On the volume of spherical Lambert cube

TL;DR

This paper investigates the volume of Lambert cubes in spherical space, providing a new formula involving a spherical analog of the Lobachevsky function, complementing existing hyperbolic volume results.

Contribution

It introduces a formula for the spherical volume of Lambert cubes using a new spherical Lobachevsky-like function, extending volume calculations to spherical geometry.

Findings

Derived a spherical volume formula for Lambert cubes.

Connected the volume formula to a spherical Lobachevsky function.

Extended understanding of polyhedral volumes in non-Euclidean spaces.

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(\alpha,\beta,\gamma). By definition, Q(\alpha,\beta,\gamma) is a combinatorial cube, with dihedral angles \alpha,\beta and \gamma assigned 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 \Lambda(x). In the present paper the spherical volume of Q(\alpha,\beta,\gamma) is defined in the terms of the function \delta(\alpha,\theta) which can be considered as a spherical analog of the Lobachevsky function \Delta(\alpha,\theta)=\Lambda(\alpha + \theta) - \Lambda(\alpha - \theta)