Cayley graphs for rotational symmetries of Platonic solids

In the previous post I used a short python program to draw a Cayley graph for the rotational symmetries of the cube.  The result was a rhombicuboctahedron, a kind of cross between a cube and an octahedron. Now,  these two platonic solids are "duals" of each other, which is to say that if you start with one and draw a node for each face,  and an edge for each pair of faces that meet, you end up with the other!

Is there a pattern here? Can we choose generators for the rotational symmetry groups of the other platonic solids so that the resulting Cayley graphs look like crosses between the original solids and their duals?

First let's see if we can find a generic representation for the rotational symmetry group.  Let's assume our solid has $n$ sided faces and the vertices are all of degree $m$. If $a$ is a clockwise rotation about one face and $b$ is a clockwise rotation about an adjoining vertex then $ab$ flips an edge about its midpoint, which implies $(ab)^2=1$.

This suggests the representation shown above.  At this stage it is possible that more relations are required.  However, by construction of a Cayley graph,  I will show that this representation leads to a group with $mv$ elements where $v$ is the number of vertices in the original platonic solid.  Since this is the size of the rotational symmetry group it follows that no further relations are needed. 

To create the Cayley graph we begin with the original solid.  Then we split each edge in two to create a separate $n$-gon for each original face.  Finally we introduce m-gons to tie the faces together at each vertex. 

The result satisfies all the relations in the representation, and no other independent relations.  Furthermore, the number of nodes is $mv$, which exactly matches the size of the rotational group. And finally, it is obvious from the method of construction that the result contains the same number of $n$-gon faces as the original platonic solid, whether you interpret the group as the rotational symmetries of the original solid,  or of its dual. 

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