On the volume of spherical Lambert cube

Abstract

The calculation of volumes of polyhedra in the three-dimensional Euclidean, spherical and hyperbolic spaces is very old and difficult problem. In particular, an elementary formula for volume of non-euclidean simplex is still unknown. One of the simplest polyhedra is the Lambert cube Q(α,β,γ). By definition, Q(α,β,γ) is a combinatorial cube, with dihedral angles α,βand γassigned to the three mutually non-coplanar edges and right angles to the remaining. The hyperbolic volume of Lambert cube was found by Ruth Kellerhals (1989) in terms of the Lobachevsky function Λ(x). In the present paper the spherical volume of Q(α,β,γ) is defined in the terms of the function δ(α,θ) which can be considered as a spherical analog of the Lobachevsky function Δ(α,θ)=Λ(α+ θ) - Λ(α- θ)