Hyperball packings related to octahedron and cube tilings in hyperbolic space

Abstract

In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular octahedron and cube tilings. These are derived from the Coxeter simplex tilings $\{p,3,4\}$ $(7\le p \in \mathbb{N})$ and $\{p,4,3\}$ $(5\le p \in \mathbb{N})$ in $3$-dimensional hyperbolic space $\mathbb{H}^3$. We determine the densest hyperball packing arrangement and its density with congruent and non-congruent hyperballs related to the above tilings in $\mathbb{H}^3$. We prove that the locally densest congruent or non-congruent hyperball configuration belongs to the regular truncated cube with density $\approx 0.86145$. This is larger than the B\"or\"oczky-Florian density upper bound for balls and horoballs. Our locally optimal non-congruent hyperball packing configuration cannot be extended to the entire hyperbolic space $\mathbb{H}^3$, but we determine the extendable densest non-congruent hyperball packing arrangement related to a regular cube tiling with density $\approx 0.84931$.