The Number of Three-Dimensional Convex Polyhedra
Edward A. Bender · American Mathematical Monthly · 1987
A convex polyhedron, or polytope, is the bounded intersection of closed half-spaces. The problems of determining the number of three dimensional convex 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 to 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. Convex (also called polytopes) are the analogues to convex polygons in higher dimensions. We can define a convex polyhedron as a bounded intersection of closed half-spaces. Alternatively, we could define a convex polyhedron to be the convex hull of a finite set of points. We will be concerned exclusively with polyhedra: those that lie in 3-space but do not lie in a plane. In the future, polyhedra will always mean three-dimensional convex Cubes, tetrahedra, and prisms are all examples of polyhedra. In an obvious way, have vertices, edges, and faces. (These can be described formally, but we need not do so here.) We will use the notation P0, P1, and P2 for the numbers of vertices, edges, and faces, respectively, of a polyhedron P. Euler's famous theorem (1752) states that