Maximal Volume Ideal Polyhedra and the Arithmetic Angle Phenomenon

Igor Rivin

Paper

Maximal Volume Ideal Polyhedra and the Arithmetic Angle Phenomenon

Abstract

We present a software suite for the analysis and optimization of ideal convex polyhedra in hyperbolic 3-space

. Using Rivin's variational characterization of ideal polyhedra, we develop efficient algorithms for checking combinatorial realizability and finding volume-maximizing configurations. Our systematic computational study reveals two striking phenomena: (1) maximal volume ideal polyhedra consistently exhibit dihedral angles that are rational multiples of

-- a property with no obvious explanation from the optimization formulation; and (2) the distribution of volumes for random configurations is well-approximated by a Beta distribution, with mean normalized volume converging to approximately

as the vertex count increases. We provide complete data for small vertex counts, including vertex positions, triangulations, and verified rational angle structures. An interactive implementation is publicly available.