NAME

Math::PlanePath::GosperReplicate -- self-similar hexagon replications

SYNOPSIS

 use Math::PlanePath::GosperReplicate;
 my $path = Math::PlanePath::GosperReplicate->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This is a self-similar hexagonal tiling of the plane. At each level the shape is the Gosper island.

                         17----16                     4  
                        /        \                       
          24----23    18    14----15                  3  
         /        \     \                                
       25    21----22    19----20    10---- 9         2  
         \                          /        \           
          26----27     3---- 2    11     7---- 8      1  
                     /        \     \                    
       31----30     4     0---- 1    12----13     <- Y=0 
      /        \     \                                   
    32    28----29     5---- 6    45----44           -1  
      \                          /        \              
       33----34    38----37    46    42----43        -2  
                  /        \     \                       
                39    35----36    47----48           -3  
                  \                                      
                   40----41                          -4  

                          ^
    -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

The points are spread out on every second X coordinate to make a a triangular lattice in integer coordinates (see "Triangular Lattice" in Math::PlanePath).

The basic pattern is the inner N=0 to N=6, then six copies of that shape are arranged around as the N=7,14,21,28,35,42 blocks. Then six copies of the N=0 to N=48 shape are replicated around, etc.

Each point represents a little hexagon, thus tiling the plane with hexagons. The innermost N=0 to N=6 are for instance,

          *     *
         / \   / \
        /   \ /   \
       *     *     *
       |  3  |  2  |
       *     *     *
      / \   / \   / \
     /   \ /   \ /   \
    *     *     *     *
    |  4  |  0  |  1  |
    *     *     *     *
     \   / \   / \   /
      \ /   \ /   \ /
       *     *     *
       |  5  |  6  |
       *     *     *
        \   / \   /
         \ /   \ /
          *     *

The FlowsnakeCentres path is this same replicating shape, but starting from a side instead of the middle and with rotations and reflections to make points adjacent. The Flowsnake curve itself is this replication too, but following edges.

Complex Base

The path corresponds to expressing complex integers X+i*Y in a base b=5/2+i*sqrt(3)/2 with a bit of scaling to fit equilateral triangles to a square grid. So for integer X,Y both odd or both even,

    X/2 + i*Y*sqrt(3)/2 = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]

where each digit a[i] is either 0 or a sixth root of unity encoded into N as base 7 digits,

     N digit     a[i]
       0          0
       1         e^(0/3 * pi * i) = 1
       2         e^(1/3 * pi * i) = 1/2 + i*sqrt(3)/2
       3         e^(2/3 * pi * i) = -1/2 + i*sqrt(3)/2
       4         e^(3/3 * pi * i) = -1
       5         e^(4/3 * pi * i) = -1/2 - i*sqrt(3)/2
       6         e^(5/3 * pi * i) = 1/2 - i*sqrt(3)/2

7 digits suffice because

     norm(b) = (5/2)^2 + (sqrt(3)/2)^2 = 7

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::GosperReplicate->new ()

Create and return a new path object.

($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.

SEE ALSO

Math::PlanePath, Math::PlanePath::GosperIslands, Math::PlanePath::QuintetReplicate, Math::PlanePath::Flowsnake, Math::PlanePath::FlowsnakeCentres

HOME PAGE

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

LICENSE

Copyright 2011, 2012 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/>.

cpanm Math::PlanePath

perl -MCPAN -e shell
install Math::PlanePath