Optimally Dense Packings for Fully Asymptotic Coxeter Tilings by Horoballs of Different Types

TL;DR

This paper determines the optimal arrangements and densities of horoballs of different types in all four fully asymptotic Coxeter tilings in hyperbolic 3-space, extending known density bounds to more general packings.

Contribution

It generalizes the projective methodology to hyperbolic spaces and proves that the B"or"oczky--Florian density upper bound applies to diverse horoball packings in Coxeter tilings.

Findings

The B"or"oczky--Florian density bound holds for all fully asymptotic Coxeter tilings.

Allowing different horoball types does not increase packing density beyond the known bound.

The results apply under prescribed symmetry groups, broadening the scope of optimal packing configurations.

Abstract

The goal of this paper to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space $\mathbb{H}^3$. Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known B\"or\"oczky--Florian density upper bound for "congruent horoball" packings of $\mathbb{H}^3$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.