aerospace Exploration Agency, and found on their website at
jda.jaxa.jp. On page 143, the three-dimensional model of the gyroscope is copyright and courtesy
of Zach Joseph Espiritu. The drawing of the precision laser gyroscope on page 144 is courtesy of
Thomas Klügel and copyright of the Bundesamt für Kartographie und Geodäsie. The photograph
of the instrument is courtesy and copyright of Carl Zeiss. The machine is located at the Funda-
mentalstation Wettzell, and its website is found at www.wettzell.ifag.de. The illustration of plate
tectonics on page 149 is from a film produced by NASA’s HoloGlobe project and can be found on
svs.gsfc.nasa.gov/cgi-bin/details.cgi?aid=1288 The graph of the temperature record on page 154 is
copyright and courtesy Jean Jouzel and Science/AAAS. The photograph of a car driving through
the snow on page 156 is copyright and courtesy by Neil Provo at neilprovo.com. The photographs
of a crane fly and of a hevring fly with their halteres on page 160 are by Pinzo, found on Wiki-
media Commons, and by Sean McCann from his website ibycter.com. The MEMS photograph
and graph is copyright and courtesy of ST Microelectronics. On page 166, Figure 123 is courtesy
560 credits

and copyright of the international Gemini project (Gemini Observatory/Association of Univer-
sities for Research in Astronomy) at www.ausgo.unsw.edu.au and www.gemini.edu; the photo-
graph with the geostationary satellites is copyright and courtesy of Michael Kunze and can be
found just before his equally beautiful film at www.sky-in-motion.de/de/zeitraffer_einzel.php?
NR=12. The photograph of the Earth’s shadow on page 167 is courtesy and copyright by Ian R.
Rees and found on his website at weaknuclearforce.wordpress.com/2014/03/30/earths-shadow.
The basilisk running over water, page 169 and on the back cover, is courtesy and copyright by the
Belgian group TERRA vzw and found on their website www.terravzw.org. The water strider pho-
tograph on page 170 is courtesy and copyright by Charles Lewallen. The photograph of the water
robot on page 170 is courtesy and copyright by the American Institute of Physics. The allometry
graph about running speed in mammals is courtesy and copyright of José Iriarte-Díaz and of
The Journal of Experimental Biology; it is reproduced and adapted with their permission from
the original article, Ref. 133, found at jeb.biologists.org/content/205/18/2897. The illustration of
the motion of Mars on page 175 is courtesy and copyright of Tunc Tezel. The photograph of the
precision pendulum clock on page 181 is copyright of Erwin Sattler OHG,Sattler OHG, Erwin
and courtesy of Ms. Stephanie Sattler-Rick; it can be found at the www.erwinsattler.de website.
The figure on the triangulation of the meridian of Paris on page 184 is copyright and courtesy of

Motion Mountain – The Adventure of Physics
Ken Alder and found on his website www.kenalder.com. The photographs of the home version of
the Cavendish experiment on page 185 are courtesy and copyright by John Walker and found on
his website www.fourmilab.ch/gravitation/foobar. The photographs of the precision Cavendish
experiment on page 186 are courtesy and copyright of the Eöt-Wash Group at the University
of Washington and found at www.npl.washington.edu/eotwash. The geoid of page 189 is cour-
tesy and copyright by the GeoForschungsZentrum Potsdam, found at www.gfz-potsdam.de. The
moon maps on page 191 are courtesy of the USGS Astrogeology Research Program, astrogeology.
usgs.gov, in particular Mark Rosek and Trent Hare. The graph of orbits on page 193 is courtesy
and copyright of Geoffrey Marcy. On page 196, the asteroid orbit is courtesy and copyrigt of
Seppo Mikkola. The photograph of the tides on page 197 is copyright and courtesy of Gilles Rég-

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
nier and found on his website www.gillesregnier.com; it also shows an animation of that tide
over the whole day. The wonderful meteor shower photograph on page 206 is courtesy and copy-
right Brad Goldpaint and found on its website goldpaintphotography.com; it was also featured on
apod.nasa.gov/apod/ap160808.html. The meteor photograph on page 206 is courtesy and copy-
right of Robert Mikaelyan and found on his website www.fotoarena.nl/tag/robert-mikaelyan/.
The pictures of fast descents on snow on page 209 are copyright and courtesy of Simone Ori-
gone, www.simoneorigone.it, and of Éric Barone, www.ericbarone.com. The photograph of the
Galilean satellites on page 210 is courtesy and copyright by Robin Scagell and taken from his
website www.galaxypix.com. On page 217, the photographs of Venus are copyrigt of Wah! and
courtesy of Wikimedia Commons; see also apod.nasa.gov/apod/ap060110.html. On page 218, the
old drawing of Le Sage is courtesy of Wikimedia. The picture of the celestial bodies on page 222
is copyright and courtesy of Alex Cherney and was featured on apod.nasa.gov/apod/ap160816.
html.
The pictures of solar eclipses on page 223 are courtesy and copyright by the Centre Na-
tional d’Etudes Spatiales, at www.cnes.fr, and of Laurent Laveder, from his beautiful site at www.
PixHeaven.net. The photograph of water parabolae on page 227 is copyright and courtesy of Oase
GmbH and found on their site www.oase-livingwater.com. The photograph of insect gear on
page 245 is copyright and courtesy of Malcolm Burrows; it is found on his website www.zoo.cam.
ac.uk/departments/insect-neuro. The pictures of daisies on page 246 are copyright and courtesy
of Giorgio Di Iorio, found on his website www.flickr.com/photos/gioischia, and of Thomas Lüthi,
found on his website www.tiptom.ch/album/blumen/. The photograph of fireworks in Chantilly
on page 249 is courtesy and copyright of Christophe Blanc and taken from his beautiful website
credits 561

at christopheblanc.free.fr. On page 258, the beautiful photograph of M74 is copyright and cour-
tesy of Mike Hankey and found on his beautiful website cdn.mikesastrophotos.com. The figure
of myosotis on page 267 is courtesy and copyright by Markku Savela. The image of the wallpa-
per groups on page page 268 is copyright and courtesy of Dror Bar-Natan, and is taken from
his fascinating website at www.math.toronto.edu/~drorbn/Gallery. The images of solid symmet-
ries on page page 269 is copyright and courtesy of Jonathan Goss, and is taken from his website
at www.phys.ncl.ac.uk/staff.njpg/symmetry. Also David Mermin and Neil Ashcroft have given
their blessing to the use. On page 292, the Fourier decomposition graph is courtesy Wikimedia.
The drawings of a ringing bell on page 292 are courtesy and copyright of H. Spiess. The image of a
vinyl record on page 293scale=1 is copyright of Chris Supranowitz and courtesy of the University
of Rochester; it can be found on his expert website at www.optics.rochester.edu/workgroups/cml/
opt307/spr05/chris. On page 296, the water wave photographs are courtesy and copyright of Eric
Willis, Wikimedia and allyhook. The interference figures on page 303 are copyright and courtesy
of Rüdiger Paschotta and found on his free laser encyclopedia at www.rp-photonics.com. On
page 308, the drawings of the larynx are courtesy Wikimedia. The images of the microanemo-
meter on page page 312 are copyright of Microflown and courtesy of Marcin Korbasiewicz. More
images can be found on their website at www.microflown.com. The image of the portable ultra-

Motion Mountain – The Adventure of Physics
sound machine on page 313 is courtesy and copyright General Electric. The ultrasound image
on page 313 courtesy and copyright Wikimedia. The figure of the soliton in the water canal on
page 316 is copyright and courtesy of Dugald Duncan and taken from his website on www.ma.
hw.ac.uk/solitons/soliton1.html. The photograph on page 319 is courtesy and copyright Andreas
Hallerbach and found on his website www.donvanone.de. The image of Rubik’s cube on page 346
is courtesy of Wikimedia. On page 327, the photographs of shock waves are copyright and cour-
tesy of Andrew Davidhazy, Gary Settles and NASA. The photographs of wakes on page 328 are
courtesy Wikimedia and courtesy and copyright of Christopher Thorn. On page 330, the pho-
tographs un unusual water waves are copyright and courtesy of Diane Henderson, Anonymous
and Wikimedia. The fractal mountain on page 334 is courtesy and copyright by Paul Martz, who

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
explains on his website www.gameprogrammer.com/fractal.html how to program such images.
The photograph of the oil droplet on a water surface on page 335 is courtesy and copyright of
Wolfgang Rueckner and found on sciencedemonstrations.fas.harvard.edu/icb. The soap bubble
photograph on page page 342 is copyright and courtesy of LordV and found on his website www.
flickr.com/photos/lordv. The photographs of silicon carbide on page 343 are copyright and cour-
tesy of Dietmar Siche. The photograph of a single barium ion on page 344 is copyright and cour-
tesy of Werner Neuhauser at the Universität Hamburg. The AFM image of silicon on page 344
is copyright of the Universität Augsburg and is used by kind permission of German Hammerl.
The figure of helium atoms on metal on page 344 is copyright and courtesy of IBM. The pho-
tograph of an AFM on page 345 is copyright of Nanosurf (see www.nanosurf.ch) and used with
kind permission of Robert Sum. The photograph of the tensegrity tower on page 349 is copy-
right and courtesy of Kenneth Snelson. The photograph of the Atomium on page 351 is courtesy
and copyright by the Asbl Atomium Vzw and used with their permission, in cooperation with
SABAM in Belgium. Both the picture and the Atomium itself are under copyright. The photo-
graphs of the granular jet on page 352 in sand are copyright and courtesy of Amy Shen, who
discovered the phenomenon together with Sigurdur Thoroddsen. The photographs of the ma-
chines on page 352 are courtesy and copyright ASML and Voith. The photograph of the bucket-
wheel excavator on page 353 is copyright and courtesy of RWE and can be found on their website
www.rwe.com. The photographs of fluid motion on page 355 are copyright and courtesy of John
Bush, Massachusetts Institute of Technology, and taken from his website www-math.mit.edu/
~bush. On page 360, the images of the fluid paradoxa are courtesy and copyright of IFE. The im-
ages of the historic Magdeburg experiments by Guericke on page 361 are copyright of Deutsche
562 credits

Post, Otto-von-Guericke-Gesellschaft at www.ovgg.ovgu.de, and the Deutsche Fotothek at www.
deutschefotothek.de; they are used with their respective permissions. On page page 362, the lam-
inar flow photograph is copyright and courtesy of Martin Thum and found on his website at www.
flickr.com/photos/39904644@N05; the melt water photograph is courtesy and copyright of Steve
Butler and found on his website at www.flickr.com/photos/11665506@N00. The sailing boat on
page 363 is courtesy and copyright of Bladerider International. The illustration of the atmosphere
on page 365 is copyright of Sebman81 and courtesy of Wikimedia. The impressive computer im-
age about the amount of water on Earth on page 371 is copyright and courtesy of Jack Cook, Adam
Nieman, Woods Hole Oceanographic Institution, Howard Perlman and USGS; it was featured on
apod.nasa.gov/apod/ap160911.html. The figures of wind speed measurement systems on page 376
are courtesy and copyright of AQSystems, at www.aqs.se, and Leosphere at www.leosphere.
fr On page 377, the Leidenfrost photographs are courtesy and copyright Kenji Lopez-Alt and
found on www.seriouseats.com/2010/08/how-to-boil-water-faster-simmer-temperatures.html.
The photograph of the smoke ring at Etna on page 380 is courtesy and copyright by Daniela
Szczepanski and found at her extensive websites www.vulkanarchiv.de and www.vulkane.net.
On page 381, the photographs of rolling droplets are copyright and courtesy of David Quéré and
taken from iusti.polytech.univ-mrs.fr/~aussillous/marbles.htm. The thermographic images of a

Motion Mountain – The Adventure of Physics
braking bicycle on page 384 are copyright Klaus-Peter Möllmann and Michael Vollmer, Fach-
hochschule Brandenburg/Germany, and courtesy of Michael Vollmer and Frank Pinno. The im-
age of page 384 is courtesy and copyright of ISTA. The images of thermometers on page 387
are courtesy and copyright Wikimedia, Ron Marcus, Braun GmbH, Universum, Wikimedia and
Thermodevices. The balloon photograph on page 390 is copyright Johan de Jong and courtesy
of the Dutch Balloon Register found at www.dutchballoonregister.nl.ballonregister, nederlands
The pollen image on page 392 is from the Dartmouth College Electron Microscope Facility and
courtesy Wikimedia. The scanning tunnelling microscope picture of gold on page 401 is cour-
tesy of Sylvie Rousset and copyright by CNRS in France. The photograph of the Ranque–Hilsch
vortex tube is courtesy and copyright Coolquip. The photographs and figure on page 419 are

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
copyright and courtesy of Ernesto Altshuler, Claro Noda and coworkers, and found on their
website www.complexperiments.net. The road corrugation photo is courtesy of David Mays and
taken from his paper Ref. 327. The oscillon picture on page 421 is courtesy and copyright by
Paul Umbanhowar. The drawing of swirled spheres on page 421 is courtesy and copyright by
Karsten Kötter. The pendulum fractal on page 425 is courtesy and copyright by Paul Nylander
and found on his website bugman123.com. The fluid flowing over an inclined plate on page 427
is courtesy and copyright by Vakhtang Putkaradze. The photograph of the Belousov-Zhabotinski
reaction on page 428 is courtesy and copyright of Yamaguchi University and found on their pic-
ture gallery at www.sci.yamaguchi-u.ac.jp/sw/sw2006/. The photographs of starch columns on
page 429 are copyright of Gerhard Müller (1940–2002), and are courtesy of Ingrid Hörnchen.
The other photographs on the same page are courtesy and copyright of Raphael Kessler, from
his websitewww.raphaelk.co.uk, of Bob Pohlad, from his websitewww.ferrum.edu/bpohlad, and
of Cédric Hüsler. On page 431, the diagram about snow crystals is copyright and courtesy by
Kenneth Libbrecht; see his website www.its.caltech.edu/~atomic/snowcrystals. The photograph
of a swarm of starlings on page 432 is copyright and courtesy of Andrea Cavagna and Physics
Today. The table of the alphabet on page 442 is copyright and courtesy of Matt Baker and found
on his website at usefulcharts.com. The photograph of the bursting soap bubble on page 476 is
copyright and courtesy by Peter Wienerroither and found on his website homepage.univie.ac.
at/Peter.Wienerroither. The photograph of sunbeams on page 479 is copyright and courtesy by
Fritz Bieri and Heinz Rieder and found on their website www.beatenbergbilder.ch. The draw-
ing on page 484 is courtesy and copyright of Daniel Hawkins. The photograph of a slide rule on
page 488 is courtesy and copyright of Jörn Lütjens, and found on his website www.joernluetjens.
credits 563

de. On page 492, the bicycle diagram is courtesy and copyright of Arend Schwab. On page 505,
the sundial photograph is courtesy and copyright of Stefan Pietrzik. On page 509 the chimney
photographs are copyright and courtesy of John Glaser and Frank Siebner. The photograph of
the ventomobil on page 517 is courtesy and copyright Tobias Klaus. All drawings are copyright
by Christoph Schiller. If you suspect that your copyright is not correctly given or obtained, this
has not been done on purpose; please contact me in this case.

