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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::KochSquareflakes \-\- four\-sided Koch snowflakes .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::KochSquareflakes; \& my $path = Math::PlanePath::KochSquareflakes\->new (inward => 0); \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This is the Koch curve shape arranged as four-sided concentric snowflakes. .PP .Vb 10 \& 61 10 \& / \e \& 63\-62 60\-59 9 \& | | \& 67 64 58 55 8 \& / \e / \e / \e \& 69\-68 66\-65 57\-56 54\-53 7 \& | | \& 70 52 6 \& / \e \& 71 51 5 \& \e / \& 72 50 4 \& | | \& 73 15 49 3 \& / / \e \e \& 75\-74 17\-16 14\-13 48\-47 2 \& | | | | \& 76 18 12 46 1 \& / / 4\-\-\-3 \e \e \& 77 19 . | 11 45 Y=0 \& \e \e 1\-\-\-2 / / \& 78 20 10 44 \-1 \& | | | \& 79\-80 5\-\-6 8\-\-9 42\-43 \-2 \& \e \e / / \& 81 7 41 \-3 \& | | \& 82 40 \-4 \& / \e \& 83 39 \-5 \& \e / \& 84 38 \-6 \& | \& 21\-22 24\-25 33\-34 36\-37 \-7 \& \e / \e / \e / \& 23 26 32 35 \-8 \& | | \& 27\-28 30\-31 \-9 \& \e / \& 29 \-10 \& \& ^ \& \-9 \-8 \-7 \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 7 8 9 10 .Ve .PP The innermost square N=1 to N=4 is the initial figure. Its sides expand as the Koch curve pattern in subsequent rings. The initial figure is on X=+/\-0.5,Y=+/\-0.5 fractions. The points after that are integer X,Y. .SH "Inward" .IX Header "Inward" The \f(CW\*(C`inward=>1\*(C'\fR option can direct the sides inward. The shape and lengths etc are the same. The angles and sizes mean there's no overlaps. .PP .Vb 10 \& 69\-68 66\-65 57\-56 54\-53 7 \& | \e / \e / \e / | \& 70 67 64 58 55 52 6 \& \e | | / \& 71 63\-62 60\-59 51 5 \& / \e / \e \& 72 61 50 4 \& | | \& 73 49 3 \& \e / \& 74\-75 17\-16 14\-13 47\-48 2 \& | | \e / | | \& 76 18 15 12 46 1 \& \e \e 4\-\-3 / / \& 77 19 |11 45 <\- Y=0 \& / / 1\-\-2 \e \e \& 78 20 7 10 44 \-1 \& | / \e | | \& 80\-79 5\-\-6 8\-\-9 43\-42 \-2 \& / \e \& 81 41 \-3 \& | | \& 82 29 40 \-4 \& \e / \e / \& 83 27\-28 30\-31 39 \-5 \& / | | \e \& 84 23 26 32 35 38 \-6 \& / \e / \e / \e | \& 21\-22 24\-25 33\-34 36\-37 \-7 \& \& ^ \& \-7 \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 7 .Ve .SS "Level Ranges" .IX Subsection "Level Ranges" Counting the innermost N=1 to N=4 square as level 0, a given level has .PP .Vb 1 \& looplen = 4*4^level .Ve .PP many points. The start of a level is N=1 plus the preceding loop lengths so .PP .Vb 3 \& Nstart = 1 + 4*[ 1 + 4 + 4^2 + ... + 4^(level\-1) ] \& = 1 + 4*(4^level \- 1)/3 \& = (4^(level+1) \- 1)/3 .Ve .PP and the end of a level similarly the total loop lengths, or simply one less than the next Nstart, .PP .Vb 2 \& Nend = 4 * [ 1 + ... + 4^level ] \& = (4^(level+2) \- 4) / 3 \& \& = Nstart(level+1) \- 1 .Ve .PP For example, .PP .Vb 5 \& level Nstart Nend (A002450,A080674) \& 0 1 4 \& 1 5 20 \& 2 21 84 \& 3 85 340 .Ve .PP The Xstart,Ystart position of the Nstart corner is a Lucas sequence, .IX Xref "Lucas Sequence" .PP .Vb 6 \& Xstart(0) = \-0.5 \& Xstart(1) = \-2 \& Xstart(2) = 4*Xstart(1) \- 2*Xstart(0) = \-7 \& Xstart(3) = 4*Xstart(2) \- 2*Xstart(1) = \-24 \& ... \& Xstart(level+1) = 4*Xstart(level) \- 2*Xstart(level\-1) \& \& 0.5, 2, 7, 24, 82, 280, 956, 3264, ... (A003480) .Ve .PP This recurrence occurs because the replications are 4 wide when horizontal but 3 wide when diagonal. .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::KochSquareflakes\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::KochSquareflakes\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::KochSquareflakes->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::KochSquareflakes\->new (inward => $bool)""" 4 .el .IP "\f(CW$path = Math::PlanePath::KochSquareflakes\->new (inward => $bool)\fR" 4 .IX Item "$path = Math::PlanePath::KochSquareflakes->new (inward => $bool)" .PD Create and return a new path object. .SS "Level Methods" .IX Subsection "Level Methods" .ie n .IP """($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 per \*(L"Level Ranges\*(R" above, .Sp .Vb 2 \& ( (4**$level \- 1)/3, \& 4*(4**$level \- 1)/3 ) .Ve .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 6 \& A003480 \-X and \-Y coordinate first point of each ring \& likewise A020727 \& A007052 X,Y coordinate of axis crossing, \& and also maximum height of a side \& A072261 N on Y negative axis (half way along first side) \& A206374 N on South\-East diagonal (end of first side) .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::KochSnowflakes .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014 Kevin Ryde .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 .