NAME

Math::PlanePath::FlowsnakeCentres -- self-similar path of hexagon centres

SYNOPSIS

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

DESCRIPTION

This path is a variation of the flowsnake curve by William Gosper which follows the flowsnake tiling the same way but follows the centres of hexagons instead of corners and across. The result is the same overall shape, but a symmetric base figure.

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

    -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9

The points are spread out on every second X coordinate to make little triangles with integer coordinates, per "Triangular Lattice" in Math::PlanePath.

The basic pattern is the seven points 0 to 6,

        4---- 5
      /        \
     3---- 2     6---
             \
        0---- 1

This repeats at 7-fold increasing scale, with sub-sections rotated according to the edge direction, and the 1, 2 and 6 sub-sections in reverse. Eg. N=7 to N=13 is the "1" part, taking the base figure in reverse, and rotated so the end points towards the "2".

The next level can be seen at the midpoints of each such group, being N=2,11,18,23,30,37,46.

                 ---- 37                 
             ----       ---    
       30----              ---           
       |                      ---       
      |                           46     
      |                              
      |        ----18                    
     |    -----      ---        
    23---               ---              
                           ---        
                           --- 11        
                      -----          
                 2 ---                   

FUNCTIONS

$path = Math::PlanePath::FlowsnakeCentres->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.

Fractional positions give an X,Y position along a straight line between the integer positions.

FORMULAS

N to X,Y

The n_to_xy() calculation follows Ed Schouten's method

    http://80386.nl/projects/flowsnake/

breaking N into base-7 digits, applying reversals from high to low according to digits 1, 2, or 6, then applying rotation and position according to the resulting digits.

Unlike Ed's code, the path here starts from N=0 at the edge of the Gosper island shape and for that reason doesn't cover the plane. An offset of N-2*7^21 and suitable X,Y offset can be applied to get the same result.

X,Y to N

The xy_to_n() calculation also follows Ed Schouten's method which is based on a nice observation that the seven cells of the base figure can be identified from their coordinates, and the centres of those figures then shrunk down to unit coordinates, thus generating digits of N from low to high.

In triangular grid style X,Y a remainder can be formed as

    m = (x + 2*y) mod 7

With the base figure 0 at 0,0 the remainders are

        4---- 6
      /        \
     1---- 3     5
             \
        0---- 2

The remainders are the same when the shape is moved by some multiple of the next level X=5,Y=1 or its rotated forms X=1,Y=3 and X=-4,Y=1 etc. Those vectors all have X+2*Y==0 mod 7.

From the m remainder an offset can be applied to move X,Y to the 0 position, leaving X,Y a multiple of the next level vector X=5,Y=1 etc. That vector can then be shrunk down with

    Xshrunk = (3*Y + 5*X) / 14
    Yshrunk = (5*Y - X) / 14

These are integers as 3*Y+5*X and 5*Y-X are always multiples of 14. For example the N=35 point at X=2,Y=6 reduces to X = (3*6+5*2)/14 = 2 and Y = (5*6-2)/14 = 2, which is then the "5" part of the base figure.

The remainders can be mapped to digits then reversals and rotations applied, from high to low, according to the edge orientation. These steps can be combined in a single lookup table with 6 states (three rotations and forward or reverse).

The key to this approach is that the base figure is symmetric around a central point, so the rotations or reversals in the path can be applied after breaking down the tiling. Can it be made to work on non-symmetric like the "across" style Math::PlanePath::Flowsnake?

SEE ALSO

Math::PlanePath, Math::PlanePath::Flowsnake, Math::PlanePath::GosperIslands

Math::PlanePath::KochCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::PeanoCurve, Math::PlanePath::ZOrderCurve

http://80386.nl/projects/flowsnake/

LICENSE

Copyright 2011 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