A A Aniçin, I.V. 539 B
Aarts Aarts, D. 543 Anonymous 45, 108, 246, 330, Babinet, Jacques
Abbott, Edwin A. 529 358, 561 life 454
Abe, F. 529 AQSystems 376, 562 Babloyantz, A. 551
Abkarian, M. 547 Aquinas, Thomas 336, 525 Baccus, S.A. 526

Motion Mountain – The Adventure of Physics
Ackermann, Rudolph 484 Archimedes Bach, Michael 18, 557
Adachi, S. 543 life 100 Bachelier, Louis 548
Adelberger, E.G. 537 Aristarchus of Samos 135, 150, Baer, Karl Ernst von 159
Adenauer, Konrad 204 533 Baez, John 556
Aetius 99, 532 Aristotle 48, 180, 530 Bagnoli, Franco 556
Ahlgren, A. 528 life 40 Bahr, D.B. 543
Ahrens, C. Donald 540 Armstrong, J.T. 530 Baille, Jean-Baptistin 537
Aigler, M. 57 Aronson, Jeff K. 555, 556 Baker, Matt 442, 562
AIP 170 Arseneau, Donald 556 Balibar, S. 546
Åkerman, N. 540 Artigas, M. 545 ballonregister, nederlands 562

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Al-Masudi 477 Ashby, M.F. 545 Banach, Stefan 57, 333
Alberti, Leon Battista 553 Ashcroft, Neil 269, 561 life 56
Alder, Ken 184, 537, 560 Ashida, H. 524 Banavar, J.R. 542
Alexander, R. McN. 536 ASML 352, 561 Bandler, Richard 241, 525
Alexander, Robert McNeill Asterix 347 Bar-Natan, Dror 268, 561
532 Ata Masafumi 556 Barber, B.P. 531
Alice 404 Atomium 351, 561 Barberini, Francesco 336
Alighieri, Dante 173 Au, W.W.L. 544 Barberi Gnecco, Bruno 556
Alizad, A. 543 Audoin, C. 554 Barbosa, Lucas 148, 557
allyhook 296, 561 Augel, Barbara 556 Barbour, Julian 528
Alsdorf, D. 540 Augel, Edgar 556 Barnes, G. 513
Alsina, Claudi 73 Ausloos, M. 544 Barone, Éric 209, 560
Altshuler, E. 550 Aussillous, P. 547 Bartel, Elmar 556
Altshuler, Ernesto 419, 420, Avogadro, Amedeo Bass, Thomas A. 533
562 life 340 Bassler, K.E. 550
Amundsen, Roald 135, 536 Axelrod, R. 524 Batista-Leyva, A.J. 550
An, K. 546 Ayers, Donald M. 554 Batty, R.S. 544
Anders, S. 537 Ayiomamitis, Anthony 69, Baudelaire, Charles
Anderson, R. 534 150, 179, 213, 217, 505, 558 life 18
Andreotti, B. 551 Azbel, M.Ya. 533 Baumbach, T. 541
Andrianne, T. 549 Bautista, Ferdinand 556
Angell, C.A. 546 Beaty, William 556
name index 565

Becker, J. 540 Boiti, M. 328, 544 Bucka, Hans 145
Beeksma, Herman 556 Bolt, Usain 78 Buckley, Michael 525
Behroozi, C.H. 527 Boltzmann, L. 549 Budney, Ryan 556
Bekenstein, Jacob 412 Boltzmann, Ludwig 254, 403 Bújovtsev, B.B. 524
Belfort, François 556 life 393 Bundesamt für Kartographie
Bellard, Fabrice 553 Bombelli, Luca 556 und Geodäsie 144, 559
Bellizzi, M.J. 536 Bonner, John T. 545 Bunn, E.F. 552
Benka, S. 532 Boone, Roggie 556 Bunyan, Paul 545
Bennet, C.L. 535 Borel, Emile 406, 550 Buridan, Jean 180, 537
Bennet-Clark, H.C. 530 Borelli, G. 525 Burrows, M. 541
Bennett, C.H. 550 Born, Max 337 Burrows, Malcolm 245, 560
B Bentley, W.A. 550
Benzenberg, Johann Friedrich
Bosch 86, 558
Boss, Alan P. 541
Burša, Milan 529
Busche, Detlev 527
136 Bourbaki, Nicolas 449 Bush, J.W.M. 546
Becker Berendsen, Marcel 91, 559 Bowden, F.P. 540 Bush, John 355, 561
Bergquist, J. 555 Bower, B. 542 Butler, Steve 362, 562
Berman, D.H. 544 Brackenbury, J. 531 Böhncke, Klaus 556

Motion Mountain – The Adventure of Physics
Bernabei, Tullio 558 Brackenbury, John 91, 559
Bernoulli, Daniel 117, 389 Bradley, James 152 C
life 361 life 152 Caesar, Gaius Julius 376, 450
Bernoulli, Johann 448, 508 Bradstock, Roald 489 Cajori, Florian 554
Bessel 146 Braginsky, V.B. 538 Caldwell, R. 531
Bessel, Friedrich Wilhelm Brahe, Tycho 175 Caldwell, R.L. 531
life 150 life 175 Calzadilla, A. 541
Bevis, M. 540 Brahm, Alcanter de Campbell, D. 550
Beyer, Lothar 556 life 449 Campbell, D.K. 550
Bielefeldt, H. 546 Brandes, John 556 Caps, H. 547

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Bieri, Fritz 479, 562 Brandt, T. 555 Carl Zeiss 144, 559
Biggar, Mark 556 Brantjes, R. 504 Carlip, Steve 556
Bilger, H.R. 534 Brasser, R. 538 Carlyle, Thomas
Binet, Jacques 165 Braun GmbH 387, 562 life 123
BIPM 100 Brebner, Douglas 556 Carnot, Sadi
Bischoff, Bernard 554 Brennan, Richard 525 life 113
Bizouard, C. 535 Brenner, M.P. 531 Carr, Jim 556
Bladerider International 363, Bright, William 553 Carroll, Lewis, or Charles
562 Brillouin, L. 549 Lutwidge Dogson 404
Blagden, Charles 408 Brillouin, Léon 399 Cartesius
Blanc, Christophe 249, 560 Bronshtein, Matvei 8 life 51
Blocken, B. 549 Brooks, Mel 395 Casati, Roberto 531
Blocken, Bert 381 Brouwer, Luitzen 50 Cassius Dio 221
Blumensath, Achim 556 Brown, Don 78 Cauchy, Augustin-Louis 273
Bocquet, Lydéric 516, 547 Brown, J.H. 542 Cavagna, Andrea 432, 562
Bode, Johann Elert 219 Brown, Robert 391 Cavendish, H. 537
Boethius 474 Browning, R.C. 543 Cavendish, Henry
Bogusch, Stephan 558 Bruce, Tom 556 life 184
Bohr, J. 529 Brunner, Esger 532 Caveney, S. 530
Bohr, Jakob 72, 558 Brush, S. 535 Caveney, Stanley 79, 558
Bohr, Niels 399 Buchmann, Alfons 556 Cayley, Arthur 273
Bohren, C.F. 545 Bucka, H. 534 Celsius, Anders
566 name index

life 408 Cook, Jack 371, 562 Demko, T.M. 535
Chabchoub, A. 544 Coolquip 412, 562 Democritus 339, 451
Chabchoub, Amin 323, 324, Cooper, Heather 537 life 339
557 Cooper, John N. 525 Denes, Marian 556
Chaineux, J. 554 Coriolis, Gustave-Gaspard 138 Descartes, René 448, 487
Chan, M.A. 535 life 110 life 51
Chandler, David G. 525 Cornell, E.A. 527 Desloge, Edward A. 534
Chandler, Seth 147 Cornu, Marie-Alfred 537 Destexhe, A. 551
Chao, B.F. 535 Corovic, Dejan 556 DeTemple, D.W. 553
Charlemagne 449 Costabel, Pierre 336 Deutsche Fotothek 361, 562
Chen, S. 93, 531 Cousteau, Jacques 368 Deutsche Post 361, 562
C Cherney, Alex 222, 560
Childs, J.J. 546
Cox, Trevor 318
Coyne, G. 545
Deutschmann, Matthias 474
Dewdney, Alexander 71
Chodas, P.W. 538 Coyne, George 545 Dewdney, Alexander K. 529
Chab choub Chomsky, Noam 528 Crane, H. Richard 534 Dicke, R.H. 538
Christlieb, N. 539 Craven, J.D. 539 Diehl, Helmut 60
Christou, A.A. 538 Crespi, Roberto 556 Diel, Helmut 558

Motion Mountain – The Adventure of Physics
Chubykalo, A.E. 212, 539 Cross, R. 489 Diel, Isolde 558
Chudnovsky, D.V. 552 Crowe, Michael J. 533 Diez, Ulrich 557
Chudnovsky, G.V. 552 Crutchfield, J. 550 DiFilippo, Frank 556
Clancy, Tom 543 Crystal, David 553 Dill, L.M. 544
Clanet, Christophe 516 Cuk, M. 538 Dio, Cassius 540
Clarke, S. 527 Dirac, Paul 448
Clausius, Rudolph 393 D Dirr, Ulrich 557
life 395 Dahlman, John 556 DiSessa, A. 532
Claveria, V. 547 Dalton, John 391 Dixon, Bernard 528
Clavius, Christophonius Danecek, Petr 556 Di Iorio, Giorgio 246, 560

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
life 65 Daniels, Peter T. 553 Dobra, Ciprian 556
Cleobulus Dante Alighieri 173 Doorne, C.W.H. van 551
life 52 Darius, J. 532 Dorbolo, S. 547
CNAM 142, 559 Darley, Vincent 557 Dorbolo, Stéphane 379
CNES 223 Darre, Daniel 556 Douady, S. 550
CNRS 401, 562 Dartmouth College Electron Dougherty, R. 528
Cohen-Tannoudji, G. 549 Microscope Facility 392, Doyle, Arthur Conan 426
Cohen-Tannoudji, Gilles 399 562 Dragon, Norbert 556
Colazingari, Elena 556 Dasari, R.R. 546 Drake, Harry 78
Coleman-McGhee, Jerdome Davidhazy, Andrew 40, 327, Drake, Stillman 529
547 355, 558, 561 Dransfeld, Klaus 523
Colladon, Daniel 375 Davidovich, Lev 451 Dreybrodt, W. 551
Collins, J.J. 525 Davis, S.D. 545 Dubelaar, N. 504
Comins, Neil F. 538 de Bree, Hans Elias 310 Dubrulle, B. 540
Compton, A.H. 534 Deaver, B.S. 555 Duillier, Nicolas Fatio de 539
Compton, Arthur 141 Dehant, V. 535 Dumont, Jean-Paul 524, 532,
Conkling, J.A. 529 Dehmelt, H. 546 552
Connors, M. 538 Dehmelt, Hans 545 Duncan, Dugald 316, 561
Conrad 358 Dehn, Max Durben, D.J. 546
Conservatoire National des life 57 Durieux, M. 548
Arts et Métiers 559 Demaine, E.D. 527 Dusenbery, David 531
Conti, Andrea 556 Demaine, M.L. 527 Dutton, Z. 527
name index 567

Dyson, Freeman 556 Eötvös, Roland von 141 Franklin, Benjamin 335
Fraser, A.B. 545
E F French, Robert M. 528
Earls, K. 531 Fabian, Werner 557 Frenzel, H. 96
Earman, John 528 Fairbanks, J.D. 555 Fresnel, Augustin 304
Eckhardt, B. 551 Faisst, H. 551 Friedman, David 227
Eddington, Arthur Falk, G. 538 Friedrich Gauss, Carl 448
life 108 Falkovich, G. 547 Fritsch, G. 544
Edelmann, H. 539 Fantoli, Annibale 545 Fromental, J.-M. 547
Edens, Harald 557 Farinati, Claudio 556 Frosch, R. 542
Edwards, B.F. 495 Farmer, J. 550 Frova, Andrea 529
D Einstein, A. 548
Einstein, Albert 17, 197, 200,
Farrant, Penelope 531
Fatemi, M. 543
Full, R. 531
Full, Robert 91, 559
201, 256, 278, 280, 392 Fatio de Duillier, Nicolas 218 Fulle, Marco 22, 558
Dyson Ekman, Walfrid Faybishenko, B.A. 551 Fumagalli, Giuseppe 532, 541
life 533 Fayeulle, Serge 540 Fundamentalstation Wettzell
Ellerbreok, L. 544 Feder, T. 552 559

Motion Mountain – The Adventure of Physics
Ellis, H.C. 370 Fedosov, D.A. 547 Furrie, Pat 556
Els, Danie 556 Fehr, C. 541
Elsevier, Louis 479 Fekete, E. 538 G
Elswijk, H.B. 546 Feld, M.S. 546 Gagnan, Emile 368
Elswijk, Herman B. 556 Fermani, Antonio 556 Galilei, Galileo 43, 69, 75, 76,
Elsässer, Martin 190, 557 Fermat, Pierre 262 136, 156, 183, 202, 208, 210,
EMBO 47 Fernandez-Nieves, A. 536 338, 339, 479, 491
EMBO Journal, Wiley & Sons Feynman, Richard P. 523 life 34, 335
93 Fibonacci, Leonardo 451 Galileo 135, 451
Emelin, Sergei 556 life 446 Galois, Evariste 273

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Emerson, Ralph Waldo 318 Figueroa, D. 541 Gans, F. 532
Emsley, John 549 Filippov, T. 543 Gardner, Martin 487
Engelmann, Wolfgang 528 Fink, Hans-Werner 546 Garrett, A. 532
Engels, Friedrich 238 Fink, M. 544 Gaspard, Pierre 394, 549
Engemann, S. 546 Finkenzeller, Klaus 556 Gasse Tyson, Neil de 539
Enquist, B.J. 542 Firpić, D.K. 539 Gastaldi, Luca 20, 57, 207, 557
Enss, Christian 548 Fischer, Ulrike 556 Gauthier, N. 542
Eötvös, R. von 538 Flach, S. 550 Gekle, S. 547
Erdős, Paul 57 Flachsel, Erwein 524 Gelb, M. 525
Eriksen, H.K. 530 Fletcher, Neville H. 544 Gelbaum, Bernard R. 528
Erwin Sattler OHG 181, 560 Flindt, Rainer 527 Gemini Observatory/AURA
Espiritu, Zach Joseph 143, Foelix, Rainer F. 547 166
508, 512, 556, 559 Fokas, A.S. 329, 544 General Electric 313, 561
Euclid, or Eukleides 35 Foreman, M. 528 Geng, Tao 264
Euler, Leonhard 73, 117, 147, Forsberg, B. 540 GeoForschungsZentrum
221, 448, 533 Fortes, L.P. 540 Potsdam 189, 560
life 228 Foster, James 541 Georgi, Renate 556
European Molecular Biology Foucault, Jean Bernard Léon Gerkema, T. 533
Organization 558 life 140 Geschwindner, Holger 78
Evans, J. 539 Fourier, Joseph 307 Ghahramani, Z. 541
Everitt, C.W. 555 Frank, Louis A. 539 Ghavimi-Alagha, B. 537
Eöt-Wash Group 186, 560 Frank, Philipp 278 Giessibl, F.J. 546
568 name index

Gilligan, James 540 Gutzwiller, Martin C. 537 Helmholtz, Hermann von 113
Glaser, John 509, 563 Gácsi, Zoltán 556 life 388
Glasheen, J.W. 536 Günther, B. 528 Helmond, Tom 556
Glassey, Olivier 556 Helmont, Johan Baptista van
Gold, Tommy 303 H 389
Goldpaint, Brad 206, 560 Haandel, M. van 537 Hembacher, S. 546
Goldrich, P. 514 Haber, John 556 Henbest, Nigel 537
Goldstein, Herbert 541 Hagen, J.G. 534 Henderson, D.M. 544
Goldstein, E. Bruce 525 Hagen, John 141 Henderson, D.W. 541
Goles, E. 551 Haigneré, Jean-Pierre 223 Henderson, Diane 330, 561
Golubitsky, M. 525 Hain, Tim 558 Henderson, Lawrence J. 549
G Gompper, G. 547
González, Antonio 556
Halberg, F. 528
Haley, Stephen 556
Henderson, Paula 556
Henon, M. 540
Gooch, Van 46 Hallerbach, Andreas 319, 561 Henson, Matthew 496
Gilligan Gordillo, J.M. 547 Halley, Edmund 192 Heraclitus
Goriely, A. 546 Halliday, David 547 life 17
Goss, Jonathan 269, 561 Halliwell, J.J. 528 Heraclitus of Ephesus 27, 270

Motion Mountain – The Adventure of Physics
Gostiaux, L. 533 Hamilton, D.P. 538 Herbert, Wally 536
Grabski, Juliusz 540 Hamilton, William 448 Hermann, Jakob 500
Gracovetsky, Serge 123, 532 Hammack, J.L. 544 Heron of Alexandria 374
Grahn, Alexander 557 Hammerl, German 561 Herrmann, F. 493, 549
Graner, F. 540 Hancock, M.J. 546 Herrmann, Friedrich 523, 532
’s Gravesande, Willem Jacob Hankey, Mike 258, 561 Herschel, William 153
111 Hardcastle, Martin 556 Hertz, H.G. 532
’s Gravesande 262 Hardy, Godfrey H. Hertz, Heinrich 254
Graw, K.-U. 514 life 178 Hertz, Heinrich Rudolf
Gray, C.G. 496, 541 Hare, Trent 560 life 236

