Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::GosperReplicate

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Module Version: 116   Source   Latest Release: Math-PlanePath-118

# 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.

# HOME PAGE

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

# LICENSE

Copyright 2011, 2012, 2013, 2014 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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