There are a ton of things to be done with rolling polyhedra (actually, we'll just limit ourselves to Platonic solids here). The stuff that comes to my mind first are:
1) How many different ways can a solid be rolled to visit each face in turn? I actually have an answer for this one, at the bottom of this page.
2) For Platonic solids that have face shapes that tile the plane (all but the dodecahedron), what is the area traced out by a rolling face tour? How many of the rolling face tours never duplicate a position on the plane?
3) Games: the list is endless
4) I could include more, but I want it to look like I actually do have some answers to some of these questions!
So - let's roll 'em. Here's the number of different complete face-tours that are possible, that never hit a face twice:
| Platonic Solid | # of tours |
|---|---|
| Tetrahedron | 6 |
| Cube | 40 |
| Octahedron | 18 |
| Dodecahedron | 6320 |
| Icosahedron | 162 |
Initially, I was surprised at the magnitude of the difference between the dodecahedron and the icosahedron. I guess the # of edges per face is much more important that the # of faces - you can see this in the cube vs. octahedron results as well.
It would be nice to derive a function for the above data:
f(e,f) = N
where:
e: # of edges per face
f: # of faces per solid
N: # of possible tours
I've only tried to work on this a bit, with no success. If you figure something out, email me.