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Gray, James 531 Harriot, Thomas 448 Hertzlinger, Joseph 557
Gray, Theodore 471 Harris, S.E. 527 Hestenes, D. 540
Greenleaf, J.F. 543 Harrison, J.J. 273 Heumann, John 556
Greenside, Henry 471 Hart, Nathan H. 530 Hewett, J.A. 537
Gregorio, Enrico 557 Hartman, W.K. 537 Hewitt, Leslie A. 523
Greier, Hans-Christian 220, Hasha, A.E. 546 Hewitt, Paul G. 523
557 Hausherr, Tilman 556 Hietarinta, J. 544
Greiner, Jochen 556 Hawkins, Daniel 66, 483, 484, Hietarinta, Jarmo 317, 329, 557
Grimaldi, Francesco 321 562 Higashi, R. 555
Grinder, John 241, 525 Hayes, Allan 556 Hilbert, David 256, 280
Groenendijk, Max 40, 558 Hays, J.D. 535 Hilgenfeldt, S. 531
Gross, R.S. 535 Heath, Thomas 533 Hillman, Chris 556
Grossmann, A. 544 Heber, U. 539 Hipparchos 146
Gruber, Werner 533 Heckel, B.R. 537 Hirano, M. 540
Grünbaum, Branko 542 Heckman, G. 537 Hirota, R. 329, 544
Guericke, Otto von 361 Hediger, Heini 529 Hise, Jason 90, 168, 557
Guglielmini, Heisenberg, Werner 266, 362, Hite, Kevin 557
Giovanni Battista 136 399, 542 Hodges, L. 539
Guillén Oterino, Antonio 91, Helden, A. van 537 Hodgkin, A.L. 315, 543
559 Heller, Carlo 45 Hoeher, Sebastian 546
Gustav Jacobi, Carl 448 Heller, E. 543 Hof, B. 551, 552
Gutierrez, G. 541 Hellinger, Bert 540 Hoffman, Donald D. 26
name index 569

Hoffman, Richard 556 IPCC 516 Joutel, F. 538
Hohenstatt, M. 546 Iriarte-Díaz, J. 536 Jouzel, J. 535
Holbrook, N.M. 545 Iriarte-Díaz, José 171, 560 Jouzel, Jean 154, 559
Hollander, E. 547 ISTA 384, 562 JPL 215
Hong, F.-L. 555 ISVR, University of Julien Brianchon, Charles 73
Hong, J.T. 551 Southampton 299, 311, 557 Jürgens, Hartmut 424, 528
Hooke, Robert 178, 288 Ivanov, Igor 556
life 176 Ivry, R.B. 527 K
Hoorn, L. van den 544 Köppe, Thomas 556
Hoornaert, Jean-Marie 132, J Kadanoff, L.P. 551
559 J. Lynch, Patrick 348 Kalvius, Georg 523
H Hopper, Arthur F. 530
Horace, in full Quintus
Jacobi, Carl 255
Jaeger-LeCoultre 131, 559
Kamerling Onnes, Heike 534
Kamiya, K. 510
Horatius Flaccus 237 Jaffe), C. Carl 348 Kamp, T. Van de 541
Hoffman Hosoi, A.E. 546 Jahn, K. 555 Kampen, N.G. van 549
Hotchner, Aaron E. 541 Jahn, O. 542 Kanada Yasumasa 553
Hoyle, R. 483 Jalink, Kim 556 Kanecko, R. 540

Motion Mountain – The Adventure of Physics
Huber, Daniel 556 James, M.C. 539 Kant, Immanuel 201
Hudson, R.P. 548 Jamil, M. 556 Kantor, Yacov 471
Huggins, E. 507 Jammer, M. 537 Kapitaniak, Tomasz 540
Hughes, David W. 541 Jancovici, B. 539 Karl, G. 496, 541
Humphreys, R. 535 Janek, Jürgen 556 Karstädt, D. 548
Humphreys, W.J. 550 Japan Aerospace Exploration Katori, H. 555
Hunklinger, Siegfried 548 Agency 559 Kawagishi, I. 531
Hunter, D.J. 530 Jarvis, Dave 88, 558 Keath, Ed 133, 559
Huxley, A.F. 315, 543 Javi, Karva 91, 559 Keesing, R.G. 527, 555
Huygens, Christiaan 180, 304 JAXA 142, 559 Kelu, Jonatan 556

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
life 103 Jeffreys, Harold 145 Kemp, David 303
Hörnchen, Ingrid 562 Jen, E. 550 Kempf, Jürgen 527
Hüsler, Cédric 429, 562 Jennings, G.M. 545 Kenderdine, M.A. 532
Jensen, Hans 553 Kendrick, E. 540
I Jesus 376 Kenji Lopez-Alt 377, 562
IBM 344 Job, Georg 523 Kennard, Earle Hesse 524
Ibn Khallikan Johansson, Mikael 556 Kepler, Johannes 176, 178
life 477 John Paul II 337 life 175
IFE 360, 561 Johnson, Ben 530 Kerswell, R. 551
ifm 358 Johnson, Michael 78 Kessler, Raphael 429, 562
Ifrah, Georges 553 Johnson, Samuel 441, 469 Keyence 54, 558
Illich, Ivan 470 Johnston, K.J. 530 Kienle, Paul 523
life 469 Jones, Gareth 550 Kirby, Jack 559
Imae, Y. 531 Jones, Quentin David 556 Kirchhoff, Gustav 304
Imbrie, J. 535 Jones, Tony 554 Kiss, Joseph 556
Ingenhousz, Jan 391 Jones, William 448 Kistler, S.F. 377, 547
Ingram, Jay 544 Jong, Johan de 390, 562 Kitaoka Akiyoshi 16, 18, 524,
INMS 45 Jong, Marc de 556 557
Innanen, K.A. 538 Joule, James 388 Kitaoka, A. 524
Inquisition 336 Joule, James P. 113 Kitaoka, Akiyoshi 524
International Tennis Joule, James Prescott Kivshar, Y.S. 550
Federation 20, 557 life 388 Klaus Tschira Foundation 557
570 name index

Klaus, Tobias 517, 563 Lambeck, K. 534 Le Verrier, Urbain 196
Kleidon, A. 532 Lambert, S.B. 535 Liaw, S.S. 551
Kluegel, T. 534 Lancaster, George 541 Libbrecht, Kenneth 422, 431,
Klügel, T. 534 Landau, Lev 451 557, 562
Klügel, Thomas 559 Landauer, R. 550 Lichtenberg, Georg Christoph
Kob, M. 542 Lang, H. de 550 life 34
Koblenz, Fachhochschule 35, Lang, Kenneth R. 87 Lide, David R. 553
558 Langangen, Ø. 530 Lim Tee Tai 379, 380, 557
Koch, G.W. 545 Lanotte, L. 547 Lim, T.T. 547
Kooijman, J.D.G. 491 Laplace, Pierre Simon 196 Linde, Johan 557
Koomans, A.A. 546 life 138 Lintel, Harald van 556
K Korbasiewicz, Marcin 561
Korteweg, Diederik 316
Larsen, J. 535
Laser Components 115, 559
Lissauer, J.J. 552
Lith-van Dis, J. van 549
Kramp, Christian 448 Laskar, J. 538, 539 Liu, C. 527
Klaus Krampf, Robert 471 Laskar, Jacques 219 Liu, R.T. 551
Krehl, P. 546 Laveder, Laurent 223, 560 Llobera, M. 533
Krehl, Peter 349 Lavenda, B. 549 Lloyd, Seth 549

Motion Mountain – The Adventure of Physics
Krijn, Marcel 556 Lavoisier, Antoine-Laurent Lock, S.J. 538
Kristiansen, J.R. 530 life 102 Lockwood, E.H. 542
Krívchenkov, V.D. 524 Laws, Kenneth 552 Lodge, Oliver
Krotkow, R. 538 Le Sage, Georges-Louis 218 life 144
Kruskal, M.D. 543 Lecar, M. 540 Lohse, D. 531, 547
Kruskal, Martin 317 Lee, C.W. 552 Lohse, Detlef 96, 559
Królikowski, Jarosław 556 Lee, S.J. 536 Lombardi, Luciano 556
Kubala, Adrian 556 Lega, Joceline 551 LordV 342, 561
Kudo, S. 531 Legendre, Adrien-Marie 448 Loschmidt, Joseph
Kumar, K.V. 530 Lehmann, Inge 146 life 341

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Kunze, Michael 166, 195, 557, Lehmann, Paul 449, 554 Lott, M. 552
560 Leibniz, Gottfried Wilhelm Lucretius 339
Kurths, Jürgen 538 110, 178, 248, 252, 255, 261, life 40
Kuzin, Pavel 556 262, 448, 527 Lucretius Carus, Titus 40, 339
Kvale, E.P. 535 life 83, 438 Luke, Lucky 347
Köller, Karl 557 Leica Geosystems 54, 558 Lévy-Leblond, Jean-Marc 120,
König, Samuel 262 Leidenfrost, Johann Gottlob 219, 523
Kötter, K. 551 377 Lüders, Klaus 524
Kötter, Karsten 421, 422, 562 Leighton, Robert B. 523 Lüthi, Thomas 246, 560
Küster, Johannes 557 Lekkerkerker, H. 543 Lütjens, Jörn 488, 562
Küstner, Friedrich 535 Lemaire, Alexis 447
life 146 Lennard, J. 449 M
Lennerz, C. 495 MacDonald, G.J.F. 534
L Leon, J.-P. 328, 544 MacDougall, Duncan 533
La Caille 179 Leonardo of Pisa MacDougalls, Duncan 126
Lagrange, Joseph Louis 262 life 446 Mach, Ernst
Lagrange, Joseph Louis 251 Leosphere 376, 562 life 103
Lagrangia, Giuseppe 448 Lesage, G.-L. 539 Macmillan, R.H. 542
Lagrangia, Leucippus of Elea Maekawa, Y. 531
Giuseppe Lodovico life 339 Magariyama, Y. 531
life 251 Leutwyler, K. 543 Magono, C. 552
Lalande 179 Lewallen, Charles 170, 560 Mahajan, S. 514
name index 571

Mahoney, Alan 556 Medenbach, Olaf 529 Moreira, N. 542
Mahrl, M. 551 Medien Werkstatt 558 Morgado, E. 528
Maiorca, Enzo 547 Meer, D. van der 547 Morgado, Enrique 47, 558
Maldacena, J. 552 Meijaard, J.P. 491 Morlet, J. 544
Malebranche 262 Meister, M. 526 Morris, S.W. 551
Malin, David 87 Melo, F. 550 Moscow State Circus 23, 558
Mallat, Stéphane 544 Mendeleyev, Moser, Lukas Fabian 556
Malley, R. 540 Dmitriy Ivanovich 108 Mould, Steve 133
Mandelbrot, Benoît 54, 333 Mendez, S. 547 Mozurkewich, D. 530
Mandelstam, Stanley 541 Mendoza, E. 551 Msadler13 433
Manfredi, Eustachio 152 Merckx, Eddy 114 Muller, R.A. 543
M Mannhart, J. 546
Manogg, P. 544
Mermin, David 269, 561
Merrit, John 556
Mulligan, J.F. 532
Munk, W.H. 534
Manu, M. 526 Mertens, K. 551 Muramoto, K. 531
Mahoney Marcus, Richard 525 Mettler-Toledo 108, 559 Murata, Y. 540
Marcus, Ron 387, 562 Mettrie, J. Offrey de la 426 Murdock, Ron 556
Marcy, Geoffrey 193, 560 Miákishev, G.Ya. 524 Mureika, J.R. 540

Motion Mountain – The Adventure of Physics
Mariotte, Edme 123 Michaelson, P.F. 555 Murillo, Nadia 556
Maris, H. 546 Michel, Stanislav 110 Murphy, Robert 448
Mark, Martin van der 409, Michelangelo 451 Murray, C.D. 538
556 Michelberger, Jürgen 88, 558 Muskens, O. 543
Markus, M. 551 Michell, John Mutka, Miles 556
Marsaglia, G. 540 life 184 Muynck, Wim de 556
Martin, A. 542 Michels, Barry 541 Måløy, K.J. 550
Martina, L. 328, 544 Micro Crystal 557 Möllmann, K.-P. 548
Martos, Antonio 556 Microcrystal 291 Möllmann, Klaus-Peter 384,
Martz, Paul 334, 561 Microflown 561 562

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Martínez, E. 550 Microflown Technologies 312 Müller, Frank 55, 558
Marx, Groucho 443 Mikaelyan, Robert 206, 560 Müller, G. 551
Matthews, R. 539 Mikkola, S. 538 Müller, Gerhard 428, 429
Mauer, J. 547 Mikkola, Seppo 196, 560 life 562
Maupertuis, Pierre Louis Milankovitch, Milutin
Moreau de life 153 N
life 136 Miller, L.M. 532 Namba, K. 531
Mayer, Julius Robert 113 Mineyev, A. 545 Namouni, F. 538
life 388 Minnaert, Marcel G.J. 98 Namouni, Fathi 556
Mayné, Fernand 556 Minski, Y.N. 527 Nanosurf 345
Mayr, Peter 556 Mirabel, I.F. 527 Nansen, Fridtjof 533
Mays, D.C. 551 Mirsky, S. 489 Napiwotzki, R. 539
Mays, David 420, 562 Mitalas, R. 527 Naples Zoo 54
McCoy, Ray 122, 559 Mitchell, J.S.B. 527 Naples zoo 558
McElwaine, J.N. 551 Mohazzabi, P. 539 Napoleon 196
McLaughlin, William 524 Mole, R. 543 NASA 45, 108, 149, 215, 364,
McLean, H. 530 Monitz, E.J. 532 516, 559
McMahon, T.A. 536 Moore, J.A. 541 Nassau, K. 98
McMahon, Thomas A. 545 Moortel, Dirk Van de 556 Nauenberg, M. 536
McMillen, T. 546 Moreau de Maupertuis, Navier, Claude
McNaught, Robert 95, 559 Pierre Louis life 363
McQuarry, George 556 life 136 Nebrija, Antonio de 553
572 name index

Needham, Joseph 541 Olveczky, B.P. 526 life 394
Nelsen, Roger B. 73 Oostrum, Piet van 556 Pesic, Peter 545
Nelson, A.E. 537 Oppenheimer, Robert 253 Peters, I.R. 547
Nelson, Edward 483 Origone, Simone 209, 560 Peterson, Peggy 71, 558
Neuhauser, W. 545 Osaka University 94, 559 Petit, Jean-Pierre 473
Neuhauser, Werner 344, 561 Osborne, A.R. 544 Pfeffer, W.T. 543
Neumaier, Heinrich 556 Osman, V. 529 Phillips, R.J. 537
Neumann, Dietrich 192, 538 Otto-von-Guericke- Phinney, S. 514
Newcomb, Simon 197, 538 Gesellschaft 361, Physics Today 432, 562
Newton, Isaac 48, 136, 178, 562 Piaget, Jean 525
458, 527 Oughtred, William 448 Piccard, Auguste
N life 34
Nicolis, Grégoire 549 P
life 375
Pieranski, P. 529
Nicoud, F. 547 Page, Don 556 Pieranski, Piotr 73, 558
Needham Niederer, U. 279, 542 Page, Janet 557 Pietrzik, Stefan 505, 558, 563
Nieman, Adam 371, 562 Pahaut, Serge 556 Pikler, Emmi 525
Niepraschk, Rolf 557 Pais, A. 542 Pikovsky, Arkady 538

Motion Mountain – The Adventure of Physics
Nieuwpoort, Frans van 556 Palmer, John D. 527 Pinker, Steven 528
Nieuwstadt, F.T.M. 551 Panov, V.I. 539 Pinno, F. 548
Nieves, F.J. de las 536 Papadopoulos, J.M. 491 Pinno, Frank 562
Nightingale, J.D. 541 Pappus 100 Pinzo 160, 559
Nitsch, Herbert 371 Park, David 524 PixHeaven.net 223
Nobel, P. 545 Parkes, Malcolm B. 554 Planck, Max 392, 398
Noda, Claro 562 Parks, David 91, 556, 558 Plato 435
Noether, Emmy Parlett, Beresford 25 life 339
life 280 Parmenides of Elea 17, 83 Plinius, in full Gaius Plinius
Nolte, John 526 Pascazio, Saverio 556 Secundus 523

