Robert Dawson - A generalized face number for regular hyperbolic honeycombs

If, seeing an icosahedron for the first time, one wanted to find out how many faces it icosahedron had, one might might use Euler's formula in combination with the vertex figure. One might look at the curvature accounted for by each face of a spherical icosahedron. Or one might simply count the faces in a systematic manner. One would, of course, expect the same answer in every case.

The first two methods generalize fairly easily to a regular hyperbolic honeycomb. The third, also, generalizes, via divergent series. We will first examine the consistency of telescoping as a method of summing the series, and then use that method, too, on honeycombs. It will be seen that while the resulting ``face numbers'' may not be positive, or even integers, the three methods--based on combinatorics, differential geometry, and divergent series--still agree.