Sponsoring The Perl Toolchain Summit 2025: Help make this important event another success Learn more

NAME

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

SYNOPSIS

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

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

SEE ALSO

Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidSpiral

HOME PAGE

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

LICENSE

Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

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/>.