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).