# 3-dimensional Cayley graph

I would like to see Cayley graphs drawn in 3-dimensional Euclidean space such that the vertices are represented by points and various shadows display the actions of the generators.

For example, suppose $G$ is a finite group generated by 2 elements $g_1,g_2\in G$. The problem is to find a function $f:G\rightarrow\mathbb{R}^3$ and 2-dimensional subspaces $V_1,V_2\subset\mathbb{R}^3$ with orthogonal projections $\pi_i:\mathbb{R}^3\rightarrow V_i$ such that each shadow $\pi_i(f(G))$ has $|G|$ points and the left regular action by $g_i$ on $G$ coincides with a Euclidean symmetry of $\pi_i(f(G))$.

Obviously this problem can be generalized in a few ways. Thus, instead of mapping $G\rightarrow\mathbb{R}^3$, one might instead consider some mapping $S\rightarrow\mathbb{R}^3$, where $S$ is a transitive $G$-set. Similarly, one may consider groups with more than 2 generators. I know very little about which Cayley graphs can be represented in this way.

The number of generators being 2 is interesting because some of my favorite groups have 2 generators. (I would like to see some finite simple groups drawn in this way.) For some reason, this problem reminds me of the proof of the Banach-Tarski paradox which uses the fact that the free group of rank 2 $F(2)$ is a subgroup of SO(3). Perhaps a map $f$ having the property described above can be regarded as some sort of approximation of the imbedding of $F(2)$ in $SO(3)$

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