/ 
Math-PlanePath-89
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::GosperReplicate

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,

N digit     a[i] complex number
-------     -------------------
  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::Flowsnake, Math::PlanePath::FlowsnakeCentres, Math::PlanePath::QuintetReplicate, Math::PlanePath::ComplexPlus

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/>.