The asymptotic number of convex polyhedra

Abstract

We obtain an asymptotic formula for the number of combinatorially distinct convex polyhedra with n edges. 1. Introduction. The number of combinatorial polyhedra has been studied by many authors, perhaps first of all by Euler. A very interesting account up to 1975 has been given by P. J. Frederico (2). By a well-known theorem of Steinitz a convex polyhedron is combinatorially equivalent to a 3-connected planar map (with more than three edges). B. Griinbaum's proof (3) is the most elegant known to these authors. Indeed Mani (5) has shown that for each 3-connected planar map M there is a convex polyhedron whose 1-skeleton is isomorphic to M and such that every automorphism of M is induced by an isometry of the convex polyhedron. The enumeration of planar maps has progressed considerably following the breakthrough of W. T. Tutte in the early sixties. Tutte (6) has given a survey of the methods and results known up to 1973. Until quite recently most of the progress concerned the enumeration of rooted planar maps. A planar map is said to be rooted when one directed edge in it is distinguished as the root and when two sides of the root are distinguished as left and right. In fact Tutte (7) has shown that the number R(n) of rooted 3-connected planar maps with n edges is asymptotic to