The volume of a spherical antiprism

TL;DR

This paper investigates the geometric properties of spherical antiprisms in three-dimensional spherical space, establishing conditions for their existence, relations between angles and edges, and deriving a volume formula.

Contribution

It provides necessary and sufficient existence conditions, relations between dihedral angles and edge lengths, and an explicit volume formula for spherical antiprisms.

Findings

Existence conditions for spherical antiprisms in S^3

Cosine relations between dihedral angles and edge lengths

Explicit integral formula for the volume

Abstract

We consider a spherical antiprism. It is a convex polyhedron with $2n$ vertices in the spherical space $\mathbb{S}^3$. This polyhedron has a group of symmetries $S_{2n}$ 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 polyhedron in $\mathbb{S}^3$. Then we find relations between its dihedral angles and edge lengths in the form of cosine rules through a property of a spherical isosceles trapezoid. Finally, we obtain an explicit integral formula for the volume of a spherical antiprism in terms of the edge lengths.