New Horoball Packing Density Lower Bound in Hyperbolic 5-space

TL;DR

This paper determines the densest known horoball packings in hyperbolic 5-space, identifying optimal arrangements related to Coxeter groups and their arithmetic properties, with a maximum density of approximately 0.59421.

Contribution

It introduces new optimal horoball packings in hyperbolic 5-space associated with Coxeter simplex tilings, including multiple arrangements and density calculations.

Findings

Optimal packing density of approximately 0.59421 achieved.

Eleven optimal arrangements found at Coxeter simplex vertices.

Densities are rational submultiples of the maximum density.

Abstract

We describe the optimal horoball packings of asymptotic Koszul type Coxeter simplex tilings of $5$-dimensional hyperbolic space where the symmetries of the packings are generated by Coxeter groups. We find that the optimal horoball packing density of $\delta_{opt}=0.59421\dots$ is realized in an entire commensurability class of arithmetic Coxeter tilings. Eleven optimal arrangements are achieved by placing horoballs at the asymptotic vertices of the corresponding Coxeter simplices that give the fundamental domains. When multiple horoball types are allowed, in the case of the arithmetic Coxeter groups, the relative packing densities of the optimal horoball types are rational submultiples of $\delta_{opt}$, corresponding to the Dirichlet-Voronoi cell densities of the packing. The packings given in this paper are so far the densest known in hyperbolic $5$-space.