| klein(6x) | XScreenSaver manual | klein(6x) |
klein - Draws a 4d Klein bottle.
klein [-display host:display.screen] [-install] [-visual visual] [-window] [-root] [-delay usecs] [-fps] [-klein-bottle bottle-name] [-figure-8] [-pinched-torus] [-lawson] [-mode display-mode] [-wireframe] [-surface] [-transparent] [-appearance appearance] [-solid] [-bands] [-colors color-scheme] [-onesided] [-twosided] [-rainbow] [-depth] [-change-colors] [-view-mode view-mode] [-walk] [-turn] [-walk-turn] [-orientation-marks] [-projection-3d mode] [-perspective-3d] [-orthographic-3d] [-projection-4d mode] [-perspective-4d] [-orthographic-4d] [-speed-wx float] [-speed-wy float] [-speed-wz float] [-speed-xy float] [-speed-xz float] [-speed-yz float] [-walk-direction float] [-walk-speed float]
The klein program shows three different Klein bottles in 4d: the figure-8 Klein bottle, the pinched torus Klein bottle, or the Lawson Klein bottle. You can walk on the Klein bottle, see it turn in 4d, or walk on it while it turns in 4d. The figure-8 Klein bottle is well known in its 3d form. The 4d form used in this program is an extension of the 3d form to 4d that does not intersect itself in 4d (which can be seen in the depth colors mode when using static colors). The pinched torus Klein bottle also does not intersect itself in 4d (which can be seen in the depth colors mode when using static colors). The Lawson Klein bottle, on the other hand, does intersect itself in 4d. Its primary use is that it has a nice appearance for walking and for turning in 3d.
The Klein bottle is a non-orientable surface. To make this apparent, the two-sided color mode can be used. Alternatively, orientation markers (curling arrows) can be drawn as a texture map on the surface of the Klein bottle. While walking on the Klein bottle, you will notice that the orientation of the curling arrows changes (which it must because the Klein bottle is non-orientable).
The program projects the 4d Klein bottle to 3d using either a perspective or an orthographic projection. Which of the two alternatives looks more appealing depends on the viewing mode and the Klein bottle. For example, the Lawson Klein bottle looks nicest when projected perspectively. The figure-8 Klein bottle, on the other hand, looks nicer while walking when projected orthographically from 4d. For the pinched torus Klein bottle, both projection modes give equally acceptable projections.
The projected Klein bottle can then be projected to the screen either perspectively or orthographically. When using the walking modes, perspective projection to the screen should be used.
There are three display modes for the Klein bottle: mesh (wireframe), solid, or transparent. Furthermore, the appearance of the Klein bottle can be as a solid object or as a set of see-through bands. Finally, the colors with with the Klein bottle is drawn can be set to one-sided, two-sided, rainbow, or depth. In one-sided mode, the Klein bottle is drawn with the same color on both "sides." In two-sided mode (using static colors), the Klein bottle is drawn with red on one "side" and green on the "other side." Of course, the Klein bottle only has one side, so the color jumps from red to green along a curve on the surface of the Klein bottle. This mode enables you to see that the Klein bottle is non-orientable. If changing colors are used in two-sided mode, changing complementary colors are used on the respective "sides." The rainbow color mode (using static colors) draws the Klein bottle with a color wheel of fully saturated rainbow colors. If changing colors are used, the color wheel's colors change dynamically. The rainbow color mode gives a very nice effect when combined with the see-through bands mode or with the orientation markers drawn. The depth color mode draws the Klein bottle with colors that are chosen according to the 4d "depth" of the points. If static colors are used, this mode enables you to see that the figure-8 and pinched torus Klein bottles do not intersect themselves in 4d, while the Lawson Klein bottle does intersect itself.
The rotation speed for each of the six planes around which the Klein bottle rotates can be chosen. For the walk-and-turn mode, only the rotation speeds around the true 4d planes are used (the xy, xz, and yz planes).
Furthermore, in the walking modes the walking direction in the 2d base square of the Klein bottle and the walking speed can be chosen.
This program is somewhat inspired by Thomas Banchoff's book "Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions", Scientific American Library, 1990.
klein accepts the following options:
The following three options are mutually exclusive. They determine which Klein bottle is displayed.
The following four options are mutually exclusive. They determine how the Klein bottle is displayed.
The following three options are mutually exclusive. They determine the appearance of the Klein bottle.
The following five options are mutually exclusive. They determine how to color the Klein bottle.
The following options determine whether the colors with which the Klein bottle is displayed are static or are changing dynamically.
The following four options are mutually exclusive. They determine how to view the Klein bottle.
The following options determine whether orientation marks are shown on the Klein bottle.
The following three options are mutually exclusive. They determine how the Klein bottle is projected from 3d to 2d (i.e., to the screen).
The following three options are mutually exclusive. They determine how the Klein bottle is projected from 4d to 3d.
The following six options determine the rotation speed of the Klein bottle around the six possible hyperplanes. The rotation speed is measured in degrees per frame. The speeds should be set to relatively small values, e.g., less than 4 in magnitude. In walk mode, all speeds are ignored. In walk-and-turn mode, the 3d rotation speeds are ignored (i.e., the wx, wy, and wz speeds). In walk-and-turn mode, smaller speeds must be used than in the turn mode to achieve a nice visualization. Therefore, in walk-and-turn mode the speeds you have selected are divided by 5 internally.
The following two options determine the walking speed and direction.
If you run this program in standalone mode in its turn mode, you can rotate the Klein bottle by dragging the mouse while pressing the left mouse button. This rotates the Klein bottle in 3D, i.e., around the wx, wy, and wz planes. If you press the shift key while dragging the mouse with the left button pressed the Klein bottle is rotated in 4D, i.e., around the xy, xz, and yz planes. To examine the Klein bottle at your leisure, it is best to set all speeds to 0. Otherwise, the Klein bottle will rotate while the left mouse button is not pressed. This kind of interaction is not available in the two walk modes.
Copyright © 2005-2020 by Carsten Steger. Permission to use, copy, modify, distribute, and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. No representations are made about the suitability of this software for any purpose. It is provided "as is" without express or implied warranty.
Carsten Steger <carsten@mirsanmir.org>, 11-jan-2020.
| 5.45 (08-Dec-2020) | X Version 11 |