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Nonnius, Peter Paschotta, Rüdiger 303, 561 Plisson, Philip 23, 558
life 65 Pasi, Enrico 556 Plutarchus 242
Norfleet, W.T. 530 Paterson, Alan L.T. 528 Pohl, Robert 524
Northwestern University 50, Patrascu, M. 527 Pohl, Robert O. 524
86, 325, 558 Paul, T. 544 Pohlad, Bob 429, 562
Novikov, V.A. 496, 541 Pauli, Wolfgang 500 Poincaré, Henri 156
Nowitzki, Dirk 78 Peano, Giuseppe 448 Poinsot, Louis 141, 165
Nuñes, Pedro Peary, Robert 135, 496, 536 Poisson, Siméon-Denis
life 65 Pěč, Karel 529 life 188
Nylander, Paul 425, 562 Pedley, T.J. 530 Polster, B. 549
Peeters, Bert 556 Pompeius, Gnaeus 242
O Peitgen, Heinz-Otto 424, 528 Preston-Thomas, H. 548
O’Keefe, R. 537 Pekár, V. 538 Price, R.H. 106
Oase GmbH 227, 560 Pelizzari, Umberto 547 Prigogine, Ilya 383, 428
Oberdiek, Heiko 556 Pempinelli, F. 328, 544 Pritchard, Carol 556
Oberquell, Brian 556 Perc, M. 532 Proença, Nuno 556
Offner, Carl 556 Perelman, Yakov 324, 375, 523 Protagoras 457
Offrey de la Mettrie, J. 426 Perini, Romano 556 Proton Mikrotechnik 358
Ogburn, P.L. 543 Perlikowski, Przemyslaw 540 Provo, Neil 156, 559
Oliver, B.M. 505 Perlman, Howard 371, 562 Pryce, Robert M. 536
Olmsted, John M.H. 528 Perrin, J. 548 Przybyl, S. 529
Olsen, K. 529 Perrin, Jean 518, 548 Ptolemy 69, 262, 529
name index 573

Purves, William 556 Rivera, A. 550 Sands, Matthew 523
Putkaradze, V. 551 Rivest, R.L. 527 Santini, P.M. 329, 544
Putkaradze, Vakhtang 427, Robertson, Will 557 Santorio Santorio
562 Robinson, John Mansley 542 life 102
Putterman, S.J. 531 Robinson, John Mansley 526 Saráeva, I.M. 524
Pythagoras 318 Robutel, P. 538 Sattler OHG, Erwin 560
Päsler, Max 541 Rodgers, Tony 26, 558 Sattler-Rick, Stephanie 560
Pérez-Mercader, J. 528 Rodin, Auguste 180 Sauer, J. 495
Pérez-Penichet, C. 550 Rodríguez, L.F. 527 Sauerbruch, Ferdinand 516
Rohrbach, Klaus 535 Saupe, Dietmar 424, 528
Q Roll, P.G. 538 Savela, Markku 267, 561
P Quinlan, G.D. 539
Quéré, D. 547
Romer, R.H. 506
Rompelberg, Fred 381
Scagell, Robin 210, 560
Schiller, Britta 556, 557
Quéré, David 381, 562 Rooy, T.L. van 546 Schiller, C. 546
Purves Rosek, Mark 560 Schiller, Christoph 274, 563
R Rosenblum, Michael 538 Schiller, Isabella 556
R. Rees, Ian 167, 560 Rossing, Thomas D. 544 Schiller, Peter 556

Motion Mountain – The Adventure of Physics
Rahtz, Sebastian 556 Rostock, University of 375 Schiller, Stephan 556
Ramanujan, Srinivasa 178 Rothacher, M. 534 Schlichting, H. Joachim 524
Ramos, O. 550 Rousset, Sylvie 562 Schlichting, H.-J. 499
Randi, James 22 Ruben, Gary 556 Schlüter, W. 534
Rankl, Wolfgang 556 Rudolff, Christoff 447, 554 Schmidt, M. 543
Rawlins, D. 536 Rudolph, Peter 556 Schmidt, T. 551
Raymond, David 472 Rueckner, Wolfgang 335, 561 Schmidt-Nielsen, K. 550
Rechenberg, Ingo 91, 531, 559 Ruga, Spurius Carvilius 442 Schneider, Jean 541
Recorde, Robert 447 Ruina, A. 491 Schneider, M. 534
life 447 Ruppel, W. 538 Schneider, Wolf 554

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Redondi, Pietro 34, 335, 336, Rusby, R.L. 548, 550 Schooley, J.F. 548
556 Rusby, Richard 550 Schreiber, K.U. 534
Rees, W.G. 523 Russel, J.S. 543 Schreiber, U. 534
Reichholf, Josef H. 552 Russel, Mark 224 Schröder, Ernst 448
Reichl, Linda 548, 549 Russell, Bertrand 253 Schultes, H. 96
Reimers, D. 539 Russell, John Scott Schwab, A.L. 491
Renselle, Doug 556 life 316 Schwab, Arend 492, 563
Reppisch, Michael 556 Russo, Lucio 523 Schwartz, Richard 525
Resnick, Robert 547 Rutherford, Ernest 24 Schwartz, Robert 137, 559
Richardson, Lewis Fray RWE 353, 561 Schwenk, K. 550
life 53 Régnier, Gilles 197, 560 Schwenkel, D. 546
Richtmyer, Floyd Karker 524 Schönenberger, C. 546
Rickwood, P.C. 529 S Schörner, E. 495
Riedel, A. 541 S.R. Madhu Rao 556 Science 154
Rieder, Heinz 479, 562 SABAM 351, 561 Science Photo Library 91
Rieflin, E. 531 Sade, Donatien de 237 Science/AAAS 559
Rieker Electronics 86, 558 Saghian, Damoon 556 Scime, E.E. 495
Rinaldo, A. 542 Sagnac, Georges Scott, Jonathan 556
Rincke, K. 523 life 141 Scriven, L.E. 377, 547
Rindt, Jochen San Suu Kyi, Aung 541 Sean McCann 160, 559
life 35 Sanctorius Sebman81 365, 562
Rivas, Martin 556 life 102 Segrè, Gino 548
574 name index

Segur, H. 544 Sluckin, T.J. 533 Strzalko, Jaroslaw 540
Seidelmann, P. Kenneth 555 Smith, George 554 Stutz, Phil 541
Seitz, M. 493 Smith, Steven B. 553 Su, Y. 539
Settles, Gary 327, 561 Smoluchowski, Marian von Suchocki, John 523
Seward, William 553 392 Sugiyama, S. 531
Sexl, Roman Snelson, Kenneth 349, 561 Suhr, S.H. 536
life 60 Socrates 435 Sum, Robert 561
Shackleton, N.J. 535 Sokal, A.D. 495 Supranowitz, Chris 293, 561
Shakespeare, William 22 Soldner, J. 538 Surdin, Vladimir 556
Shalyt-Margolin, A.E. 549 Soldner, Johann 201 Surry, D. 530
Shapiro, A.H. 534 Solomatin, Vitaliy 556 Sussman, G.J. 539
S Shapiro, Asher 140
Sharma, Natthi L. 532
Sonett, C.P. 535
Song, Y.S. 536
Sutton, G.P. 541
Swackhamer, G. 540
Shaw, George Bernard Sotolongo-Costa, O. 550 Swatch Group 557
Segur alphabet of 446 Spaans, Piet 358 Sweetser, Frank 556
Sheldon, Eric 556 Speleoresearch & Films/La Swenson, C.A. 548
Shen, A.Q. 546 Venta 58, 558 Swift, G. 550

Motion Mountain – The Adventure of Physics
Shen, Amy 351, 352, 561 Spence, J.C.H. 546 Swinney, H.L. 550
Sheng, Z. 546 Spencer, R. 527 Swope, Martha 552
Shenker, S.H. 552 Spiderman 347 Szczepanski, Daniela 380, 562
Shephard, G.C. 542 Spiess, H. 292, 561 Szilard, L. 549
Shinjo, K. 540 Spinelli, Matthew 87, 558 Szilard, Leo 399
Shirham, King 32 Spiropulu, M. 537
Short, J. 555 SRAM 115, 559 T
Siart, Uwe 557 Sreedhar, V.V. 542 T. Schmidt, Klaus 530
Siche, Dietmar 343, 561 Sreenivasan, K.P. 547 Taberlet, N. 551
Siebner, Frank 509, 563 ST Microelectronics 160, 559 Tabor, D. 540

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Siegel, Lee 491 Stalla, Wolfgang 346 Taimina, D. 541
Sierra, Bert 556 Stanaway, F.F. 555 Tait, Peter 448
Sigwarth, J.B. 539 Stanford, D. 552 Takamoto, M. 555
Sillett, S.C. 545 Stasiak, A. 529 Talleyrand 183
Sills, K.R. 527 Stedman, G.E. 534 Tamman, Gustav 345
Silva 35, 558 Stefanski, Andrzeij 540 Tan, G. 529
Silverman, M. 531 Steiner, Kurt 547 Tan, R. 529
Silverman, Mark P. 521, 539 Stengel, Ingrid 527 Tarko, Vlad 556
Simanek, Donald 526 Stephenson, Richard 535 Tarski, Alfred 333
Simmons, J.A. 544 Sternlight, D.B. 536 life 57
Simon, Julia 556 Steur, P.P.M. 548 Tartaglia, Niccolò 451
Simon, M.I. 531 Stewart, I. 525 life 33
Simonson, A.J. 505 Stewart, Ian 58 Taylor, E.F. 541
Simoson, A.J. 540 Stewart, S.T. 538 Taylor, G.J. 537
Simplicius 436 Stieber, Ralph 559 Taylor, John R. 555
Singh, C. 532 Stokes, Georges Gabriel Taylor, Rhys 155, 557
Singleton, Douglas 556 life 363 Technical University
Sissa ben Dahir 32 Stong, C.L. 530 Eindhoven 381
Sitti, M. 536 Story, Don 556 Teeter Dobbs, Betty Jo 527
Sitti, Metin 170 Stroock, A.D. 545 Tegelaar, Paul 556
Slabber, André 556 Strunk, C. 523 Telegdi, V.L. 550
Slansky, Peter C. 532 Strunk, William 554 Tennekes, Henk 38, 558
name index 575

TERRA 169 U Waleffe, F. 551
Tetzlaff, Tim 558 Ucke, C. 499 Walgraef, Daniel 551
Tezel, Tunc 175, 560 Ucke, Christian 524 Walker, J. 547
Thaler, Jon 556 Uffink, J. 549 Walker, Jearl 377, 524, 547
Theon 95 Uguzzoni, Arnaldo 556 Walker, John 185, 560
Theophrastus 40, 44 Ulam, Stanislaw 415 Wallis, John 448
Thermodevices 387, 562 Ullman, Berthold Louis 554 Walter, Henrik 541
Thies, Ingo 556 Umbanhowar, P.B. 550 Ward, R.S. 543
Thinktank Trust 108, 559 Umbanhowar, Paul 421, 562 Warkentin, John 557
Thomas Aquinas 336, 525 University of Rochester 293, Weber, R.L. 551
Thomson, W. 546 561 Webster, Hutton 554
T Thomson-Kelvin 113
Thomson-Kelvin, W. 351
Universität Augsburg 344
Universum 387, 562
Wedin, H. 551
Wegener, Alfred 147, 535
Thomson-Kelvin, William Upright, Craig 556 Wehner, R. 536
TERRA life 388 USGS 371, 562 Wehus, I.K. 530
Thorn, Christopher 328, 561 Weierstall, U. 546
Thoroddsen, S.T. 546 V Weierstrass, Karl 448

Motion Mountain – The Adventure of Physics
Thoroddsen, Sigurdur 351, 561 Vagovic, P. 541 Weil, André 449
Thorp, Edward O. 533 Vandewalle, N. 547 Weiss, M. 530
Thum, Martin 362, 562 Vanier, J. 554 Weiss, Martha 556
Titius, Johann Daniel 219 Vannoni, Paul 556 Wells, M. 540
Titze, I.R. 542 Vareschi, G. 510 Weltner, K. 547
Titze, Ingo 542 Veillet, C. 538 Weninger, K.R. 531
Tonzig, Giovanni 524 Velikoseltsev, A. 534 West, G.B. 542
Toparlar, Y. 549 Verne, Jules 493, 506 Westerweel, J. 551
Topper, D. 537 Vernier, Pierre Weyand, P.G. 536
Torge, Wolfgang 529 life 65 Weyl, Hermann

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Torricelli, Evangelista 95 Vestergaard Hau, L. 527 life 49
Toschek, P.E. 546 Victor Poncelet, Jean 73 Wheeler, John 475
Townsend, Paul 556 Vincent, D.E. 537 Wheeler, T.D. 545
Tracer 35, 558 Virgo, S.E. 545 White, E.B. 554
Trefethen, L.M. 534 Vitali, Giuseppe 55 White, M. 536
Tregubovich, A.Ya. 549 Viviani, Vincenzo 141 Whitney, Charles A. 87
Trevorrow, Andrew 556 Vlaev, S.J. 212, 539 Widmann, Johannes 447
Truesdell, C. 547 Vogel, Steven 532, 551 Widmer-Schnidrig, R. 542
Truesdell, Clifford 34 Voith 352, 561 Wiegert, P.A. 538
Truesdell, Clifford A. 527 Volin, Leo 556 Wienerroither, Peter 476, 562
Tsang, W.W. 540 Vollmer, M. 548 Wierda, Gerben 556
Tschichold, J. 554 Vollmer, Michael 384, 562 Wierzbicka, Anna 528, 556
Tschira, Klaus 557 Voltaire 221, 458 Wijk, Mike van 556
Tsuboi, Chuji 529 life 255 Wikimedia 35, 45, 115, 130, 218,
Tsukanov, Alexander 23, 558 Vorobieff, P. 551 232, 292, 296, 302, 308, 313,
Tucholsky, Kurt 271 Voss, Herbert 556 328, 330, 346, 387, 558–562
Tuijn, C. 489 Vries, Gustav de 316 Wikimedia Commons 559,
Tuinstra, B.F. 507 560
Tuinstra, F. 507 W Wilder, J.W. 495
Tuppen, Lawrence 556 Wagon, Stan 58, 528 Wiley, Jack 546
Turner, M.S. 536 Wah 217, 560 Wilk, Harry 529
Wald, George 400, 549 Wilkie, Bernard 531
576 name index

Williams, G.E. 535 Wulfila 445 Zaccone, Rick 557
Willis, Eric 296, 561 Zakharian, A. 535
Wilson, B. 544 Y Zalm, Peer 556
Wiltshire, D.L. 534 Yamafuji, K. 546 Zanker, J. 526
Wisdom, J. 539 Yamaguchi University 428, Zecherle, Markus 556
Wise, N.W. 555 562 Zedler, Michael 556
Wittgenstein, Ludwig 16, 24, Yang, C.C. 551 Zeh, Heinz-Dieter 528
27, 48, 85, 157, 238, 402, 437 Yatsenko, Dimitri 482 Zenkert, Arnold 527
Wittlinger, M. 536 Yeoman, Donald 215 Zeno of Elea 15, 17, 66, 83
Wittlinger, Matthias 171 Young, Andrew 556 Zheng, Q. 546
Wolf, G.H. 546 Young, James A. 531 Ziegler, G.M. 57
W Wolf, H. 536
Wolfsried, Stephan 58, 558
Young, Thomas 111
Yourgray, Wolfgang 541
Zimmermann, H.W. 549
Zimmermann, Herbert 399
Wolpert, Lewis 525 Yukawa Hideki 451 Zurek, Wojciech H. 528
Williams Wong, M. 531 Zweck, Josef 533
Woods Hole Oceanographic Z Zwieniecki, M.A. 545
Institution 371, 562 Zabunov, Svetoslav 164 Zürn, W. 542

Motion Mountain – The Adventure of Physics
Wright, Joseph 557 Zabusky, N.J. 543
Wright, S. 536 Zabusky, Norman 317

Symbols abjad 445 physical 248
(, open bracket 448 abugida 445 principle 242
), closed bracket 448 Academia del Cimento 77 principle of least 190
+, plus 447 acausality 239, 240 quantum of 399
−, minus 447 acceleration 85 action, quantum of, ℏ

