Hyperball packings in hyperbolic $3$-space

J. Szirmai

Abstract: In earlier works we have investigated the densest packings and the thinnest coverings by congruent hyperballs based on the regular prism tilings 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 sufficient to determine this density upper bound locally, e.g. in truncated tetrahedra. Thus we prove that if the truncated tetrahedron is regular, then the density of the densest packing is $\approx 0.86338$. This is larger than the B\"or\"oczky-Florian density upper bound for balls and horoballs. Our locally optimal hyperball packing configuration cannot be extended to the entire hyperbolic 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 by a regular truncated tetrahedron tiling as well.

Keywords: Hyperbolic geometry; hyperball packings; packing density; Coxeter tilings.

Classification (MSC2000): 52C17; 52C22, 52B15

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