Math::PlanePath::OctagramSpiral -- integer points drawn around an octagram
use Math::PlanePath::OctagramSpiral; my $path = Math::PlanePath::OctagramSpiral->new; my ($x, $y) = $path->n_to_xy (123);
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
.
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
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.
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.
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
Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidSpiral
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of Math-PlanePath.
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