Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings

TL;DR

This paper investigates the densest arrangements of hyperballs in hyperbolic spaces derived from regular simplex tilings, establishing density bounds for both congruent and non-congruent packings.

Contribution

It introduces new density bounds for hyperball packings based on truncated regular simplex tilings in hyperbolic spaces, including both congruent and non-congruent cases.

Findings

Determined the densest hyperball packings for specific tilings.

Established the smallest density upper bounds for non-congruent hyperball packings.

Provided explicit density calculations for hyperball packings in hyperbolic spaces.

Abstract

In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular tetrahedron tilings. These are derived from the Coxeter simplex tilings $\{p,3,3\}$ $(7\le p \in \mathbb{N})$ and $\{5,3,3,3,3\}$ in $3$ and $5$-dimensional hyperbolic space. We determine the densest hyperball packing arrangements related to the above tilings. We find packing densities using congruent hyperballs and determine the smallest density upper bound of non-congruent hyperball packings generated by the above tilings.