Motion Mountain – The Adventure of Physics
⋅, multiplied by 448 angular 118 physics and 8
×, times 448 animal record 169 actuator 233
:, divided by 448 centrifugal 161 table 233
<, is smaller than 448 centripetal 161, 179 addition 273
=, is equal to 447 dangers of 96 additivity 37, 43, 51, 106
>, is larger than 448 due to gravity, table of of area and volume 56
!, factorial 448 values 182 adenosine triphosphate 532
[], measurement unit of 449 effects of 85 aeroplane
⌀, empty set 449 highest 96 flight puzzle 96
=,̸ different from 448 of continents 490 speed unit 103

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
√ , square root of 447 sensors, table 86 speed, graph of 38
@ , at sign 449 table of values 84 toilet 380
Δ, Laplace operator 448 tidal 199 aerostat 375
∩, set intersection 448 accelerometers 477 AFM see microscope, atomic
∪, set union 448 photographs of 86 force
∈, element of 448 accents aggregate
∞, infinity 448 in Greek language 445 of matter 257
∇, nabla/gradient of 448 accumulation 356 overview 259
⊗, dyadic product 448 accuracy 437, 459 table 259
⊂, subset of/contained in 448 limits to 460 air 516
⊃, superset of/contains 448 why limited? 437 composition table 516
⟨ |, bra state vector 448 Acetabularia 46 jet, supersonic 370
| ⟩, ket state vector 448 Acinonyx jubatus 36 pressure 110
|𝑥|, absolute value 448 Ackermann steering 484 resistance 80
∼, similar to 448 acoustical thermometry 548 Airbus 479
action 248–265, 276 airflow instruments 309
A as integral over time 251 alchemy 34
𝑎𝑛 448 definition 251 Aldebaran 87, 261
𝑎𝑛 448 is effect 252 aleph 445
a (year) 41 is not always action 248 algebraic surfaces 473
abacus 446 measured values 250 Alice 404
aberration 135, 152 measurement unit 252 Alpha Centauri 261
578 subject index

alphabet erythrocephalus 91 athletics 171
Cyrillic 445 Apis mellifera 105 and drag 235
first 441 apnoea 371–372 atmosphere 159, 364
Greek 443, 444 apogee 465 angular momentum 118
Hebrew 445 Apollo 502 composition 516
Latin 441 apple composition table 516
modern Latin 443 and fall 100 layer table 366, 367
phonemic 445 standard 458 of the Moon 411
Runic 445 trees 458 atmospheric pressure 465
story of 442 approximation atom
syllabic 445 continuum 89 manipulating single 344
A alphasyllabaries 445
Alps 189
Aquarius 213
aqueducts 370
Atomic Age 385
atomic clock 45
weight of the 219 Arabic digit see number atomic force microscope 495
alphabet Altair 261 system, Indian atomic mass unit 463
alveoli 347, 368 Arabic number see number Atomium 351
Amazon 148 system, Indian atoms

Motion Mountain – The Adventure of Physics
Amazon River 321, 379 Arabidopsis 46 and breaking 338
Amoeba proteus 91 Arcturus 261 and Galileo 336
ampere area are not indivisible 406
definition 453 additivity 56 arranging helium 344
amplitude 289 existence of 56, 58 explain dislocations 343
anagyre 133 argon 516 explain round crystal
analemma 212, 505 Aries 213 reflection 343
angels 21, 99 Aristarchus of Samos 533 explain steps 343
angle 118 Aristotle 500 Galileo and 335–338
and knuckles 68 arithmetic sequence 477 Greeks thinkers and 339

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
in the night sky 487 arm image of silicon 344
plane 67 swinging 122 in ferritic steel 351
solid 67 Armillaria ostoyae 53, 105 Lego and 339
angular momentum 117 arrow of time 48 photo of levitated 344
aspect of state 118 artefact 92 ATP 109, 532
conservation 190 ash 443 atto 455
pseudo-vector 162 associativity 272 attraction
values, table 118 Asterix 125 types of physical 258
anharmonicity 315 asteroid 261 Atwood machine 501
anholonomic constraints 254 difficulty of noticing 205 auricola 324
Antares 87, 261 falling on Earth 205 austenitic steels 349
anti-bubbles 379 puzzle 212 Avogadro’s number 341, 376,
antigravity 194, 206 Trojan 194 461
device 207 Astrid, an atom 545 from the colour of the sky
antimatter 106 astrology 178, 213 341
Antiqua 445 astronaut see cosmonaut axioms 35
antisymmetry 275 astronomer axis
ape smallest known 192 Earth’s, motion of 146
puzzle 163 astronomical unit 220, 466 Earth’s, precession 153
apex angle 487 astronomy axle
aphelion 465 picture of the day 473 impossibility in living
Aphistogoniulus at-sign 449 beings 490
subject index 579

B 378 definition 392, 399
Babylonians 221, 450 Bessel functions 150 minimum entropy 408
background 26, 27, 49 Betelgeuse 87, 261 physics and 8
bacterium 92, 114 beth 445 bone
badminton smash bets and jumping 338
record 36, 482 how to win 141 human 348
Balaena mysticetus 371 bicycle book
Balaenoptera musculus 105, research 491 and physics paradox 438
320 riding 100 definition 441
ball stability, graph of 492 information and entropy
rotating in mattress 167 weight 391 398
B balloon
inverted, puzzle 390
bifurcation 423
billiards 103
boom
sonic, due to supersonic
puzzle 389 bimorphs 233 327
Babylonians rope puzzle 375 biographies of boost 282
Banach measure 56 mathematicians 473 bore in river 329
Banach–Tarski paradox or biological evolution 418 bottle 109, 129

Motion Mountain – The Adventure of Physics
theorem 57, 334 biology 233 empty rapidly 245
banana BIPM 453 bottom quark
catching puzzle 120 bird mass 462
barycentre 216 fastest 36 boundaries 98
baryon number density 467 singing 328 boundary layer 507
basal metabolic rate 114 speed, graph of 38 planetary 367
base units 453 bismuth 41 bow, record distance 78
Basiliscus basiliscus 169 bit brachistochrone 243, 508
basilisk 169 to entropy conversion 464 braid pattern 428
bath-tub vortex 140, 534 black holes 239 brain 409

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
bathroom scales 158, 347 block and tackle 30, 31 and physics paradox 438
bce 450 blood stem 25, 525
bear circulation, physics of 367 brass instruments 309
puzzle 63 dynamic viscosity 367 bread 336, 480
beauty 18, 415, 423, 424 supply 89 breath
origin of 415 board divers 121 physics of 367
becquerel 455 boat 109 breathings 445
beer 379 sailing 363 Bronshtein cube 8
beer mat 63 Bode’s rule 220 Bronze Age 385
beetle body brooms 207, 509
click 169 connected 89 brown dwarfs 261
before the Common Era 450 definition 26 Brownian motion 391, 394
behaviour 20 extended, existence of 333 typical path 393
belief extended, non-rigid 244 browser 470
collection 22 fluids 367 bubble
belief systems 99 rigid 243 soap 377, 476
bell Bohr magneton 463 soap and molecular size
resonance 291 Bohr radius 463 342
vibration patterns 292 Boltzmann constant 390 bucket
Belousov-Zhabotinski Boltzmann constant 𝑘 399, experiment, Newton’s 158
reaction 428 461 puzzle 19
Bernoulli equation 361, 362, and heat capacity 394 bullet
580 subject index

speed 36 wheel angular momentum measure of 251
speed measurement 61, 481 118 measuring 248–265
Bunsen burner 386 carbon dioxide 391 quantum of, precise value
Bureau International des Carlson, Matt 472 461
Poids et Mesures 453 Carparachne 92 types of 20
bushbabies 169 carriage chaos 424
butterfly effect 425 south-pointing 244, 509, and initial conditions 425
button 541 in magnetic pendulum 425
future 404 cars 437 chapter sign 449
Cartesian 51 charge 281
C cartoon physics, ‘laws’ of 98 elementary 𝑒, physics and
B c. 450
caesium 41
cat 109
falling 120
8
positron or electron, value
calculating prodigies 447 Cataglyphis fortis 171 of 461
Bunsen calculus catenary 487 charm quark
of variation 255 caterpillars 79 mass 462
calendar 449, 450 catholicism 336 chaturanga 32

Motion Mountain – The Adventure of Physics
Gregorian 450 causality cheetah 36, 84
Julian 450 of motion 239 chemistry 436
modern 450 cavitation 96, 313 chess 32
caliper 54, 65 and knuckle cracking 331 chest operations 516
Callisto 210 cavity resonance 329 child’s mass 126
calorie 457 CD childhood 36, 49
Calpodes ethlius 79 angular momentum 118 Chimborazo, Mount 189
camera 66 Cebrennus villosus 90, 91, 531 chocolate 373
Cancer 213 celestrocentric system 480 does not last forever 333
candela Celsius temperature scale 385 chocolate bars 333

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
definition 454 Celtic wobble stone 133 Chomolungma, Mount 189,
candle 411 cement kiln 491 378
in space 520 cementite 350 circalunar 192
motion 32 centi 455 circle packing 487
canoeing 170 centre of gravity 212 circular definition
canon centre of mass 212 in physics 438
puzzle 158 centrifugal acceleration 161, Clay Mathematics Institute
Canopus 261 180 363
cans of peas 127 centripetal acceleration 161 click beetles 169
cans of ravioli 127 cerebrospinal fluid 367 clock 44–48, 110, 183
capacity 357 Ceres 214 air pressure powered 110
Capella 261 cerussite 58 definition 42
capillary Cetus 213 exchange of hands 64
human 372 CGPM 454 precision 44
Capricornus 213 chain puzzle 524
capture fountain 133 puzzles 64
in universal gravity 215 hanging shape 69 summary 438
car challenge types, table 46
engine 388 classification 9 clock puzzles 64
on lightbulb 375 chandelier 183 clockwise
parking 65 change rotation 48
weight 391 and transport 24 closed system 403, 405
subject index 581

cloud table of astronomical 464 cortex 25
mass of 418 table of basic physical 461 cosmological constant 466
Clunio 192 table of cosmological 466 cosmonaut 107, 108, 369, 502
coastline table of derived physical space sickness 504
infinite 53 463 cosmonauts
length 53 constellations 87, 213 health issues 204
CODATA 555 constraints 254 cosmos 27
coffee machines 402 contact 436 coulomb 455
coin and motion 99 countertenors 331
puzzle 62, 205, 346 container 49 crackle 490
collision continent crackpots 474
C and acceleration 233
and momentum 110
acceleration of 490
motion of 147
cradle
Mariotte 125
and motion 99 continuity 43, 50, 51, 106 Newton 125
cloud open issues in 436 equation 231 Crassula ovata 273
comet 192 in symmetry creation 17
as water source 370 transformations 276 of motion 108

Motion Mountain – The Adventure of Physics
Halley’s 192 limits of 74 crest
mini 215 continuum 37 of a wave 301
origin of 214 as approximation 89 cricket bowl 482
comic books 347 mechanics 245 cricket chirping
Commission Internationale physics 354 and temperature 406
des Poids et Mesures 453 convection 405 crooks 432
compass 244 Convention du Mètre 453 cross product 115
completeness 43, 51, 106 conventions crossbow 78
complexity 398 used in text 441 crust
Compton tube 141 cooking 436 growth of deep sea

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Compton wavelength 463 cooperative structures 422 manganese 36
Compton wheel 141 coordinates 51, 75 crying
computer 409 generalized 254, 256 sounds of 331
concatenation 272 Copernicus 533 crystal
concave Earth theory 480 Corallus caninus 387 and straight line 59
concepts 437 cords, vocal 307 class 269
conditions Coriolis acceleration 138, 139, symmetry table 269
initial, definition of 237 496, 534 cube
conductance quantum 463 Coriolis effect 138, 496, 533 Bronshtein 8
Conférence Générale des and navigation 159–161 physics 8
Poids et Mesures 453, 458 and rivers 159 Rubik’s 346
configuration space 77 Coriolis force 295 cumulonimbus 322
Conférence Générale des cork 109, 125, 129 curiosity 18, 241
Poids et Mesures 454 corn starch 429, 430 curvature 64
conic sections 192 corner film curve
conservation 17, 106, 286 lower left 25–27 of constant width 33
of momentum 108 corpuscle 87 curvimeter 55
of work 31 corrugation cycle 315
principles 134 road 420 limit 423
constant corrugations menstrual 192, 505
gravitational 177 road 420 cycloid 264, 508
constants corrugations, road 422 cyclotron frequency 463
582 subject index

Cyrillic alphabet 445 devil 24 Dolittle
Devonian 200 nature as Dr. 255
D diagram donate
∂, partial differential 448 state space 78 to this book 10
d𝑥 448 diamond doublets 276
daleth 445 breaking 347 Dove prisms 531
damping 234, 289, 301 die throw 239 down quark
dancer Diet Coca Cola 379 mass 462
angular momentum 118 differential 82 drag 234
rotations 79 diffraction 301 coefficient 234, 437
Davey–Stewartson equation diffusion 409, 411 viscous 235
C day
328 digamma 444
digits
drift
effect 457
length in the past 41 Arabic see number system, dromion 329
Cyrillic length of 145, 200 Indian film of motion 329
mean solar 151 history of 446 drop
sidereal 151, 464 Indian 446 of rain 235

Motion Mountain – The Adventure of Physics
time unit 455 dihedral angles 57 Drosophila melanogaster 46
death 280, 438 dilations 279 duck
and energy consumption dimensionality 37, 51 swimming 322
125 dimensionless 463 wake behind 326
conservation and 110 dimensions 50 duration
energy and 125 number of 304 definition 41
mass change with 126 dinosaurs 205 Dutch Balloon Register 562
origin 15 diptera 159 duvet 405
rotation and 116 direction 37 dwarf
death sentence 336 disappearance stars 261

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
deca 455 of motion 108 dwarf planets 214
decay 409 dispersion 301 d𝑥 448
deci 455 and dimensionality 304 dyadic product 448
deer 25 dissection dynabee 163
definition of volumes 57 dynamics 189, 226
circular, in physics 438 dissipative systems 423
degree distance 52 E
angle unit 455 measurement devices, e, natural exponential 448,
Celsius 455 table 55 468
delta function 305 values, table 53 e.g. 451
Denier 458 distinguish 25 ear 50, 53, 303, 340, 542
denseness 43, 51 distinguishability 37, 43, 51, as atomic force
derivative 82 106 microscope 345
definition 82 distribution 305 growth 61
description 75 Gaussian 459 human 324
accuracy of 437 Gaussian normal 394 illustration 325
Desmodium gyrans 46 normal 459 problems 324
details divergence 188 wave emission 303
and pleasure 19 diving 371 Earth 405
determinism 239, 240, 426 DNA 53, 458 age 41, 465
deviation DNA (human) 261 angular momentum 118
standard, illustration 459 DNA, ripping apart 227 average density 465
subject index 583

axis tilt 153 electric effects 233 definition 395, 398
crust 126 electrodynamics 354 flow 404
density 185 electromagnetism 186 measuring 396
dissection 58 electron see also positron quantum of 399
equatorial radius 465 classical radius 463 random motion and 396
flat 490 g-factor 463 specific, value table 397
flattened 136, 159 magnetic moment 463 state of highest 412
flattening 465 mass 462 to bit conversion 464
from space 473 speed 36 values table 397
gravitational length 464 electron charge see also environment 26
hollow 60 positron charge Epargyreus clarus 79, 530
E humming 331
mass 464
electron volt
value 464
ephemeris time 42
eponym 452
mass measurement 184 element of set 37, 43, 51, 106 equation
earthquake mass, time variation 205 elephant differential 481
motion around Sun 152 and physics 415 Equator
normal gravity 465 sound use 322 wire 224

Motion Mountain – The Adventure of Physics
radius 465 elevator equilibrium 402
rise 219 space 224, 345 thermal 402
rotation 135–150, 456 ellipse 192 Eris 214
change of 145 as orbit 182 eros 186
rotation data, table 151 email 470 error
rotation speed 145 emergence 426 example values 436
shadow of 165 emission in measurements 459
shape 188–189, 212 otoacoustic 303 random 459
speed 36 EMS98 162 relative 459
speed through the Encyclopedia of Earth 474 systematic 459

