Volume of three dimensional spherical polyhedra

Abstract

To determine the volume of a given polyhedron in the three dimensional Euclidean space is a very old and difficult problem. Probably, the first result in this direction belongs to N. Tartaglia (1494) who found a closed formula for the volume of the Euclidean tetrahedron in terms of the length of its edges. Now this result is contained in the Cayley-Menger formula, for the volume of an Euclidean simplex in terms of the length of its edges. In our time, I. Kh. Sabitov [Sb] has shown that the volume of any Euclidean polyhedron is a root of an algebraic equation whose coefficients are integer polynomials depending only on the length of its edges and the combinatorial type of polyhedron. In hyperbolic and spherical cases, the situation is more complicated. The first results in this direction were obtained by Lobachevsky and J. Bolyai who gave formulas for hyperbolic tetrahedra with three consequent right angles (orthoschemes). Later, the problem was solved by Schläfli for the spherical tetrahedron. The volume of the hyperbolic tetrahedron in many particular cases was investigated by E. Vinberg [Vi]. The general formula for the volume of the tetrahedron remained to be unknown for a long time. Jist recently, Y. Choi, H. Kim [ChK], J. Murakami, U. Yano [MY] and A.Ushijirna [U] were successful in determining such formula. Their approach allowed to express the volume of tetrahedron in term of 16 Dilogarithm or Lobachevsky functions, D. Derevnin, A. Mednykh [DM1] suggested an elementary integral formula for the volume of hyperbolic tetrahedron. It was discovered by J. Milnor [Mill] and by D. Derevnin, A. Mednykh and M. Pashkevich [DMP] that in the case when all faces of tetrahedron are mutually congruent the volume formula can be obtained in a very explicit way.