The Optimal Hyperball Packings Related to the Smallest Compact Arithmetic 5-Orbifolds

Jeno Szirmai

The smallest three hyperbolic compact arithmetic 5-orbifolds can be derived from two compact Coxeter polytopes that are combinatorially simplicial prisms (i.e. complete orthoschemes of degree $d=1$) in five dimensional hyperbolic space $\mathbf{H}^5$ (see \cite{BE} and \cite{EK}). The corresponding hyperbolic tilings are generated by reflections through their delimiting hyperplanes. These involve our studies for the relating densest hyperball (hypersphere) packings with congruent hyperballs. The analogous problem was discussed in \cite{Sz06-1} and \cite{Sz06-2} in hyperbolic spaces $\mathbf{H}^n$ ($n=3,4$). In this paper we extend this procedure to determine the optimal hyperball packings of the above 5-dimensional prism tilings. We compute their metric data and the densities of their optimal hyperball packings, moreover, we formulate a conjecture for the candidates for the densest hyperball packings in 5-dimensional hyperbolic space $\mathbf{H}^5$.