The Number of Three-Dimensional Convex Polyhedra

A convex polyhedron, or polytope, is the bounded intersection of closed half-spaces. The problems of determining the number of three dimensional convex polyhedra as a function of the number of faces or edges or both have been around for over 150 years. Except for Steinitz's conversion of polyhedra to “planar maps”, little was done on the problem until the work on “rooted” planar maps in the 1960's. Recently the original (unrooted) questions have been answered asymptotically. We will retrace the steps that led to this result.