Hyperball packings related to octahedron and cube tilings in hyperbolic space

TL;DR

This paper investigates the densest arrangements of hyperballs in hyperbolic 3-space related to octahedron and cube tilings, finding configurations that surpass classical density bounds and identifying the most optimal packings.

Contribution

It introduces new hyperball packing configurations in hyperbolic space, determining their densities and showing some exceed known bounds for ball packings.

Findings

Densest hyperball packing density is approximately 0.86145.

Locally optimal non-congruent hyperball packing density is about 0.84931.

Some hyperball packings surpass the B"or"oczky-Florian density upper bound.

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 $\approx 0.84931$.