The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds

TL;DR

This paper investigates the densest hyperball packings in 5-dimensional hyperbolic space related to specific compact arithmetic orbifolds, extending previous work in lower dimensions and proposing a conjecture for optimal packings.

Contribution

It extends the study of hyperball packings to 5-dimensional hyperbolic space, providing metric data, densities, and a conjecture for the densest packings associated with certain orbifolds.

Findings

Computed metric data for 5D hyperbolic prism tilings

Determined densities of optimal hyperball packings in 5D

Formulated a conjecture for the densest hyperball packings in H^5

Abstract

The smallest three hyperbolic compact arithmetic 5-orbifolds can be derived from two compact Coxeter polytops which are combinatorially simplicial prisms (or complete orthoschemes of degree $d=1$) in the five dimensional hyperbolic space $\mathbf{H}^5$ (see \cite{BE} and \cite{EK}). The corresponding hyperbolic tilings are generated by reflections through their delimiting hyperplanes those involve the study of the relating densest hyperball (hypersphere) packings with congruent hyperballs. The analogous problem was discussed in \cite{Sz06-1} and \cite{Sz06-2} in the hyperbolic spaces $\mathbf{H}^n$ $(n=3,4)$. In this paper we extend this procedure to determine the optimal hyperball packings to the above 5-dimensional prism tilings. We compute their metric data and the densities of their optimal hyperball packings, moreover, we formulate a conjecture for the candidate of the densest hyperball packings in the 5-dimensional hyperbolic space $\mathbf{H}^5$.