If the graph $G=(V,E)$ has a lot of symmetries, then using **spectral embeddings** might give you nice drawings that reflects these symmetries. The success of this method (e.g. whether the drawing is planar, whether all vertices are on a sphere) depends on a lot of factors, some of which are not completely clear to me. However, what I can tell you is that it works for the graphs of all uniform 3-polytopes (so, e.g., the dodecahedron). 

I explain the most straight forward way to do it, some tweaks might be neccessary for the general case:

> **Costruction.** Let $\theta$ be an eigenvalue of (the adjacency matrix of) $G$, and $v_1,v_2,v_3\in\Bbb R^n$ three ortho-normal eigenvectors to $\theta$. Construct the matrix $M:=(v_1,v_2,v_3)\in\Bbb R^{n\times 3}$ with the $v_i$ as columns. The rows of that matrix are a 3-dimensional embedding of the vertices of $G$.

Usually, you should take $\theta_2$, i.e., the second-largest eigenvalue of the adjacency matrix of $G$. Surprisingly, this eigenvalue has multiplicity three for most very symmetric graphs that come from 3-polytopes. This means, you cannot do anything wrong by choosing just any orthonormal basis of eigenvectors.