Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

It is easy to tell when a cellulation of the 2-sphere arises as the face structure of a convex polyhedron. It is necessary and sufficient that each cell is embedded and that the intersection of two cells is a cell; such a cellulation is called strongly regular. This fact is part of a collection of results that say a lot about the combinatorial structure of convex polyhedra in R^3.

In comparison convex polytopes in R^4 are poorly understood. It is known that not every strongly regular cellulation can be made convex. But it is plausible to conjecture that strongly regular cellulations are at least numerically equivalent to convex polytopes, in the sense of having the same number of faces in every dimension. One measure of the numerical profile of a polytope or cellulation is fatness, defined as the number of edges plus ridges divided by the number of vertices plus facets. I will describe a family of cellulations of the 3-sphere with unbounded fatness. It is reasonable to conjecture that fatness is bounded for convex 4-polytopes. If so, the numerics of a strongly regular cellulation of the 3-sphere can be very different from that of any convex polytope.

(Joint work with David Eppstein and Gunter Ziegler)

arXiv reference: math.CO/0204007