A Characterization of Ideal Polyhedra in Hyperbolic 3-Space

Abstract

The goals of this paper are to provide a characterization of dihedral angles of convex ideal (those with all vertices on the sphere at infinity) polyhedra in H3, and also of those convex polyhedra with some vertices on the sphere at infinity and some in the finite part of H3. These characterizations are given in, respectively, Theorems 0.1 and 10.5. The first theorem is proved in detail, while the proof of the second (which is similar) is only outlined. The results of this paper grow out of the general framework of the author's doctoral dissertation [13], as published in [20]. A lot of the language, and some of the auxilary results come from there as well, so familiarity with the latter reference is very helpful. The necessity of the conditions postulated in Theorem 0.1 has been shown in [14]; hence only the sufficiency direction will be shown in the current paper. In 1832, Jakob Steiner asked for a combinatorial characterization of convex polyhedra inscribed in the sphere. This was considered intractable it took almost a hundred years to find a single example of an uninscribable combinatorial type (by Steinitz [24]). However, Theorem 0.1 gives such a characterization. Section 11 is dedicated to a (very brief) historical survey and some purely graph-theoretic and computational-geometric consequences of Theorem 0.1. In order to state this theorem, suppose that a convex ideal polyhedron P in H3 is given. Let P* denote the Poincar' dual of P, and assign to each edge e* of P* a weight w(e*) equal to the exterior dihedral angle at the corresponding edge e of P. Then the following result holds: