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This phenomenon can be understood as follows:

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive hyperbolic group (and non virtually cyclic).

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ (and orbit of (0,0)), the Cayley graph of $\Gamma$ (Gromov hyperbolic geometry) is projected (in a manner not too degenerate) on $\mathbb{Z}^2$ (euclidean geometry).
Then the emergence of explosive and fractal-like structures on the spheres is not surprising.

So any (non virtually cyclic) hyperbolic group would generate beautiful fractal-like pictures.

Example: $\Gamma = \langle a,b \rangle < Sym(\mathbb{Z}^{2})$ such that

• $a: (m,n) \to (m+1,m+n)$
• $b: (m,n) \to (m+n,n+1)$

To be confirmed: $\Gamma$ acts transitively on $\mathbb{Z}^{2}$, and is isomorphic to the free group $\mathbb{F}_2$ (so $\Gamma$ is hyperbolic).

The cardinalities of the spheres of radii $r=0,\dots,14$ about $(0,0)$ are :
$1,4,11,20,47,100,238,512,1124,2540,5569,12101,26208,56720,122600$

The entire sphere of radius $14$:

The entire ball of radius $14$ with a rainbow gradient according to the spheres:

Remark: I don't know if your group $G$ is hyperbolic, but we could have the same understanding for all the finitely generated groups of exponential growth.

This phenomenon can be understood as follows:

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive hyperbolic group (and non virtually cyclic).

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ (and orbit of (0,0)), the Cayley graph of $\Gamma$ (Gromov hyperbolic geometry) is projected (in a manner not too degenerate) on $\mathbb{Z}^2$ (euclidean geometry).
Then the emergence of explosive and fractal-like structures on the spheres is not surprising.

So any (non virtually cyclic) hyperbolic group would generate beautiful fractal-like pictures.

Example: $\Gamma = \langle a,b \rangle < Sym(\mathbb{Z}^{2})$ such that

• $a: (m,n) \to (m+1,m+n)$
• $b: (m,n) \to (m+n,n+1)$

To be confirmed: $\Gamma$ acts transitively on $\mathbb{Z}^{2}$.
Remark: $\Gamma$ is neither free nor hyperbolic nor torsion-free (see the comments below).

The cardinalities of the spheres of radii $r=0,\dots,14$ about $(0,0)$ are :
$1,4,11,20,47,100,238,512,1124,2540,5569,12101,26208,56720,122600$

The entire sphere of radius $14$:

The entire ball of radius $14$ with a rainbow gradient according to the spheres:

Remark: I don't know if your group $G$ is hyperbolic, but we could have the same understanding for all the finitely generated groups of exponential growth.

This phenomenon can be understood as follows:

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive hyperbolic group (and non virtually cyclic).

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ (and orbit of (0,0)), the Cayley graph of $\Gamma$ (Gromov hyperbolic geometry) is projected (in a manner not too degenerate) on $\mathbb{Z}^2$ (euclidean geometry).
Then the emergence of explosive and fractal-like structures on the spheres is not surprising.

So any (non virtually cyclic) hyperbolic group would generate beautiful fractal-like pictures.

Example: $\Gamma = \langle a,b \rangle < Sym(\mathbb{Z}^{2})$ such that

• $a: (m,n) \to (m+1,m+n)$
• $b: (m,n) \to (m+n,n+1)$

To be confirmed: $\Gamma$ acts transitively on $\mathbb{Z}^{2}$, and is isomorphic to the free group $\mathbb{F}_2$ (so $\Gamma$ is hyperbolic).

The cardinalities of the spheres of radii $r=0,\dots,13$ about $(0,0)$ are :
$1,4,11,20,47,100,238,512,1124,2540,5569,12101,26208,56720$

The entire sphere of radius $13$:

Remark: I don't know if your group $G$ is hyperbolic, but we could have the same understanding for all the finitely generated groups of exponential growth.

This phenomenon can be understood as follows:

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive hyperbolic group (and non virtually cyclic).

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ (and orbit of (0,0)), the Cayley graph of $\Gamma$ (Gromov hyperbolic geometry) is projected (in a manner not too degenerate) on $\mathbb{Z}^2$ (euclidean geometry).
Then the emergence of explosive and fractal-like structures on the spheres is not surprising.

So any (non virtually cyclic) hyperbolic group would generate beautiful fractal-like pictures.

Example: $\Gamma = \langle a,b \rangle < Sym(\mathbb{Z}^{2})$ such that

• $a: (m,n) \to (m+1,m+n)$
• $b: (m,n) \to (m+n,n+1)$

To be confirmed: $\Gamma$ acts transitively on $\mathbb{Z}^{2}$, and is isomorphic to the free group $\mathbb{F}_2$ (so $\Gamma$ is hyperbolic).

The cardinalities of the spheres of radii $r=0,\dots,14$ about $(0,0)$ are :
$1,4,11,20,47,100,238,512,1124,2540,5569,12101,26208,56720,122600$

The entire sphere of radius $14$:

The entire ball of radius $14$ with a rainbow gradient according to the spheres:

Remark: I don't know if your group $G$ is hyperbolic, but we could have the same understanding for all the finitely generated groups of exponential growth.

This phenomenon can be understood as follows:

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive hyperbolic group (and non virtually cyclic).

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ and the orbit of $(0,0)$, the Cayley graph (Gromov hyperbolic geometry) of $\Gamma$ is projected (in a manner not too degenerate) on $\mathbb{Z}^2$ (euclidean geometry).
So the emergence of (explosive and) fractal-like structures on the spheres is not surprising. The same computation with several such hyperbolic groups would generate many beautiful fractal-like pictures.

I don't know if your group $G$ is hyperbolic, but we can have the same understanding.

Example: $\Gamma = \langle a,b \rangle < Sym(\mathbb{Z}^{2})$ such that

• $a: (m,n) \to (m+1,m+n)$
• $b: (m,n) \to (m+n,n+1)$

To be confirmed: $\Gamma$ acts transitively on $\mathbb{Z}^{2}$, and is isomorphic to the free group $\mathbb{F}_2$ (so $\Gamma$ is hyperbolic).

The cardinalities of the spheres of radii $r=0,\dots,13$ about $(0,0)$ are :
$1,4,11,20,47,100,238,512,1124,2540,5569,12101,26208,56720$

The entire sphere of radius $13$:

This phenomenon can be understood as follows:

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive hyperbolic group (and non virtually cyclic).

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ (and orbit of (0,0)), the Cayley graph of $\Gamma$ (Gromov hyperbolic geometry) is projected (in a manner not too degenerate) on $\mathbb{Z}^2$ (euclidean geometry).
Then the emergence of explosive and fractal-like structures on the spheres is not surprising.

So any (non virtually cyclic) hyperbolic group would generate beautiful fractal-like pictures.

Example: $\Gamma = \langle a,b \rangle < Sym(\mathbb{Z}^{2})$ such that

• $a: (m,n) \to (m+1,m+n)$
• $b: (m,n) \to (m+n,n+1)$

To be confirmed: $\Gamma$ acts transitively on $\mathbb{Z}^{2}$, and is isomorphic to the free group $\mathbb{F}_2$ (so $\Gamma$ is hyperbolic).

The cardinalities of the spheres of radii $r=0,\dots,13$ about $(0,0)$ are :
$1,4,11,20,47,100,238,512,1124,2540,5569,12101,26208,56720$

The entire sphere of radius $13$:

Remark: I don't know if your group $G$ is hyperbolic, but we could have the same understanding for all the finitely generated groups of exponential growth.