This phenomenon could be "understood" as follows:
Let $\Gamma < Sym(\mathbb{Z}^{2})$ be an hyperbolic group (non virtually cyclic), acting transitively on $\mathbb{Z}^{2}$.
Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$, the Cayley graph of $\Gamma$ (with its Gromov hyperbolic geometry) is projected (in a not too much degenerated manner) on $\mathbb{Z}^2$ (with its euclidean geometry).
So the emergence of a fractal structure on the spheres is not surprising.
The same computation for several such hyperbolic groups could generate many beautiful pictures.
I don't know if your group $G$ is hyperbolic, but I think we can have the same understanding.
Here is (perhaps) an easier 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 $\mathbb{F}_2$ (so $\Gamma$ is hyperbolic).