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Hyperball packings in hyperbolic $3$-space

In earlier works \cite{Sz06-1}, \cite{Sz06-2}, \cite{Sz13-3} and \cite{Sz13-4} we have investigated the densest packings and the least dense coverings by congruent hyperballs (hyperspheres) to 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 (hypersphere) packings in $3$-dimensional hyperbolic space that can be derived from truncated simplex tilings (e.g. \cite{S14}, \cite{MPSz}). It is clear, that in order to get a density upper bound for the above hyperball packings, it is sufficient to determine the density upper bound locally, e.g. in truncated simplices. Thus we study hyperball packings in truncated simplices, i.e. truncated tetrahedra and prove that if the truncated tetrahedron is regular, then the density of the densest packing is $\approx 0.86338$. This is larger than the Böröczky-Florian density upper bound for balls and horoballs (horospheres) \cite{B--F64} but our locally optimal hyperball packing configuration cannot be extended to the entirety of hyperbolic space $\mathbb{H}^3$. But our regular truncated tetrahedron construction, under the extended Coxeter group $[3, 3, 7]$ with top density $\approx 0.82251$, seems to be good enough (Table 1). Moreover, we show that the densest known hyperball packing, related to the regular $p$-gonal prism tilings \cite{Sz06-1}, can dually be realized by regular truncated tetrahedron tilings as well.