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
universe 155 energy 110, 111, 231, 278 total 459
stops rotating 189 as change per time 111 ESA 473
earthquake 162, 189 conservation 187, 190, 281 escape velocity 212
and Moon 192 conservation and time 113 Escherichia coli 92
energy 162 consumption in First et al. 451
information 535 World 113 eth 443, 553
starting 347 definition 111 ethel 443
triggered by humans 347 flows 108 ethics 242
eccentricity 194 from action 262 Eucalyptus regnans 338
of Earth’s axis 153 kinetic 110 Eucharist 336
echo 303 observer independence Euclidean space 51
ecliptic 175 278 Euclidean vector space 35, 37
illustration of 222 of a wave 300 Euler’s polyhedron formula
effort 229 rotational, definition 119 348
everyday 229 scalar generalization of 278 Euler’s wobble 147
egg thermal 388 Europa 210
cooking 406 values, table 112 evaporation 409
eigenvalue 275 engine evening
Ekman layer 533 car 388 lack of quietness of 322
elasticity 233 enlightenment 201 event
Elea 339 entropy 357 definition 41
Electric Age 385 characteristic in nature 399 Everest, Mount 189, 378
584 subject index

everything flows 17 familiarity 25 minimum, value table 401
evolution 27 family names 451 of everything 17
and running 123 fantasy of time 48
and thermodynamics 432 and physics 439 fluid
biological 21 farad 455 and self-organization 418
cosmic 21 Faraday’s constant 463 ink motion puzzle 413
equation, definition 239 farting mechanics 244
Exa 455 for communication 323 motion examples 355
exclamation mark 449 fear of formulae 32 motion know-how 369
exclusion principle 258 Fedorov groups 267 fluids
exobase 366 femto 455 body 367
E exoplanet
discoveries 216
Fermi coupling constant 461
Fermi problem 407
flute 288
fly
exosphere Fermilab 66 common 186
every thing details 366 ferritic steels 349 focal point 489
expansion Fiat Cinquecento 375 focus 489
of gases 384 fifth foetus 313

Motion Mountain – The Adventure of Physics
of the universe 156 augmented 309 folds, vocal 307
expansions 279 perfect 309 fool’s tackle 31
Experience Island 16, 22 figures footbow 78
exponential notation 66 in corners 25 force 118
exponents filament 386 acting through surfaces 231
notation, table 66 film central 104
extension 352 Hollyoood 333 centrifugal 189
and waves 291 Hollywood, and action 248 definition 104
extrasolar planets 261 lower left corner 27 definition of 228
eye 25 fine-structure constant 461, is momentum flow 228

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
blinking after guillotine 462 measurement 228
102 finite 395 normal 236
of fish 378 fire 386 physical 228
eye motion 50 pump 395 use of 227–236
eyelid 65 firework 67 values, table 227
fish Ford and precision 437
F eyes of 378 forest 25
𝜑𝑥 448 farting 323 forget-me-not 267
𝑓(𝑥) 448 flagella 92 formulae
𝑓󸀠 (𝑥) 448 flagellar motor 114 ISO 553
F. spectabilis 46 flame 407 liking them 32
F. suspensa 46 flame puzzle 161 mathematical 473
F. viridissima 46 flatness 60 Forsythia europaea 46
faeces 79 flattening Foucault’s pendulum
Falco peregrinus 36 of the Earth 136, 159 web cam 141
fall 180 fleas 169 fountain
and flight are independent flies 25 chain 133
76 flip film 25, 26 Heron’s 374
is not vertical 136 explanation of 75 water 95
is parabolic 77 floor Fourier analysis 292
of light 201 are not gavitational 226 Fourier decomposition 292
of Moon 180 flow 354, 356 Fourier transformation 326
subject index 585

fourth centre 153 geostationary satellites 194
augmented 309 collision 104 Gerridae 170
perfect 309 rotation 155 geyser 417
fractal size 53 ghosts 31, 99
landscapes 334 galaxy cluster 260 Giant’s Causeway 59, 429
fractals 54, 83, 333 galaxy group 260 giants 338
frame galaxy supercluster 260 Gibbs’ paradox 283
of refeence 266 Galilean physics 29, 34, 226, Giga 455
Franz Aichinger 556 435 gimel 445
frass 79 highlight 247 giraffe 368
freak wave 323 in six statements 29 glass
F free fall
speed of 36
research in 435
Galilean satellites 210
and π 61
is not a liquid 345
frequency 293 Galilean space-time Global Infrasound Network
fourth values for sound, table 290 summary 74 325
friction 109, 110, 234, 402, 404 Galilean time global warming 323, 405
between planets and the definition 43 gluon 462

Motion Mountain – The Adventure of Physics
Sun 110 limitations 44 gnomonics 42
dynamic 234 Galilean transformations 278 gods
importance of 402 Galilean velocity 35 and conservation 134
not due to gravity 226 galvanometer 233 and energy 111
picture of heat 384 Ganymede 210 and freedom 256
produced by tides 199 gas 389 and Laplace 196
static 234, 236 as particle collection 393 and motion 525
sticking 234 ideal 384 and Newton 34, 527
froghopper 84 gas constant and walking on water 170
Froude number 322 ideal 385, 390 gold 480

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
fuel consumption 437 gas constant, universal 463 surface atoms 401
full width at half maximum gasoline golden rule of mechanics 31
459 dangers of 370 golf balls 482
funnel 378, 379 gauge Gondwanaland 147
puzzle 378 change 272 gorilla test for random
Futhark 443 symmetry 280, 283 numbers 239
Futhorc 443 theory 49 Gosper’s formula 409
futhorc 445 Gaussian distribution 394, 459 Gothic alphabet 445
future gearbox Gothic letters 445
button 404 differential 244 GPS 42
fixed 239 gears grace 23, 122, 430, 525
remembering 404 in nature 244 gradient 187
𝑓𝑥 448 Gemini 213 granular jet 351
geocentric system 216 gravitation see also universal
G geocorona 366 gravitation, 173, 233
γ, Euler’s constant 468 geodesics 236 and measurement 439
Gaia 424 geoid 188 and planets 174–178
gait geological maps 473 as momentum pump 228
animal 501 geology of rocks 473 Earth acceleration value
human 123 geometric sequence 477 182
galaxy geometry essence of 218
and Sun 153 plane 73 properties of 182–186
586 subject index

sideways action of 183 clip 234 hips 122
summary of universal 225 diameter 53 hoax
universal see universal growth 36 overview 474
gravitation, 177 halfpipe 264 hodograph 77, 179, 210
value of constant 184 haltere hole
gravitational acceleration, and insect navigation 159 system or not? 26
standard 84 Hamilton’s principle 248 through the Earth 221
gravitational constant 177 hammer drills 128 hole, deep 348
geocentric 464 hands of clock 64 Hollywood
heliocentric 465 hank 458 films 333
gravitational constant 𝐺 461 hard discs films and action 248
G physics and 8
gravitational field 188
friction in 236
heartbeat 183
thriller 255
westerns 531
gravitons 219 heat 233 holonomic 255
gravitational gravity see gravitation ‘creation’ 387 holonomic systems 254
as limit to motion 173 engine 388 holonomic–rheonomic 254
centre of 212 in everyday life 395 holonomic–scleronomic 254

Motion Mountain – The Adventure of Physics
inside matter shells 217 in physics 395 Homo sapiens 46
is gravitation 173 Hebrew abjad homogeneity 43, 51
lack of 204 table 445 homomorphism 275
wave 295 Hebrew alphabet 445 homopause 366
gray (unit) 455 hecto 455 homosphere
Greek alphabet 443, 444 helicity details 366
greenhouse effect 405 of stairs 48 honey bees 105
Greenland 41 helicopter 378, 474 Hopi 528
Gregorian calendar 65, 450 helicoseir 348 horse 114, 228
grim reaper 61 heliocentric system 216 power 114

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
group helium 260, 386, 409 speed of 78
Abelian, or commutative danger of inhaling 318 horse power 114
273 superfluid 375 hour 455
crystal 269 helix hourglass puzzle 62
crystallographic point 269 in Solar System 154 Hubble parameter 467
mathematical 272 hemispheres Hubble space telescope 473
growth 21, 246 Magdeburg 361 Hudson Bay 41
as self-organization 418 henry 455 humour 426
human 36 heptagon physics 474
Gulf Stream 534 in everyday life 512 Huygens’ principle 304
guns and the Coriolis effect heresy 336 illustration of 304
139 Hermitean 275 illustration of consequence
gymnasts 121 Heron’s fountain 374 305
gynaecologist 313 herpolhode 164, 165 hydrofoils 363
gyroscope 120, 141 herring Hydroptère 36
laser 144 farting 323 hyperbola 192
vibrating Coriolis 159 hertz 455
heterosphere I
H details 366 i, imaginary unit 448
hafnium carbide 386 hiccup 431 i.e. 451
Hagedorn temperature 386 Higgs mass 462 ibid. 451
hair Himalaya age 41 IBM 561
subject index 587

ice ages 153, 535 navigation 160 ionosphere
iceberg 105 speed, graph of 38 details 366
icicles 347 water walking 299 shadow of 165
icosane 397 instant 34 IPA 441
ideal gas 390 definition 41 Iron Age 385
ideal gas constant 519 single 17 irreducible 276
ideal gas relation 390 instant, human 41 irreversibility 383
idem 451 insulation power 405 of motion 239
Illacme plenipes 96 integral irreversible 402
illness 367 definition 252 Island, Experience 16
illusions integration 56, 251 ISO 80000 447
I of motion 16
optical 473
integration, symbolic 473
interaction 27, 227
isolated system 405
isomorphism 275
image 98 symmetry 285 isotomeograph 141
ice definition 26 interface Issus coleoptratus 244
difference from object 98 waves on an 294 Istiophorus platypterus 36, 372
imagination interference 301 Isua Belt 41

Motion Mountain – The Adventure of Physics
and symmetry 271 illustration 303 IUPAC 555
impenetrability 106 interferometer 141 IUPAP 555
of matter 34 International Astronomical
inclination Union 466 J
of Earth’s axis 153 International Earth Rotation Jarlskog invariant 461
indeterminacy relation 314 Service 456, 535, 537 jerk 85, 237, 490
of thermodynamics 399 International Geodesic Union Jesus 170
index of refraction 510 466 joints
Indian numbers 446 International Latitude Service cracking 331
individuality 28, 237 146, 535 Josephson frequency ratio 463

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
induction 281–283 International Phonetic joule 455
inertia 99 Alphabet 441 juggling 202
moment of 164, 285 internet 470 record 489
inf. 451 list of interesting websites robot 489
infinite 471 jump 169
coastline 53 interval height of animals 80
infinity 37, 43, 51, 106 musical 309 long 530
and motion 17 intonation Jupiter 217, 261
in physics 83 equal 309 angular momentum 118
information 398 just 309 moons of 221
erasing of 409 well-tempered 309 properties 465
quantum of 399 invariance 106, 272
infrasound 325 mirror 284 K
infusion 368 parity 284 k-calculus 478
initial condition inverse element 272 Kadomtsev–Petviashvili
unfortunate term 240 inversion equation 329
initial conditions 237 at circle 244, 508 Kapitza pendulum 319
injectivity 275 spatial 284 Kaye effect 379
inner world theory 480 invisibility kefir 260
Inquisition 336 of loudspeaker 409 kelvin
insect of objects 89 definition 453
breathing 411 Io 200, 210 Kepler’s laws 176
588 subject index

ketchup of nature 255 animal 125
motion 36 lea 458 lift
kilo 455 lead 375 for ships 131
kilogram leaf into space 345
definition 453 falling 379 light 233
kilotonne 112 leap day 449 deflection near masses 201
kinematics 75, 226 learning 25 fall of 201
summary 96 best method for 9 mill 233
kinetic energy 110 mechanics 229 slow group velocity 527
Klitzing, von – constant 463 without markers 9 year 464, 466
knot 74 without screens 9 light speed
K knowledge
humorous definition 507
Lebesgue measure 58
lecture scripts 472
measurement 61
lightbulb
knuckle leg 183 below car 375
ketchup angles 68 number record 96 temperature 386
cracking 331 performance 169 lighthouse 36, 71
koppa 444 Lego 339, 344 lightning speed 36

Motion Mountain – The Adventure of Physics
Korteweg–de Vries equation legs 31 Lilliputians 320
307, 317, 329 advantages 168 limbic system 526
Kuiper belt 214, 215, 260, 370 and plants 31 limit cycle 423
Kurdish alphabet 443 efficiency of 170 limits
Kármán line 364 in nature 89 to precision 460
on water 169 line
L vs. wheels 168 straight, in nature 59
ladder 129 Leidenfrost effect 377 Listing’s ‘law’ 167
puzzles 70 length 50, 52, 81, 439 litre 455
sliding 129 assumptions 56 living thing, heaviest 105

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
Lagrange issues 56 living thing, largest 53
equations of motion 255 measurement devices, lizard 169
Lagrangian table 55 ln 2 468
average 255 puzzle 56 ln 10 468
examples, table 257 scale 411 local time 42
is not unique 255 values, table 53 locusts 169
points in astronomy 502 Leo 213 logarithms 176
Lagrangian (function) Leptonychotes weddellii 371 long jump 78, 530
251–253 letters record 112
Lake Nyos 391 origin of 442 Loschmidt’s number 341, 376,
laminarity 361 Leucanthemum vulgare 246 463
languages on Earth 446 levitation 194 lottery
Laplace operator 188 lex parismoniae 262 and temperature 384
laser gyroscopes 144 Libra 213 rigging 383
laser loudspeaker 409 libration 191 loudspeaker
Latin 451 Lagrangian points 194 invisible 409
Latin alphabet 441 lichen growth 36 laser-based 409
laughter 426 lidar 375 loudspeaker, invisible 409
lawyers 100 life love
laziness everlasting 110 physics of 186
cosmic, principle of 243, shortest 41 Love number 151
253 lifespan low-temperature physics 391
subject index 589

luggage 224 definition 102 self-adjoint 275
lumen 455 definition implies singular 275
lunar calendar 42 momentum conservation skew-symmetric 275
Lunokhod 502 104 symmetric 275
lux 455 deflects light 201 transposed 275
Lyapounov exponent 424 gravitational, definition matter 86, 99
Lyapunov exponent 202 impenetrability of 34
largest 432 gravitational, puzzle 212 quantity 106
lymph 367 identity of gravitational quantity of 102
and inertial 202–204 shell, gravity inside 217
M inertial 99 mattress
L M82, galaxy 84
mach 103
inertial, definition 202
inertial, puzzle 212
with rotating steel ball 167
meaning
Mach number 159 is conserved 102 of curiosity 19
luggage Mach’s principle 103 measures motion difficulty measurability 37, 43, 51, 106
Machaeropterus deliciosus 290 101 measure 439
machine 92 measures quantity of measurement

Motion Mountain – The Adventure of Physics
example of high-tech 352 matter 102 and gravitation 439
moving giant 353 negative 106, 107 comparison 456
servants 113 no passive gravitational definition 37, 453, 456
Magdeburg hemispheres 360, 203 error definition 459
361 of children 126 errors, example values 436
magic 474 of Earth, time variation irreversibility 456
Magna Graecia 339 205 meaning 456
magnetic effects 233 point 87 process 456
magnetic flux quantum 463 properties, table 106 mechanics 243
magnetic thermometry 548 sensors, table 107 classical 226

