This phenomenon could be "understood" as follows:    

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive **[hyperbolic group][1]** ( and non virtually cyclic).  

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$, the **[Cayley graph][2]** 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][3]** structure on the spheres is not surprising. The same computation with several such hyperbolic groups could generate many beautiful fractal-like 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).     

 


  [1]: http://en.wikipedia.org/wiki/Hyperbolic_group
  [2]: http://en.wikipedia.org/wiki/Cayley_graph
  [3]: http://en.wikipedia.org/wiki/Fractal