Kevin Ryde > Math-PlanePath > Math::PlanePath::GosperReplicate

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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 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.

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 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.

The FlowsnakeCentres 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 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 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,

    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,

     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

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::Flowsnake, Math::PlanePath::FlowsnakeCentres, Math::PlanePath::QuintetReplicate, Math::PlanePath::ComplexPlus

HOME PAGE ^

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

LICENSE ^

Copyright 2011, 2012, 2013, 2014 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/>.

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