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
magnetism 104 values, table 105 continuum 245
magnetization mass ratio definition 226
of rocks 149 muon–electron 463 fluid 244
magneton, nuclear 464 neutron–electron 464 learning 229
magnetosphere 366 neutron–proton 464 quantum 226
details 366 proton–electron 463 Medicean satellites 210
magnitude 81, 277 match medicines 31
man 255 lighting 276 Mega 455
wise old 255 math forum 473 megatonne 112
manakin math problems 473 memory 25, 31, 402
club winged 290 math videos 473 menstrual cycle 538
many-body problem 194–197, mathematicians 473 Mentos 379
435 mathematics 35 Mercalli scale 162
marble problem of the week 473 Mercury 196
oil covered 109 matrix mercury 342
marker adjoint 275 meridian
bad for learning 9 anti-Hermitean 275 and metre definition 183
Mars 456 anti-unitary 275 mesopause 366
martensitic steels 349 antisymmetric 275 mesopeak 366
mass 101, 106, 231, 439 complex conjugate 275 Mesopotamia 441
centre of 119, 212 Hermitean 275 mesosphere
concept of 101 orthogonal 275 details 366
590 subject index

metabolic rate 125 invariance of everyday fall of 180
metabolic scope, maximal 533 motion 284 hidden part 191
metallurgy symmetry 285 illusion 69
oldest science 436 mixing matrix orbit 192
meteorites 205 CKM quark 461 path around Sun 211
meteoroid 261 PMNS neutrino 461 phase 216
meteors 325 mogul, ski 319 properties 465
metre mol 341 size illusion 69
definition 453 definition 341 size, angular 69
stick 49 from the colour of the sky size, apparent 179
metre sticks 438 341 weight of 219
M metricity 37, 43, 51, 106
micro 455
molar volume 463
molecules 340
moons
limit for 506
microanemometer 310 moment 42 observed 260
metab olic microphone 310 of inertia 116, 118, 278, 285 moped 133
microscope 321 of inertia, extrinsic 118 morals 242
atomic force 236, 344, 345, of inertia, intrinsic 118 morning

Motion Mountain – The Adventure of Physics
495 momentum 102, 108, 231 quietness of 322
atomic force, in the ear 345 angular, extrinsic 119, 120 moth 363
lateral force 236 angular, intrinsic 119 motion 27, 438
microwave background as a substance 228 and change 20
temperature 467 as change per distance 111 and collision 99
mile 456 as fluid 109 and contact 99
military change 228 and dimensions 189
anemometer 311 conservation 108 and infinity 17
sonic boom 326 conservation follows from and measurement units
underwater motion 372 mass definition 104 454

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
underwater sound 322, 324 flow is force 228–236 Aristotelian view 229
useful website 457 flows 108 as an illusion 17
milk carton from action 262 as change of state of
puzzle 370 of a wave 300 permanent objects 28
Milky Way total 102 as illusion 17
age 466 values, table 104 as opposite of rest 17
angular momentum 118 momentum conservation based on friction 234
mass 466 and force 231 books on, table 21
size 466 and surface flow 231 conditions for its existence
milli 455 momentum flow 231 17
mind momentum, angular 119 creation 108
change 21 money detector 32
minerals 473 humorous definition 507 disappearance 108
minimum monster wave 323 does not exist 16
of curve, definition 253 month 449 drift-balanced 526
Minion Math font 557 Moon 190–192 everyday, is
minute 455 angular momentum 118 mirror-invariant 30
definition 466 atmosphere 411 existence of 16
Mir 223 calculation 180 faster than light 404
mirror dangers of 200 global descriptions
cosmic 473 density 465 242–247
invariance 30, 284 density and tides 199 harmonic 288
subject index 591

has six properties 29–30, Motion Mountain 16 nerve
33 aims of book series 7 signal propagation 315
illusions, figures showing helping the project 10 signal speed 36
16 supporting the project 10 Neurospora crassa 46
in configuration space 424 motor neutral element 272
infinite 439 definition 232 neutrino 353
is and must be predictable electrostatic 233 masses 462
242 linear 233 PMNS mixing matrix 461
is change of position with table 233 neutron
time 75 type table 233 Compton wavelength 464
is conserved 29 motor bike 100 magnetic moment 464
M is continuous 29
is due to particles 24
mountain
surface 333
mass 464
star 260
is everywhere 15 moustache 54 newspaper 550
Motion is fundamental 15, 33, 454 movement 22 Newton
is important 15 multiplet 273, 274 his energy mistake 111
is lazy 30 multiplication 272 newton 455

Motion Mountain – The Adventure of Physics
is mysterious 15 muon Newtonian physics 34, 226
is never inertial 174 g-factor 463 NGC 2240 386
is never uniform 173 muon magnetic moment 463 Niagara 426
is part of being human 15 muon mass 462 Niagara Falls 426
is relative 26, 29 Musca domestica 46, 84 nitrogen 374
is reversible 30 music 307–310 Noether charge 281
is simple 264 notes and frequencies 310 Noether’s theorem 280
is transport 24 search online 474 noise 341, 392, 457
limits of 439 Myosotis 267 physical 315
linear–rotational myosotis 266 shot 340

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
correspondence table 118 mystery thermometry 548
manifestations 16 of motion 15 nonholonomic constraints
minimizes action 30 254
mirror invariance 284 N nonius 65
non-Fourier 526 N. decemspinosa 531 norm 81
of continents 36, 147 nabla 448 normal distribution 394
parity invariance 284 nail puzzle 33 normality 552
passive 22, 234 names North Pole 48, 147, 189
polar 146 of people 451 notation 441
possibly infinite? 31 Nannosquilla decemspinosa 91 scientific 441
predictability 435 nano 455 nuclear explosions 322
predictability of 226–247 Nanoarchaeum equitans 260 nuclear magneton 464
relative, through snow 156 NASA 456 nuclei 233
reversal invariance 30 NASA 473 nucleon 353
reversibility 284 natural unit 462 null vector 81
simplest 24 nature 27 number
stationary fluid 361 is lazy 30, 255, 264 Arabic see number system,
turbulent fluid 361 Navier–Stokes equations 363 Indian
types of 20 needle on water 379 Indian 446
unlimited 439 negative vector 81 real 51
volitional 22 neocortex 526 Number English 458
voluntary 234, 238 Neptune 213, 217 number system
592 subject index

Greek 446 orbit and drag 236
Indian 446 elliptical 177 parachute
Roman 446 order 43, 51, 106, 415 needs friction 234
numbers appearance examples 418 paradox
and time 43 appearance, mathematics about physics books 438
necessity of 277 of 423 hydrodynamic 360
nutation 152 of tensor 279 hydrostatic 360
Nyos, Lake 391 table of observed parallax 135, 150
phenomena 415 parallelepiped 116
O order appearance parenthesis 449
object 26, 27, 98 examples 415 parity
N definition 40
defintion 26
order parameter 422
orgasm
invariance 30, 284
inversion 284
difference from image 98 against hiccup 431 parking
numbers immovable 100 orientation car statistics 428
invisibility 89 change needs no challenge 66
movable 100 background 120 mathematics 483

Motion Mountain – The Adventure of Physics
summary 438 origin parallel 65, 483
obliquity 153, 505 human 15 parsec 464
oboe 313 Orion 87 part
observable ornament everything is made of 401
definition 28 Hispano-Arabic 274 of systems 27
physical 276 orthogonality 81 particle 87
observables oscillation 288 elementary 353
discrete 277 damped 289 particle data group 471
observation definition 288 parts 85
physical 271 harmonic and anharmonic pascal 455

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
sequence and time 40 289 passim 451
ocean harmonic or linear 288 passive motion 238
angular momentum 118 oscillons 421 past
origin of 370 osmosis 233, 409 of a system 237
oceanography 326 oxygen 374 path 75
octave 309 bottle 372 pattern
definition 318 ozone layer 366 braid 428
octet 276 in corners 25
odometer 53 P random 25
ohm 455 𝜑𝑥 448 Paul trap 344
oil 109, 377 π and glasses 61 pea
oil film experiment 334 π and gravity 182 dissection 58
Olympic year counting 450 π, circle number 448, 468, 552 in can 127
Oort cloud 214, 215, 260 paerlite 350 pearls 346
op. cit. 451 painting puzzle 33 Peaucellier-Lipkin linkage
openness 106 paper 244, 508
operator 188 cup puzzle 407 pee
Ophiuchus 213 aeroplanes 474 research 428
optical radiation boat contest 375 Peirce’s puzzle 63
thermometry 548 parabola 69, 77, 182, 192 pencil 95
optics picture of the day 472 of safety 95 invention of 34
optimist 253 parachte puzzle 62
subject index 593

pendulum 183 scourge of 451 pollution 516
and walking 122 physics cube 8 polyhedron formula 348
inverse 319 pico 455 pool
short 496 piezoelectricity 233 filled with corn starch and
penguins 122 pigeons 122 water 430
people names 451 ping-pong ball 162 pop 490
peplosphere 367 pinna 324 Pororoca 321
perigee 465 Pioneer satellites 84 Porpoise Cove 41
perihelion 197, 465 Pisces 213 positions 81
shift 153, 194 Pitot–Prandtl tube 35 positivity 106
period 289 Planck constant positron charge
P periodic table
with videos 474
value of 461
planet 110
specific 463
value of 461
permanence 25 and universal gravitation possibility of knots 51
pendulum of nature 17 174 postcard 62
permeability distance values, table 220 stepping through 481
vacuum 463 dwarf 214 potassium 315, 411

Motion Mountain – The Adventure of Physics
permittivity gas 261 potential
vacuum 463 minor 261 gravitational 186
permutation orbit periods, Babylonian sources of 188
symmetry 272 table 221 potential energy 187
perpetuum mobile 107, 110 planet–Sun friction 110 power
first and second kind 110 planetoid 261 humorous definition 507
of the first kind 388 planetoids 200, 260 in flows 357
of the second kind 413 plants physical 113, 229
Peta 455 and sound 323 sensors, table 115
phase 289, 293 plate tectonics 149 values, table 114

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
velocity 293 plats ppm 516
phase space 27, 238 and legs 31 pralines 373
diagram 77 play 22 praying effects 262
PhD Pleiades star cluster 175 precession 153, 165
enjoying it 430 pleural cavity 359, 516 Earth’s axis 153
Philaenus spumarius 84 Pleurotya ruralis 91 equinoctial 146
photoacoustic effect 409, 410 plumb-line 59 of a pendulum 141
photon Pluto 213 precision 18, 35, 75, 437, 459
mass 462 pneumothorax 516 limits to 460
number density 467 point measuring it 31
Physeter macrocephalus 371 in space 34, 89 why limited? 437
physicists 472 mass 87 predictability
physics 15 mathematical 51 of motion 226–247, 435
circular definition in 438 particle 87 prefixes 455, 555
continuum 354 point-likeness 89 SI, table 455
everyday 29 poisons 30 prefixes, SI 455
Galilean 29 Poisson equation 188 preprints 470, 471
Galilean, highlight 247 Polaris 261 pressure 231, 340, 389
map of 8 polarization 301 air, strength of 361
outdated definition 22 polhode 148, 164, 165 definition 357
problems 471 pollen 392 measured values 359
school 523 pollutants 516 puzzle 359
594 subject index

second definition 362 gyromagnetic ratio 464 necessity in nature 277
principle magnetic moment 464 quantum
conservation 134 mass 463 of entropy 399
extremal 248 specific charge 463 of information 399
of cosmic laziness 253 pseudo-scalar 284 Quantum Age 385
of gauge invariance 272 pseudo-tensor 284 quantum mechanics 226
of laziness 243 pseudo-vector 162, 284 quantum of action
of least action 136, 190, PSR 1257+12 261 precise value 461
243, 253, 254 PSR 1913+16 41 quantum of circulation 463
history of 261 psychokinesis 22 quantum theory 26, 337
of relativity 272, 478 pulley 233 quark
P of the straightest path 236
variational 242, 248, 254
pulsar period 41
pulse 303
mixing matrix 461
quartets 276
principle of thermodynamics circular 305 quartz 291
principle second 403 spherical 305 oscillator film 291
prism 531 puzzle quasar 260
prize about clocks 64 quasi-satellite 204

Motion Mountain – The Adventure of Physics
one million dollar 363 ape 163 quietness
problem bear 63 of mornings 322
collection 471 bucket 19 quint
many-body 194–197, 435 coin 62, 205 false 309
process five litre 62 Quito 140
change and action in a 252 flame 161 quotations
in thermodynamics 402 hourglass 62 mathematical 474
sudden 241 knot 74
Procyon 261 length 56 R
prodigy liquid container 370 R-complex 525

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
calculating 447 milk carton 370 radian 67, 454
product nail 33 radiation 27, 99, 405
dyadic 448 painting 33 as physical system 26
outer 448 reel 19 thermometry 548
vector 115 snail and horse 63 radio speed 36
pronunciation sphere 206 rain drops 79, 235
Erasmian 444 sunrise 324 rain speed 36
proof 35 train 92 rainbow 278
propagation turtle 285 randomness 239
velocity 293 pyramid 116 rank
propeller 490 Pythagoras’ theorem 67 of tensor 279
in living beings 89 Ranque–Hilsch vortex tube
property Q 412
emergent 426 Q-factor 289 ratio between the electron
intrinsic 28 qoppa 444 magnetic moment and the
invariant 266 quanta Bohr magneton 436
permanent 28 smallest 339 rattleback 133
protestantism 336 quanti, piccolissimi 335–337 ravioli 127
proton quantity ray form 216
age 41 conserved 17 reactance
Compton wavelength 464 extensive 108, 354 inertive 309
g factor 464 extensive, table 356 reaper, grim 61
subject index 595

recognition 25 Galilean 82 around Earth 482
record is relative 157 geometry of 72
human running speed 78 Reuleaux curves 478 motion of hanging 430
vinyl 293 reversibility 383 rosette groups 267
reducible 276 of everyday motion 284 Rossby radius 159
reed instruments 309 of motion 239 Roswell incident 322
reel reversible 402 rotation
puzzle 19 Reynolds number absolute or relative 158
reflection definition 430 and arms 158
of waves 300 Rhine 379 as vector 120
refraction 263, 301 rhythmites, tidal 535 cat 120
R and minimum time 263
and sound 322
Richter magnitude 162
riddle 16
change of Earth 145
clockwise 48
illustration 321 Riemann integral 510 frequency values, table 117
reco gnition refractive index 263 Rigel 261 in dance 79
Regulus 261 right-hand rule 115 of the Earth 135–150, 456
relation ring rate 116

Motion Mountain – The Adventure of Physics
among nature’s parts 27 planetary 506 reluctance 116
indeterminacy 314 ring laser 534 sense in athletic stadia 48
relativity 243 rings snake 121
and ships 156 astronomical, and tides 198 speed 116
challenges 158 ripple tethered 89, 90, 168
Galilean 157 water 299 roulette and Galilean
Galilean, summary 172 road mechanics 127
Galileo’s principle of 157 corrguated or washboard Rubik’s cube 346
of rotation 158 422 rugby 111
principle 478 corrugations 420 rum 410

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
reluctance Roadrunner 347 Runge–Lenz vector 500
to rotation 116 robot 349, 541 Runic script 443
representation 275 juggling 489 running
definition 275 walking 264 backwards 405
faithful 275 walking on water 170 human 123
of observable 279 walking or running 170 on water 536
unitary 275 Roche limit 506 reduces weight 209
reproducibility 270 rock Rutherford, Ernest 24
reproduction 15 magnetization 149 Rydberg constant 436, 463
repulsion self-organization 429
types of physical 258 rocket S
research launch site puzzle 158 Sagarmatha, Mount 189, 378
in classical mechanics 435 rocket motor 233 Sagittarius 153, 213
resolution rogue wave 323 Sagnac effect 142, 498
of measurements is finite roll sailfish 36, 372
29 speed 32 sailing 109
resonance 291–293 roller coasters 224 and flow 363
definition 291 rolling 121 Salmonella 92
direct 318 motion 91 sampi 444
parametric 319 puzzle 33 san 444
rest wheels 121 sand 420
as opposite of motion 17 rope Cuban 421
596 subject index

granular jets in 352 selenite 58 leaving river 373
self-organization 419 self-adjoint 275 lift 131
singing 420 self-organization 246, 415–434 mass of 131
table of patterns in 420 self-similarity 54 pulling riddle 375
Saraeva 524 sensor speed limit 322
Sarcoman robot 489 acoustic vector 310 wake behind 326
satellite sequence 43, 51, 106 shit 530
artificial 88 Sequoiadendron giganteum shock wave
definition 88 105 in supersonic motion 326,
geostationary 163 serpent bearer 213 327
limit 506 Serpentarius 213 shoe size 457
S of the Earth 204
satellites 215
servant
machines 113
shoelaces 50, 402
shore birds 536
observable artificial 473 set shortest measured time 41
Saraeva Saturn 198, 215 connected 83 shot
saucer sexism in physics 339 noise 340
flying 322 sextant 537 small lead 375

Motion Mountain – The Adventure of Physics
scalar 81, 270, 277 SF6 318 shoulders 122
true 284 sha 444 shutter 66
scalar product 51, 81 shadow shutter time 67
scale invariance 338 and attraction of bodies SI
scales 218 prefixes 458
bathroom 158, 347 book about 531 table of 455
scaling Earth, during lunar eclipse units 453, 461
‘law’ 44 179 SI prefixes
allometric 37 motion 98 infinite number 458
school physics 523 of ionosphere 165 SI units

