The size of hyperbolic Coxeter simplex
Coxeter simplices of finite volume with dihedral angles submultiples of \(\pi\) or zero exist in hyperbolic \(n\)-space only up to \(n=9\) and for \(n>2\), there are finitely many. Extending and combining their earlier work, jointly and separately, the authors determine the volumes of all these hyperbolic simplices. Dissection methods using orthoschemes and Schäfli's reduction method are shown to give formulae for these volumes in the form of iterated integrals. These are used to give accurate numerical values for the volumes but also, in most cases, to give closed formulae involving the evaluation of some reasonably tractable function at a suitable value. The associated reflection groups in many cases are arithmetic so that these functions can be arithmetically defined and commensurability relationships between the groups deduced. Furthermore, since the Gauss-Bonnet formula holds in even dimensions, the main difficult cases are in odd dimensions. Complete tables are given.
scientific article; zbMATH DE number 1394481
scientific article