Let $P$ be a polyhedron which satisfies the following three conditions:
- $P$ is built out of regular hexagons and regular pentagons.
- Three faces meet at each vertex.
- $P$ is topologically a sphere.
An easy Euler characteristic argument tells you that $P$ has exactly twelve pentagonal faces.
An example of a polyhedron like this is a soccer ball. In this case, the pentagonal faces are arranged with some nice symmetry.
Another (trivial) example is the regular dodecahedron, which again has some very nice symmetry.
Here's my question: Is this symmetry forced? What, if anything, can be said in general about the symmetry of a polyhedron which satisfies the above three conditions?