NAME

Math::PlanePath::OctagramSpiral -- integer points drawn around an octagram

SYNOPSIS

 use Math::PlanePath::OctagramSpiral;
 my $path = Math::PlanePath::OctagramSpiral->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path makes a spiral around an octagram (8-pointed star),

          29          25                 4
           | \       / |
          30 28    26 24    ...56-55     3
           |   \  /    |         /
    33-32-31  7 27  5 23-22-21 54        2
      \       |\  / |      /  /
       34  9- 8  6  4- 3 20 53           1
         \  \        /  /  /
          35 10  1--2 19 52          <- Y=0
         /  /           \  \
       36 11-12 14 16-17-18 51          -1
      /       |/  \ |         \
    37-38-39 13 43 15 47-48-49-50       -2
           |   /  \    |
          40 42    44 46                -3
           |/        \ |
          41          45                -4
                 ^
    -4 -3 -2 -1 X=0 1  2  3  4  5 ...

Each loop is 16 longer than the previous. The 18-gonal numbers 18,51,100,etc fall on the horizontal at Y=-1.

The inner corners like 23, 31, 39, 47 are similar to the "SquareSpiral" path, but instead of going directly between them the octagram takes a detour out to make the points of the star. Those excursions make each loops 8 longer (1 per excursion), hence a step of 16 here as compared to 8 for the "SquareSpiral".

N Start

The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start, in the same pattern. For example to start at 0,

    n_start => 0
          28          24
          29 27    25 23   ... 55 54
    32 31 30  6 26  4 22 21 20 53
       33  8  7  5  3  2 19 52
          34  9  0  1 18 51
       35 10 11 13 15 16 17 50
    36 37 38 12 42 14 46 47 48 49
          39 41    43 45
          40          44

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

"$path = Math::PlanePath::OctagramSpiral->new ()": Create and return a new octagram spiral object.
"($x,$y) = $path->n_to_xy ($n)": Return the X,Y coordinates of point number $n on the path.
For "$n < 1" the return is an empty list, it being considered the path starts at 1.
"$n = $path->xy_to_n ($x,$y)": Return the point number for coordinates "$x,$y". $x and $y are each rounded to the nearest integer, which has the effect of treating each N in the path as centred in a square of side 1, so the entire plane is covered.

FORMULAS

X,Y to N

The symmetry of the octagram can be used by rotating a given X,Y back to the first star excursion such as N=19 to N=23. If Y is negative then rotate back by 180 degrees, then if X is negative rotate back by 90, and if Y>=X then by a further 45 degrees. Each such rotation, if needed, is counted as a multiple of the side-length to be added to the final N. For example at N=19 the side length is 2. Rotating by 180 degrees is 8 side lengths, by 90 degrees 4 sides, and by 45 degrees is 2 sides.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A125201> (etc)

    n_start=1 (the default)
      A125201    N on X axis, from X=1 onwards, 18-gonals + 1
      A194268    N on diagonal South-East
    n_start=0
      A051870    N on X axis, 18-gonal numbers
      A139273    N on Y axis
      A139275    N on X negative axis
      A139277    N on Y negative axis
      A139272    N on diagonal X=Y
      A139274    N on diagonal North-West
      A139276    N on diagonal South-West
      A139278    N on diagonal South-East, second 18-gonals

HOME PAGE

<http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.

2021-01-23

perl v5.32.0