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,

     w6 = e^(i*pi/3)            sixth root of unity
        = 1/2 + i*sqrt(3)/2

     N digit     a[i] complex number
     -------     -------------------
       0          0
       1         w6^0 =  1
       2         w6^1 =  1/2 + i*sqrt(3)/2
       3         w6^2 = -1/2 + i*sqrt(3)/2
       4         w6^3 = -1
       5         w6^4 = -1/2 - i*sqrt(3)/2
       6         w6^5 =  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.

Level Methods

($n_lo, $n_hi) = $path->level_to_n_range($level)

Return (0, 7**$level - 1).

FORMULAS ^

Rotations

The digits positions 1,2,3,4,5,6 go around +60deg each, so the N for rotation by +60 is each digit +1, cycling around.

    rot+60(N)  = 0, 2, 3, 4, 5, 6, 1, 14, 16, 17, ... decimal
               = 0, 2, 3, 4, 5, 6, 1, 20, 22, 23, ... base7

    rot+120(N) = 0, 3, 4, 5, 6, 1, 2, 21, 24, 25, ... decimal
               = 0, 3, 4, 5, 6, 1, 2, 30, 33, 34, ... base7

    rot180(N)  = 0, 4, 5, 6, 1, 2, 3, 28, 32, 33, ... decimal
               = 0, 4, 5, 6, 1, 2, 3, 40, 44, 45, ... base7

    rot-120(N) = 0, 5, 6, 1, 2, 3, 4, 35, 40, 41, ... decimal
               = 0, 5, 6, 1, 2, 3, 4, 50, 55, 56, ... base7

    rot-60(N)  = 0, 6, 1, 2, 3, 4, 5, 42, 48, 43, ... decimal
               = 0, 6, 1, 2, 3, 4, 5, 60, 66, 61, ... base7

X,Y Extents

The maximum X in a given level N=0 to 7^k-1 can be calculated from the replications. A given high digit 1 to 6 has sub-parts located at b^k*w6^(d-1). Those sub-parts are all the same, so the one with maximum real(b^k*w6^(d-1)) contains the maximum X.

    N_xmax_digit(j) = d=1to6 where real(w6^(d-1) * b^j) is maximum
                    = 1,1,6,6,6,5,5,5,4,4,4,3,3,3,3,2,2, ...

                 k-1
    N_xmax(k) = digits N_xmax_digit(j)    low digit j=0
                 j=0
              = 0, 1, 8, 302, 2360, 16766, 100801, ...  decimal
              = 0, 1, 11, 611, 6611, 66611, 566611, ...  base7

                k-1
    z_xmax(k) = sum  w6^d[j] * b^j
                j=0      each d[j] with real(w6^d[j] * b^j) maximum
          = 0, 1, 7/2+1/2*sqrt3*i, 10-sqrt3*i, 57/2-3/2*sqrt3*i,...

    xmax(k) = 2*real(z_xmax(k))
            = 0, 2, 7, 20, 57, 151, 387, 1070, 2833, 7106, ...

For computer calculation these maximums can be calculated from the powers. The parts resulting can also be written in terms of the angle

    arg(b) = atan(sqrt(3)/5) = 19.106... degrees

For successive k, if adding this pushes the b^k angle past +30deg then the preceding digit goes past -30deg and becomes the new maximum X. Write the angle as a fraction of 60deg (pi/3),

    F = atan(sqrt(3)/5) / (pi/3)  = 0.318443 ...

This is irrational since b^k is never on the X or Y axes. That can be seen since 2/sqrt3*imag(b^k) mod 7 goes in a repeating pattern 1,5,4,6,2,3. Similarly 2*real(b^k) mod 7 so not on the Y axis, and also anything on the Y axis would have 3*k fall on the X axis.

Digits low to high are successive steps back cyclically 6,5,4,3,2,1 so that (with mod giving 0 to 5),

    N_xmax_digit(j) = (-floor(F*j+1/2) mod 6) + 1

The +1/2 is since initial direction b^0=1 is angle 0 which is half way between -30 and +30 deg.

Similarly for the location, using conj(w6) for rotation back

    z_xmax_exp(j) = floor(F*j+1/2)
                  = 0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5, ...
    z_xmax(k) = sum(j=0,k-1, conj(w6)^z_xmax_exp(j) * b^j)

By symmetry the maximum extent is the same in 60deg, 120deg, etc directions, suitably rotated. The N in those cases has the digits 1,2,3,4,5,6 cycled around for the rotation. In PlanePath triangular X,Y coordinates direction 60deg means when sum X+3*Y is a maximum, etc.

If the +1/2 in the floor is omitted then the effect is to find the maximum point in direction +30deg. In the PlanePath coordinates this means maximum sum S = X+Y.

    N_smax_digit(j) = (-floor(F*j) mod 6) + 1
                    = 1,1,1,1,6,6,6,5,5,5,4,4,4,3,3, ...

                 k-1
    N_smax(k) = digits N_smax_digit(j)    low digit j=0
                 j=0
              = 0, 1, 8, 57, 400, 14806, 115648, ...     decimal
              = 0, 1, 11, 111, 1111, 61111, 661111, ...  base7
    and also N_smax() + 1

    z_smax_exp(j) = floor(F*j)
                  = 0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6, ...
    z_smax(k) = sum(j=0,k-1, conj(w6)^z_smax_exp(j) * b^j)
              = 0, 1, 7/2+1/2*sqrt3*i, 9+3*sqrt3*i, 19+12*sqrt3*i, ...
    and also z_smax() + w6^2

    smax(k) = 2*real(z_smax(k)) + imag(z_smax(k))*2/sqrt3
            = 0, 2, 8, 24, 62, 172, 470, 1190, 3202, 8740, ...
              coordinate sum X+Y max

In the base figure, points 1 and 2 have the same X+Y=2 and this remains so in subsequent levels, so that for k>=1 N_smax(k) and N_smax(k)+1 are equal maximums.

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, 2015, 2016 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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