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
science of sundials 48 definition 453
main result 400 of the Earth 165 prefixes 455
time line 19 shadows 98 supplementary 454
scientist shampoo, jumping 379 siemens 455
time line 19 shape 51, 87 sievert 455
Scorpius 213 and atoms 352 signal
scourge deformation and motion decomposition in
of physics 451 90 harmonic components 292
screw of the Earth 212 types, table 306
and friction 234 sharp s 443 Simon, Julia 530
in nature 244 Shaw, George Bernard sine curve 288
scripts alphabet of 446 singing 307
complex 445 sheep singlets 276
scuba diving 368 Greek 444 singular 275
sea 297 shell sink vortex 534
seal gravity inside matter 217 siren 320
Weddell 371 Shell study 532 Sirius 87, 261
season 175 shift size 49
second 455 perihelion 153 skateboarding 264
definition 453, 466 ship 105 ski moguls 319, 420
Sedna 214 and relativity 157 skin 89
seiche 300 critical speed 322 skipper 79
subject index 597

sky sonic boom 326 specific mechanical energy
colour and density sonoluminescence 96 489
fluctuations 545 soprano spectrum
colour and molecule male 331 photoacoustic 410
counting 341 soul 126 speed 82
moving 146 sound 294 critical ship 322
nature of 125 channel 322 infinite 439
slide rule 487, 488 frequency table 290 limit 440
slingshot effect 210 in Earth 331 lowest 31
Sloan Digital Sky Survey 473 in plants 323 of light 𝑐
smallest experimentally intensity table 311 physics and 8
S probed distance 66
smartphone
measurement of 310–311
none in the high
of light at Sun centre 36
values, table 36
bad for learning 9 atmosphere 364 speed of sound 103, 145
sky smiley 449 speed 36 speed record
smoke 393 threshold 311 human running 78
ring picture 380 source term 188 under water 372

Motion Mountain – The Adventure of Physics
snake South Pole 188 speed, highest 36
rotation 121 south-pointing carriage 244 sperm
snap 490 space 438 motion 36
snorkeling absolute 141 sphere
dangers of 515 absoluteness of 51 puzzle 206
snowflake 550 as container 52 swirled, self-organized 421
self-organization film 422 definition 49 Spica 261
speed 36 elevator 224, 345 spider
snowflakes 550 empty 188, 301 jumping 370
soap Euclidean 51 leg motion 370

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
bubble bursting 31 Galilean 51 rolling 90
bubbles 377 is necessary 51 water walking 170
bubbles and atoms 342 lift 345 spin
soccer ball 92 physical 49 group 266
sodar 375 point 89 spinal engine 123
sodium 315, 385, 411, 527 properties, table 51 spine
solar data 473 relative or absolute 52 human 348
Solar System 260 sickness 504 spinning top
angular momentum 118 white 449 angular momentum 118
as helix 154 Space Station spinors 277
formation 146 International 216 spiral
future of 219 Space Station, International logarithmic 511
motion 153 204 spirituality 241
small bodies in 261 space travel 161 Spirograph Nebula 506
Solar System simulator 473 space-time 27 split personality 240
solitary wave Galilean, summary 74 sponsor
definition 316 relative or absolute 52 this book 10
in sand 420 space-time diagram 77 spoon 133
soliton 300, 317 Spanish burton 31 sport
film 317 spatial inversion 279 and drag 235
solitons 316 special relativity before the spring 288
sonar 328 age of four 271 constant 288
598 subject index

spring tides 199 radiation constant 464 Sun–planet friction 110
sprint Steiner’s parallel axis theorem sunbeams 479
training 171 118 sundial 45, 505, 527
squark 556 steradian 67, 454 story of 48
St. Louis arch 487 stick sundials 42
stadia 48 metre 49 sunrise
stainless steel 350 stigma 444 puzzle 324
staircase Stirling’s formula 409 superboom 326
formula 170 stone supercavitation 372
stairs 48 falling into water 299 superfluidity 375
stalactites 347, 427 skipping 374 supergiants 261
S stalagmites 36
standard apple 458
stones 24, 40, 76, 80, 87, 109,
133, 136, 176, 178, 179, 182,
superlubrication 234
supermarket 48
standard clock 210 207, 210, 221, 304, 324, 402 supernovae 96
spring standard deviation 437, 459 straight superposition 301
illustration 459 lines in nature 59 support
standard kilogram 101, 131 straight line this book 10

Motion Mountain – The Adventure of Physics
standard pitch 290 drawing with a compass surface
standing wave 303 244 area 50
star straightness 50, 59, 60, 64 surface tension 235
age 41 strange quark dangers of 372
dark 261 mass 462 wave 295
twinkle 88 stratopause 366 surfing 321
twinkling 88 stratosphere surjectivity 275
star classes 261 details 366 surprises 281
starch stream in nature 241
self-organization 429 dorsal 29 swan

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
stargazers 473 ventral 29 wake behind 326
stars 87, 261 streets 500 swarms 433
state 27, 28 strong coupling constant 461 self-organization in
allows to distinguish 28 structure starling 432
definition 28 highest human-built 53 swell 297
in flows 356 stuff swimming
initial 237 in flows 354 olympic 322
of a mass point, complete subgroup 273 swimming speed
237 submarines 322 underwater 372
of a system 237 subscripts 445 swing 318
of motion 27 Sun 105, 110, 217, 261 Swiss cheese 333
physical 27 angular momentum 118 switching
state space 78 density and tides 199 off the lights 305
states 438 stopping 347 syllabary 445
stationary 253 will rise tomorrow 189 symbols
statistical mechanics 246 Sun size, angular 69 mathematical 447
statistical physics Sun’s age 465 other 447
definition 383 Sun’s lower photospheric symmetry 272
steel 349 pressure 466 and talking 270
stainless 350 Sun’s luminosity 465 classification, table 267
types table 350 Sun’s mass 465 colour 266
Stefan–Boltzmann black body Sun’s surface gravity 466 comparison table 277
subject index 599

crystal, full list 269 telephone speed 36 thermometer 387
discrete 272, 284 teleportation thermometry 548
external 272 impossibility of 109 thermopause 366
geometric 266 telescope 321 thermosphere
internal 272 television 25 details 366
low 266 temperature 357, 384, 385 thermostatics 389
mirror 285 absolute 385 third, major 309
of interactions 285 and cricket chirping 406 thorn 443, 553
of Lagrangian 279 granular 427 thought
of wave 285 introduction 383 processes 434
parity 284 lowest 385 thriller 255
S permutation 272
scale 266
lowest in universe 385
measured values table 385
throw
and motion 24
summary on 286 negative 385, 413 record 53
symmetry types in nature 281–283 scale 548 throwing
wallpaper, full list 268 tensegrity importance of 24
symmetry operations 272 example of 349 speed record 36, 64

Motion Mountain – The Adventure of Physics
Système International structures 348 thumb 69
d’Unités (SI) 453 tensor 277, 278 tide
system and rotation 282 and Coriolis effect 139
conservative 187, 234, 254 definition 278 and friction 199
definition 28 moment of inertia 285 and hatching insects 192
dissipative 234, 236, 254 order of 279 as wave 296
extended 291 product 448 importance of 197
geocentric 216 rank 279 once or twice per day 224
heliocentric 216 rank of 278, 279 slowing Moon 191, 535
motion in simple 426 web sites 542 spring 192

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
physical 26 Tera 455 time 438, 439
positional number 446 tesla 455 coordinate, universal 42
writing 445 testicle 530 absolute 43
system, isolated 405 testimony 530 absoluteness of 43
tether 89 arrow of 48
T tetrahedron 65, 116 deduced from clocks 42
Talitrus saltator 46 Theophrastus 40, 44 deduction 42
talking theorem definition 41
and symmetry 270 Noether’s 280 definition of 281
tantalum 41 thermoacoustic engines 412 flow of 48–49
tau mass 462 thermodynamic degree of interval 43
Taurus 213 freedom 394 is absolute 43
tax collection 453 thermodynamics 354, 383, 389 is necessary 43
teaching first law 113 is unique 43
best method for 9 first principle 388 measured with real
teapot 376 indeterminacy relation of numbers 43
technology 446 399 measurement, ideal 44
tectonics 149 second law 113 proper 42
plate 535 second principle 403 properties, table 43
tectonism 192 second principle of 409 relative or absolute 52
teeth growth 418 third principle 385 reversal 48, 284
telekinesis 262 third principle of 400 translation 279
600 subject index

travel, difficulty of 156 tripod 233 unicycle 99, 350
values, table 41 tritone 309 uniqueness 43, 51
time-bandwidth product 314 Trojan asteroids 194, 319 unit 405
Titius’s rule 220 tropical year 464 astronomical 464
TNT energy content 464 tropopause 367 natural 462
Tocharian 530 troposphere unitary 275
tog 405 details 367 units 51, 453
toilet truth 336 non-SI 456
aeroplane 380 tsunami 299 provincial 456
research 428 tuft 546 SI, definition 453
tokamak 386 tumbleweed 90 universal gravitation 177
T tonne, or ton 455
top quark
tunnel
through the Earth 221
origin of name 180
universal time coordinate 457
mass 462 turbopause 366 universality of gravity 195
time-bandwidthtopology 51 turbulence 362, 426 universe 27, 238
torque 118 in pipes 430 accumulated change 264
touch 98 is not yet understood 436 angular momentum 118

Motion Mountain – The Adventure of Physics
tower Turkish mathematics 443 description by universal
leaning, in Pisa 76 turtle gravitation 211
toys puzzle 285 two-dimensional 71
physical 473 twinkle unpredictability
train 105 of stars 88 practical 239
motion puzzle 158 typeface up quark
puzzle 92 italic 441 mass 462
trajectory 75 Tyrannosaurus rex 169 urination
transformation time for 379
of matter and bodies 20 U URL 470

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
of motion in engines 109 U(1) 512 UTC 42
symmetry 272 udeko 455
translation invariance 43, 51 Udekta 455 V
transport 20 UFOs 322 vacuum
and change 24 ultracentrifuges 209 human exposure to 161
is motion 24 ultrasound impedance 463
transubstantiation 336 imaging 290, 311, 313 permeability 463
travel motor 233 permittivity 463
space, and health 161 umbrella 79 variability 25
tree 458 unboundedness 37, 43, 51, 106, variance 459
family, of physical 276 variation 27
concepts 27 uncertainty Varuna 260
growth 36 relation in vases, communicating 359
height, limits to 338 thermodynamics 399 vector 278
leaves and Earth rotation relative 459 axial 284
145 total 459 definition 80
pumping of water in 545 underwater polar 284
strength 338 sound 322, 324 product 115
triangle speed records 372 space 81
geometry 73 swimming 372 space, Euclidean 81
triboscopes 236 wings 363 Vega 261
Trigonopterus oblongus 244 Unicode 446 at the North pole 146
subject index 601

velars 444 assumptions 58 group velocity of 322
velocimeters 477 vomit comet 503 photographs of types 296
velocity 35, 82, 439 vortex solitary 316
angular 116, 118 at wing end 363 spectrum, table 297
as derivative 83 in bathtub or sink 534 thermal capillary 321
escape 212 in fluids 382 types and properties 298
first cosmic 212 in the sea 382 watt 455
Galilean 35 ring film 380 wave 291
in space 37 ring leapfrogging 379 capillary 299
is not Galilean 37 rings 379 circular 305, 307
measurement devices, tube 412 cnoidal 329
V table 39
of birds 37
Vulcan 197
Vulcanoids 261
crest 323
deep water 295
phase 293 dispersion, films 299
velars propagation 293 W emission 303
proper 36 W boson energy 300
properties, table 37 mass 462 freak 323

Motion Mountain – The Adventure of Physics
second cosmic 212 wafer gravity water 295–300
third cosmic 212 consecrated 344 group 303
values, table 36 silicon 344 harmonic 293
wave 293 wake 297 highest sea 320
vena cava 367 angle 326, 327 in deep water 327
vendeko 455 behind ship or swan 326, infra-gravity 296
Vendekta 455 328 internal 300
ventomobil 517 walking linear 293, 307
Venturi gauge 362 and friction 402 long water 295
Venus 217 and pendulum 122 long-period 296

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
vernier 65 angular momentum 118 longitudinal 294
vessels, communicating 359 human 122 momentum 300
video maximum animal speed monster 323
bad for learning 9 501 motion 300
recorder, dream 404 on water 169, 300 plane 307
videos 25 robots 264 pulse 303
Vietnam 224 speed 183 reflection 300
viewpoint-independence 270 wallpaper groups 268 rogue 323
viewpoints 266 washboard roads 422 sea, energy 326
vinyl record 293 water 545 secondary 304
Virgo 213 dead 294 shallow 298
vis viva 110 origin of oceans 370 shallow water 295
viscosity patterns in 427 shock 326
dynamic 341, 430 ripple 299 short water 295
kinematic 430, 519 speed records under 372 sine 293
voice strider 170 solitary 316
human 307 tap 426 solitary, in sand 420
void 339 triple point 385 speed
volcanoes 535 walking on 169, 300 measured values, table 294
volt 455 walking robot 170 spherical 305, 307
volume 57 water wave 295, 329 standing 303, 307
additivity 56 formation of deep 295 summary 331
602 subject index

surface tension 295–300 wheel work 31, 110, 254
symmetry 285 axle, effect of 490 conservation 31
thermal capillary 321 biggest ever 92 definition in physics 111
trans-tidal 296 in bacteria 92 physical 229
transverse 294 in living beings 89–92 world 15, 27
ultra-gravity 296 rolling 121 question center 474
velocity 293 ultimate 23, 558 World Geodetic System 466,
water see water wave, 295, vs. leg 168 537
329 whelps 329 writing 441
wave equation whip system 445
definition 305 cracking 349 wrong 437
W wave motion
six main properties and
speed of 36
whirl 154
wyn 443

effects 302 white colour X
wave wavelength 293, 301 does not last 403 xenno 455
wavelet transformation 326 Wien’s displacement constant Xenta 455
waw 444 464 xylem 359

Motion Mountain – The Adventure of Physics
weak mixing angle 461 wife 530
weather 159, 431 Wikimedia 562 Y
unpredictability of 239 will, free 241–242 yard 458
web wind generator year
largest spider 53 angular momentum 118 number of days in the past
list of interesting sites 471 wind resistance 234 200
world-wide 470 wine 336 yo-yo 182
web cam arcs 428 yocto 455
Foucault’s pendulum 141 bottle 109, 129 yogh 443
weber 455 puzzle 62 yot 444

copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net
week 450 wing Yotta 455
days, order of 221 and flow 363
Weib 530 underwater 363 Z
weight 202, 228 Wirbelrohr 412 Z boson
of the Moon 219 wire mass 462
weightlessness at Equator 224 zenith 163
effects on cosmonauts 204 WMAP 503 angle 179
weightlessness, feeling of 504 wobble zepto 455
weko 455 Euler’s 147 zero
Wekta 455 falsely claimed by Indian digit 446
whale Chandler 147, 535 zero gravity 503
blue 320 women 192, 530 Zetta 455
constellation 213 dangers of looking after zippo 425
ears 324 167 zodiac 175
sounds 322 wonder illustration of 222
sperm 371 impossibility of 388
MOTION MOUNTAIN
The Adventure of Physics – Vol. I
Fall, Flow and Heat

Is nature really as lazy as possible?
How do animals and humans move?
What is the most fantastic voyage possible?
Can we achieve levitation with the help of gravity?
Is motion predictable?
What is the smallest entropy value?
How do patterns and rhythms appear in nature?
How can you detect atoms and measure their size?
Which problems in everyday physics are unsolved?

Answering these and other questions on motion,
this series gives an entertaining and mind-twisting
introduction into modern physics – one that is
surprising and challenging on every page.
Starting from everyday life, the adventure provides
an overview of the modern results in mechanics,
heat, electromagnetism, relativity,
quantum physics and unification.

Christoph Schiller, PhD Université Libre de Bruxelles,
is a physicist and physics popularizer. He wrote this
book for his children and for all students, teachers and
readers interested in physics, the science of motion.

Pdf file available free of charge at
www.motionmountain.net