This phenomenon can 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 $\mathbb{Z}^{2}$ and the orbit of $(0,0)$, the  **[Cayley graph][2]** (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][3]**-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).     

 


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