On the volumes of hyperbolic 5-orthoschemes and the trilogarithm (Q1209302)

The author considers simplices in a 5-dimensional hyperbolic space. The aim of this note is to calculate the volumina of simplices belonging to a special class of asymptotic 5-orthoschemes. A general \(d\)-orthoscheme is a \(d\)-dimensional simplex where the most dihedral angles (well-defined located) are orthogonal. Only \(d\) dihedral angles \((\alpha_ 1,\alpha_ 2,\dots,\alpha_ d)\) are not necessarily orthogonal ones. We call these dihedral angles essential ones. An orthoscheme is called 1- or 2- asymptotic if 1 or all the 2 principal vertices lie in the infinity. \textit{P. Müller} [``Über Simplexinhalte in nichteuklidischen Räumen'', Diss. Univ. Bonn 1954] had already principally shown that the volume of a 5-dimensional hyperbolic asymptotic orthoscheme is a function of the trilogarithm. The reviewer had given a formula to calculate the volume of an arbitrary non-Euclidean orthoscheme [Arch. Math. 11, 298-309 (1960; Zbl 0095.152)]. This formula contains of course polylogarithm with degree at most three. In this note the volumina for all 5-dimensional hyperbolic orthoschemes are given where the essential dihedral angles \(\alpha_ i(i=1,\dots,5)\) suffices the following conditions: \(\alpha:=\alpha_ 1=\alpha_ 4\); \(\beta:=\alpha_ 2=\alpha_ 5\); \(\gamma:=\alpha_ 3\); \(\cos^ 2\alpha+\cos^ 2\beta+\cos^ 2\gamma=1\). These orthoschemes are 2- asymptotic. Their volumina are expressed by the Lobachevskian function of order three. This function is related to the function of the trilogarithm. Especially here we find for the first time explicitly (i) the volumina of the three hyperbolic Coxeter 5-orthoschemes (they are rational multiples of Riemann's function \(\zeta(3))\) and (ii) the volumina of those five characteristic simplices for regular hyperbolic star-honeycombs (being necessarily of infinite density) which are 2- asymptotic orthoschemes. Their volumina are linear combinations of \(\zeta(3)\) and of the Lobachevskian function of order three \(\Lambda_ 3(\pi/5)\). For the proof the author uses Schläfli's differential form for the volume of a non-Euclidean orthoscheme, the results of N. I. Lobatchevsky in dimensions three, and results in scissors congruence of \textit{H. E. Debrunner} [Geom. Dedicata 33, No. 2, 123-152 (1990; Zbl 0699.51012)].