Non-Euclidean Octahedra with mm2-Symmetry

摘要

The problem of calculating the volumes of polyhedra is a very old and complicated one; although it goes back to the time of antique mathematics, it is still topical today. The formula for the volume of a triangular pyramid was obtained by Archimedes and, in the 16th century, Tartaglia expressed the volume of a Euclidean tetrahedron in terms of the squares of the lengths of its edges. In the spherical and hyperbolic cases, the situation is more complicated. Thus, the formula for the volume of an arbitrary non-Euclidean tetrahedron remained unknown for a long time. Only in 1999 and in the following decade, this problem was solved definitively in the papers of Cho and Kim, Murakami and Yano, Murakami and Ushijima, Derevnin and Mednykh, and Murakami. It should be mentioned that the formula for the volume of an arbitrary non-Euclidean tetrahedron was first written by the Italian duke Sforza in 1906. Unfortunately, his formula contains a multivalued function, and it was not indicated in which branch yields the volume. Because of this, the Sforza formula was completely forgotten for a long time and interest to it reappeared only after a discussion between Mednykh and Montesinos during a conference in Spain in August 2006.