The volume of a compact hyperbolic antiprism

TL;DR

This paper investigates the properties and volume formulas of compact hyperbolic antiprisms, a class of convex polyhedra in hyperbolic space with specific symmetries, providing existence conditions, angle-length relations, and explicit volume formulas.

Contribution

It establishes existence criteria, derives relations between angles and edges, and provides exact volume formulas for hyperbolic antiprisms, advancing understanding of their geometric structure.

Findings

Necessary and sufficient conditions for existence in hyperbolic space.

Relations between dihedral angles and edge lengths via cosine rule.

Explicit integral formulas for the volume in terms of edge lengths.

Abstract

We consider a compact hyperbolic antiprism. It is a convex polyhedron with $2n$ vertices in the hyperbolic space $\mathbb{H}^3$. This polyhedron has a symmetry group $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e. rotation to the angle $\pi/n$ followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in $\mathbb{H}^3$. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formulas expressing the volume of a hyperbolic antiprism in terms of the edge lengths.