Maximal Volume Ideal Polyhedra and the Arithmetic Angle Phenomenon

TL;DR

This paper introduces a software suite for analyzing ideal convex polyhedra in hyperbolic space, revealing that maximal volume configurations have dihedral angles as rational multiples of pi and that volume distributions follow a Beta distribution.

Contribution

It develops efficient algorithms for ideal polyhedra analysis and uncovers novel geometric phenomena related to volume maximization and angle rationality.

Findings

Maximal volume ideal polyhedra have dihedral angles that are rational multiples of π.

Volume distribution of random configurations approximates a Beta distribution.

Normalized mean volume approaches approximately 0.69 as vertex count increases.

Abstract

We present a software suite for the analysis and optimization of ideal convex polyhedra in hyperbolic 3-space $\mathbb{H}^3$. 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 $\pi$ -- 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 $\ln 2 \approx 0.69$ 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.