SIMPLE SOLUTIONS

# MATH::PLANEPATH::GOSPERREPLICATE(3PM) - man page online | library functions

Chapter
2016-01-11
```Math::PlanePath::GosperReplicatUsermContributed Perl DocumenMath::PlanePath::GosperReplicate(3pm)

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*b^2 + a*b + a

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.

Level Methods
"(\$n_lo, \$n_hi) = \$path->level_to_n_range(\$level)"
Return "(0, 7**\$level - 1)".

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

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

Copyright 2011, 2012, 2013, 2014, 2015 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