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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::SquareReplicate \-\- replicating squares .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::SquareReplicate; \& my $path = Math::PlanePath::SquareReplicate\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path is a self-similar replicating square, .PP .Vb 5 \& 40\-\-39\-\-38 31\-\-30\-\-29 22\-\-21\-\-20 4 \& | | | | | | \& 41 36\-\-37 32 27\-\-28 23 18\-\-19 3 \& | | | \& 42\-\-43\-\-44 33\-\-34\-\-35 24\-\-25\-\-26 2 \& \& 49\-\-48\-\-47 4\-\- 3\-\- 2 13\-\-12\-\-11 1 \& | | | | | | \& 50 45\-\-46 5 0\-\- 1 14 9\-\-10 <\- Y=0 \& | | | \& 51\-\-52\-\-53 6\-\- 7\-\- 8 15\-\-16\-\-17 \-1 \& \& 58\-\-57\-\-56 67\-\-66\-\-65 76\-\-75\-\-74 \-2 \& | | | | | | \& 59 54\-\-55 68 63\-\-64 77 72\-\-73 \-3 \& | | | \& 60\-\-61\-\-62 69\-\-70\-\-71 78\-\-79\-\-80 \-4 \& \& ^ \& \-4 \-3 \-2 \-1 X=0 1 2 3 4 .Ve .PP The base shape is the initial N=0 to N=8 section, .PP .Vb 3 \& 4 3 2 \& 5 0 1 \& 6 7 8 .Ve .PP It then repeats with 3x3 blocks arranged in the same pattern, then 9x9 blocks, etc. .PP .Vb 7 \& 36 \-\-\- 27 \-\-\- 18 \& | | \& | | \& 45 0 \-\-\- 9 \& | \& | \& 54 \-\-\- 63 \-\-\- 72 .Ve .PP The replication means that the values on the X axis are those using only digits 0,1,5 in base 9. Those to the right have a high 1 digit and those to the left a high 5 digit. These digits are the values in the initial N=0 to N=8 figure which fall on the X axis. .PP Similarly on the Y axis digits 0,3,7 in base 9, or the leading diagonal X=Y 0,2,6 and opposite diagonal 0,4,8. The opposite diagonal digits 0,4,8 are 00,11,22 in base 3, so is all the values in base 3 with doubled digits aabbccdd, etc. .SS "Level Ranges" .IX Subsection "Level Ranges" A given replication extends to .PP .Vb 3 \& Nlevel = 9^level \- 1 \& \- (3^level \- 1) <= X <= (3^level \- 1) \& \- (3^level \- 1) <= Y <= (3^level \- 1) .Ve .SS "Complex Base" .IX Subsection "Complex Base" This pattern corresponds to expressing a complex integer X+i*Y in base b=3, .PP .Vb 1 \& X+Yi = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0] .Ve .PP using complex digits a[i] encoded in N in integer base 9, .PP .Vb 11 \& a[i] digit N digit \& \-\-\-\-\-\-\-\-\-\- \-\-\-\-\-\-\- \& 0 0 \& 1 1 \& i+1 2 \& i 3 \& i\-1 4 \& \-1 5 \& \-i\-1 6 \& \-i 7 \& \-i+1 8 .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::SquareReplicate\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::SquareReplicate\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::SquareReplicate->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, 9**$level \- 1)\*(C'\fR. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::CornerReplicate, Math::PlanePath::LTiling, Math::PlanePath::GosperReplicate, Math::PlanePath::QuintetReplicate .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 .