Horoball packings and their densities by generalized simplicial density function in the hyperbolic space

TL;DR

This paper investigates the densest horoball packings within fully asymptotic tetrahedra in hyperbolic 3-space, introducing a generalized density function and demonstrating that the classical upper bound does not apply in this context.

Contribution

It introduces a generalized simplicial density function in hyperbolic space and shows that the known density upper bound is not valid for fully asymptotic tetrahedra with horoballs of different types.

Findings

The densest packing density is approximately 0.875.

The classical B"or"oczky--Florian bound (~0.853) can be exceeded in this setting.

The locally densest packing may not extend to the entire hyperbolic space.

Abstract

The aim of this paper to determine the locally densest horoball packing arrangements and their densities with respect to fully asymptotic tetrahedra with at least one plane of symmetry in hyperbolic 3-space $\bar{\mathbf{H}}^3$ extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a fully asymptotic tetrahedron. We allow horoballs of different types at the various vertices. Moreover, we generalize the notion of the simplicial density function in the extended hyperbolic space $\bar{\mathbf{H}}^n, ~(n \ge 2)$, and prove that, in this sense, {\it the well known B\"or\"oczky--Florian density upper bound for "congruent horoball" packings of $\bar{\mathbf{H}}^3$ does not remain valid to the fully asymptotic tetrahedra.} The density of this locally densest packing is $\approx 0.874994$, may be surprisingly larger than the B\"or\"oczky--Florian density upper bound $\approx 0.853276$ but our local ball arrangement seems not to have extension to the whole hyperbolic space.