Hyperball packings in hyperbolic $3$-space

Abstract

In earlier works we have investigatedthe densest packings and the thinnest coverings by congruent hyperballs based on the regular prismtilings in $n$-dimensional hyperbolic space $\HYN$ ($ 3 \le n \in \mathbb{N})$.In this paper we study a large class of hyperball packings in $\HYP$that can be derived from truncated tetrahedron tilings.In order to get an upper bound for the density of the above hyperball packings, it is sufficientto determinethis density upper bound locally, e.g. in truncated tetrahedra.Thus we prove that if the truncated tetrahedron is regular, then the densityof the densest packing is $\approx 0.86338$.This is larger than the Böröczky-Florian density upper bound for balls and horoballs.Our locally optimal hyperball packing configuration cannot be extended to the entirehyperbolic space $\mathbb{H}^3$, but we describe a hyperball packing construction,by the regular truncated tetrahedron tiling under the extended Coxeter group $[3, 3, 7]$with maximal density $\approx 0.82251$.Moreover, we show that the densest known hyperball packing, related to the regular $p$-gonal prism tilings, can be realized bya regular truncated tetrahedron tiling as well.