Because of Euler we know that for polyhedra without holes we have
V+F=E+2. As a result we can't have a polyhedron made entirely of
hexagons, and need at least 12 pentagons or an equivalent set of
shapes. Etc.
But things change if we allow intersecting faces. Clearly we have
a lot of scope for variations, but we can always retain limits to
make things reasonable.
But ...
Can we make a "polyhedron" consisting only of hexagonal "faces"?
@ColinTheMathmo I don't know the answer, but I remember that Branko Gruenbaum was very interested in polyhedra where regularity conditions are maintained at the potential expense of allowing faces to self intersect.
Maybe you know all this...
@bremner I didn't know that, thanks. Someone has also pointed me at this:
@ColinTheMathmo Oh right. I was guessing you had convex faces in mind. In fact even self intersecting faces can make some sense. Consider e.g. a 5-cycle 13524.
