Volumes of ideal hyperbolic drums

TL;DR

This paper derives a volume formula for ideal hyperbolic antiprisms, expanding the understanding of hyperbolic 3-manifolds and their volume calculations, with potential applications in geometric topology.

Contribution

It provides the first explicit volume formula for ideal hyperbolic antiprisms, generalizing previous results on hyperbolic prisms and contributing to the study of hyperbolic 3-manifolds.

Findings

Derived a volume formula for ideal hyperbolic antiprisms

Extended known volume calculations from prisms to antiprisms

Facilitated construction of hyperbolic 3-manifolds with rational volume sums

Abstract

Milnor computed the volumes of ideal hyperbolic prisms as part of an effort to construct 3-manifolds whose volumes are finite rational sums of the Lobachevsky function evaluated at rational multiples of pi. Motivated by these results and with an eye to related applications, we prove a volume formula for arbitrary ideal hyperbolic antiprisms, also called drums.