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 Böröczky-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 $\approx 0.84931$.

33 pages, 12 figures. arXiv admin note: text overlap with arXiv:1510.03208, arXiv:1505.03338