Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group generated by the permutation $$ a: \ (m,n) \ \mapsto \ (m-n,m) $$ of order $6$ and the involutions $$ b: \ (m,n) \ \mapsto \ \begin{cases} (m,2n+1) & \text{if} \ n \equiv 0(\text{mod} \ 2), \\ (m,(n-1)/2) & \text{if} \ n \equiv 1(\text{mod} \ 4), \\ (m,n) & \text{if} \ n \equiv 3(\text{mod} \ 4) \\ \end{cases} $$ and $$ c: \ (m,n) \ \mapsto \ \begin{cases} (m,2n+3) & \text{if} \ n \equiv 0(\text{mod} \ 2), \\ (m,(n-3)/2) & \text{if} \ n \equiv 3(\text{mod} \ 4), \\ (m,n) & \text{if} \ n \equiv 1(\text{mod} \ 4). \\ \end{cases} $$ Drawing spheres of radius $r$ about $(0,0)$ for "large enough" $r$ reveals fractal-like structures.

**Added on July 28, 2014:** A video showing more pictures is now available
on YouTube here. The video starts with a sequence showing entire spheres
of small radii, i.e. from $r = 8$ to $r = 24$, and continues with pictures
showing smaller parts of spheres of larger radii up to $r = 45$.
Monochrome pictures show only one sphere, respectively,
a part thereof; colored pictures show multiple spheres in different colors.

**Added on March 16, 2014:** Pictures of the spheres of radius $30$ and $36$ can be downloaded here:

Radius 30 (3487 x 3079 pixels, 111KB), Radius 36 (10375 x 9103 pixels, 693KB).

Sample snippets of the large -- about $200$ megapixels at $r = 38$ to about $3$ gigapixels at $r = 45$ -- pictures are (black pixel = belongs to sphere, white pixel = doesn't belong to sphere):

The images above show parts of the spheres of radius $38$, $40$ and $45$.

Question:How can the observed patterns be explained?

*Remark 1:* The cardinalities of the spheres of radii $r = 0, \dots, 45$
about $(0,0)$ are

```
1, 2, 4, 8, 14, 26, 39, 68, 114, 188, 289, 404, 560, 827, 1341, 2052, 3158, 4540,
6091, 8630, 12241, 17739, 27727, 41846, 61234, 86647, 117806, 163795, 233939, 340659,
523862, 768739, 1110855, 1569204, 2148377, 2994661, 4287462, 6195498, 9389566,
13568954, 19542862, 27619364, 38048372, 53304607, 76433012, 109839303.
```

The *entire* sphere of radius $20$ looks as follows (the overall
shape of the larger spheres is roughly similar):

**Added:** At a scale of about $1:100$, the entire sphere of radius $45$ looks as follows:

*Remark 2:* In the notation of this and this question, $b$ and $c$
induce on the second coordinate the *class transpositions* $\tau_{0(2),1(4)}$
and $\tau_{0(2),3(4)}$, respectively. Further, in the notation of (1)
and (2) we have $G < {\rm RCWA}(\mathbb{Z}^2)$.

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