Provided by: libmath-planepath-perl_113-1_all bug

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 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/>.