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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::GosperReplicate \-\- self\-similar hexagon replications .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::GosperReplicate; \& my $path = Math::PlanePath::GosperReplicate\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is a self-similar hexagonal tiling of the plane. At each level the shape is the Gosper island. .PP .Vb 10 \& 17\-\-\-\-16 4 \& / \e \& 24\-\-\-\-23 18 14\-\-\-\-15 3 \& / \e \e \& 25 21\-\-\-\-22 19\-\-\-\-20 10\-\-\-\- 9 2 \& \e / \e \& 26\-\-\-\-27 3\-\-\-\- 2 11 7\-\-\-\- 8 1 \& / \e \e \& 31\-\-\-\-30 4 0\-\-\-\- 1 12\-\-\-\-13 <\- Y=0 \& / \e \e \& 32 28\-\-\-\-29 5\-\-\-\- 6 45\-\-\-\-44 \-1 \& \e / \e \& 33\-\-\-\-34 38\-\-\-\-37 46 42\-\-\-\-43 \-2 \& / \e \e \& 39 35\-\-\-\-36 47\-\-\-\-48 \-3 \& \e \& 40\-\-\-\-41 \-4 \& \& ^ \& \-7 \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 7 .Ve .PP The points are spread out on every second X coordinate to make a a triangular lattice in integer coordinates (see \*(L"Triangular Lattice\*(R" in Math::PlanePath). .PP The base pattern is the inner N=0 to N=6, then six copies of that shape are arranged around as the blocks N=7,14,21,28,35,42. Then six copies of the resulting N=0 to N=48 shape are replicated around, etc. .PP Each point represents a little hexagon, thus tiling the plane with hexagons. The innermost N=0 to N=6 are for instance, .PP .Vb 10 \& * * \& / \e / \e \& / \e / \e \& * * * \& | 3 | 2 | \& * * * \& / \e / \e / \e \& / \e / \e / \e \& * * * * \& | 4 | 0 | 1 | \& * * * * \& \e / \e / \e / \& \e / \e / \e / \& * * * \& | 5 | 6 | \& * * * \& \e / \e / \& \e / \e / \& * * .Ve .PP The further replications are the same arrangement, but the sides become ever wigglier and the centres rotate around. The rotation can be seen at N=7 X=5,Y=1 which is up from the X axis. .PP The \f(CW\*(C`FlowsnakeCentres\*(C'\fR path is this same replicating shape, but starting from a side instead of the middle and traversing in such as way as to make each N adjacent. The \f(CW\*(C`Flowsnake\*(C'\fR curve itself is this replication too, but following edges. .SS "Complex Base" .IX Subsection "Complex Base" The path corresponds to expressing complex integers X+i*Y in a base .PP .Vb 1 \& b = 5/2 + i*sqrt(3)/2 .Ve .PP with some scaling to put equilateral triangles on a square grid. So for integer X,Y with X and Y either both odd or both even, .PP .Vb 1 \& X/2 + i*Y*sqrt(3)/2 = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0] .Ve .PP where each digit a[i] is either 0 or a sixth root of unity encoded into N as base 7 digits, .PP .Vb 2 \& r = e^(i*pi/3) \& = 1/2 + i*sqrt(3)/2 sixth root of unity \& \& N digit a[i] complex number \& \-\-\-\-\-\-\- \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& 0 0 \& 1 r^0 = 1 \& 2 r^2 = 1/2 + i*sqrt(3)/2 \& 3 r^3 = \-1/2 + i*sqrt(3)/2 \& 4 r^4 = \-1 \& 5 r^5 = \-1/2 \- i*sqrt(3)/2 \& 6 r^6 = 1/2 \- i*sqrt(3)/2 .Ve .PP 7 digits suffice because .PP .Vb 1 \& norm(b) = (5/2)^2 + (sqrt(3)/2)^2 = 7 .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::GosperReplicate\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::GosperReplicate\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::GosperReplicate->new ()" Create and return a new path 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$n\fR on the path. Points begin at 0 and if \f(CW\*(C`$n < 0\*(C'\fR then the return is an empty list. .SS "Level Methods" .IX Subsection "Level Methods" .ie n .IP """($n_lo, $n_hi) = $path\->level_to_n_range($level)""" 4 .el .IP "\f(CW($n_lo, $n_hi) = $path\->level_to_n_range($level)\fR" 4 .IX Item "($n_lo, $n_hi) = $path->level_to_n_range($level)" Return \f(CW\*(C`(0, 7**$level \- 1)\*(C'\fR. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::GosperIslands, Math::PlanePath::Flowsnake, Math::PlanePath::FlowsnakeCentres, Math::PlanePath::QuintetReplicate, Math::PlanePath::ComplexPlus .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 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 .