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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::OctagramSpiral \-\- integer points drawn around an octagram .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::OctagramSpiral; \& my $path = Math::PlanePath::OctagramSpiral\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a spiral around an octagram (8\-pointed star), .PP .Vb 10 \& 29 25 4 \& | \e / | \& 30 28 26 24 ...56\-55 3 \& | \e / | / \& 33\-32\-31 7 27 5 23\-22\-21 54 2 \& \e |\e / | / / \& 34 9\- 8 6 4\- 3 20 53 1 \& \e \e / / / \& 35 10 1\-\-2 19 52 <\- Y=0 \& / / \e \e \& 36 11\-12 14 16\-17\-18 51 \-1 \& / |/ \e | \e \& 37\-38\-39 13 43 15 47\-48\-49\-50 \-2 \& | / \e | \& 40 42 44 46 \-3 \& |/ \e | \& 41 45 \-4 \& \& ^ \& \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 ... .Ve .PP Each loop is 16 longer than the previous. The 18\-gonal numbers 18,51,100,etc fall on the horizontal at Y=\-1. .PP The inner corners like 23, 31, 39, 47 are similar to the \f(CW\*(C`SquareSpiral\*(C'\fR path, but instead of going directly between them the octagram takes a detour out to make the points of the star. Those excursions make each loops 8 longer (1 per excursion), hence a step of 16 here as compared to 8 for the \&\f(CW\*(C`SquareSpiral\*(C'\fR. .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern. For example to start at 0, .PP .Vb 1 \& n_start => 0 \& \& 28 24 \& 29 27 25 23 ... 55 54 \& 32 31 30 6 26 4 22 21 20 53 \& 33 8 7 5 3 2 19 52 \& 34 9 0 1 18 51 \& 35 10 11 13 15 16 17 50 \& 36 37 38 12 42 14 46 47 48 49 \& 39 41 43 45 \& 40 44 .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::OctagramSpiral\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::OctagramSpiral\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::OctagramSpiral->new ()" Create and return a new octagram spiral object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. .Sp For \f(CW\*(C`$n < 1\*(C'\fR the return is an empty list, it being considered the path starts at 1. .ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4 .el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4 .IX Item "$n = $path->xy_to_n ($x,$y)" Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR. \f(CW$x\fR and \f(CW$y\fR 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. .SH "FORMULAS" .IX Header "FORMULAS" .SS "X,Y to N" .IX Subsection "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. .SH "OEIS" .IX Header "OEIS" Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include .Sp .RS 4 (etc) .RE .PP .Vb 3 \& 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 .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidSpiral .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .