Coverings with congruent and non-congruent hyperballs generated by doubly truncated Coxeter orthoschemes

TL;DR

This paper investigates hyperball coverings in hyperbolic 3-space generated by doubly truncated Coxeter orthoschemes, identifying configurations with minimal density and comparing congruent and non-congruent cases, revealing densities below known bounds.

Contribution

It extends previous work by analyzing non-congruent hyperball coverings and identifies new locally optimal configurations with densities below established bounds.

Findings

The thinnest hyperball covering density is approximately 1.26829 for the {7,3,7} tiling.

Locally optimal coverings in {u,3,7} orthoschemes have a minimum density around 1.26454.

Hyperball coverings in the hyperbolic plane can approximate the universal lower bound density with non-congruent hypercycles.

Abstract

After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings \cite{SzJ1}, we consider the corresponding covering problems. In \cite{MSSz} the authors gave a partial classification of supergroups of some hyperbolic space groups whose fundamental domains will be integer parts of truncated tetrahedra, and determined the optimal congruent hyperball packing and covering configurations belonging to some of these classes. In this paper we compliment these results with the investigation of the non-congruent covering cases, and the remainig congruent cases. We prove, that between congruent and non-congruent hyperball coverings the thinnest belongs to the $\{7,3,7\}$ Coxeter tiling with density $\approx 1.26829$. This covering density is smaller than the conjectured lower bound density of L.~Fejes~T\'oth for coverings with balls and horoballs. We also study the local packing arrangements related to $\{u,3,7\}$ $(6< u < 7, ~ u\in \mathbb{R})$ doubly truncated orthoschemes and the corresponding hyperball coverings. We prove, that these coverings are achieved their minimum density at parameter $u\approx 6.45953$ with covering density $\approx 1.26454$ which is smaller then the above record-small density, but this hyperball arrangement related to this locally optimal covering can not be extended to the entire $\mathbb{H}^3$. Moreover we see, that in the hyperbolic plane $\mathbb{H}^2$ the universal lower bound of the congruent circle, horocycle, hypercycle covering density $\frac{\sqrt{12}}{\pi}$ can be approximated arbitrarily well also with non-congruent hypercycle coverings generated by doubly truncated Coxeter orthoschemes.