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

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Module Version: 126

# 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```

Points are spread out on every second X coordinate to make 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 can be taken as a little hexagon, so that all points tile 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 N=7 at 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 segments across hexagons.

## Complex Base

The path corresponds to expressing complex integers X+i*Y in a base

`    b = 5/2 + i*sqrt(3)/2`

with coordinates scaled to put equilateral triangles on a square grid. So for integer X,Y on the triangular grid (X,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 base-7 digits of N,

```     w6 = e^(i*pi/3)            sixth root of unity, b = 2 + w6
= 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`

## Rotate Numbering

Parameter `numbering_type => 'rotate'` applies a rotation in each sub-part according to its location around the preceding level.

The effect can be illustrated by writing N in base-7. Part 10-16 is the same as the middle 0-6. Part 20-26 has a rotation by +60 degrees. Part 30-36 has rotation by +120 degrees, and so on.

```                         22----21
/     /           numbering_type => 'rotate'
31    36    23    20    26          N shown in base-7
/  \     \     \        /
32    30    35    24----25    13----12
\        /                 /        \
33----34     3---- 2    14    10----11
/        \     \
46----45     4     0---- 1    15----16
\     \
41----40    44     5---- 6    64----63
\        /                 /        \
42----43    55----54    65    60    62
/        \     \     \  /
56    50    53    66    61
/     /
51----52```

Notice this means in each part the 11, 21, 31, etc, points are directed away from the middle in the same way, relative to the sub-part locations.

Working through the expansions gives the following rule for when an N is on the boundary of level k,

```    write N in k many base-7 digits  (empty string if k=0)
if any 0 digit then non-boundary
ignore high digit and all 1 digits
if any 4 or 5 digit then non-boundary
if any 32, 33, 66 pair then non-boundary```

A 0 digit is the middle of a block, or 4 or 5 digit the inner side of a block, for k>=1, hence non-boundary. After that the 6,1,2,3 parts variously expand with rotations so that a 66 is enclosed on the clockwise side and 32 and 33 on the anti-clockwise side.

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::GosperReplicate->new ()`
`\$path = Math::PlanePath::GosperReplicate->new (numbering_type => \$str)`

Create and return a new path object. The `numbering_type` parameter can be

```    "fixed"        (default)
"rotate"```
`(\$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

## Axis Rotations

In the fixed numbering, digit positions 1,2,3,4,5,6 go around +60deg each, so the N for rotation of X,Y by +60 degrees is each digit +1.

```    N          = 0, 1, 2, 3, 4, 5, 6, 10, 11, 12

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

etc```

In the rotate numbering, just adding +1 (etc) at the high digit alone is rotation.

## 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.

# HOME PAGE

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

# LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 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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