Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space

Robert T. Kozma, Jenő Szirmai

TL;DR

The paper determines the exact optimal horoball packing densities for all Koszul-type, noncompact Coxeter simplex tilings in hyperbolic 3-space with ideal vertices, showing that the universal simplicial density bound $d_3(\infty) = \left(2\sqrt{3}\,\Lambda\left(\frac{\pi}{3}\right)\right)^{-1}$ is realized by multiple tilings, notably in the $[3,3,6]$ commensurability class. The authors develop a framework based on the Busemann function and a projective Cayley–Klein model to parametrize horoballs, compute local densities via horospherical triangulations, and extend these to global tilings, yielding explicit densities in terms of $\Lambda$, Catalan's constant $C$, and Dirichlet $L$-function values such as $L(2,\chi_{-3})$. The key contributions include a complete catalog of optimal densities for 23 Koszul simplex tilings, new connections between hyperbolic geometry, special function values, and cusp-manifold volumes, and a methodological advance using horoball types to distinguish density realizations beyond Böröczky-type bounds. The results have implications for cusp-volume estimates and the geometric understanding of densest packings in hyperbolic 3-space, demonstrating that extremal configurations are realized by several explicit Coxeter tilings and are tied to deep number-theoretic quantities.

Abstract

We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic $3$-space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of packings that attain the universal simplicial density upper bound \[ d_3(\infty) \;=\; \left( 2 \sqrt{3}\,Λ\!\left(\tfracπ{3}\right) \right)^{-1} \;\approx\; 0.853276, \] where $Λ$ denotes the Lobachevsky function. These results show that extremal packing densities in $\mathbb{H}^3$ are realized by multiple explicit Coxeter tilings and are closely tied to special values of $L$-functions and hyperbolic manifold volumes.

Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space

TL;DR

The paper determines the exact optimal horoball packing densities for all Koszul-type, noncompact Coxeter simplex tilings in hyperbolic 3-space with ideal vertices, showing that the universal simplicial density bound is realized by multiple tilings, notably in the commensurability class. The authors develop a framework based on the Busemann function and a projective Cayley–Klein model to parametrize horoballs, compute local densities via horospherical triangulations, and extend these to global tilings, yielding explicit densities in terms of , Catalan's constant , and Dirichlet -function values such as . The key contributions include a complete catalog of optimal densities for 23 Koszul simplex tilings, new connections between hyperbolic geometry, special function values, and cusp-manifold volumes, and a methodological advance using horoball types to distinguish density realizations beyond Böröczky-type bounds. The results have implications for cusp-volume estimates and the geometric understanding of densest packings in hyperbolic 3-space, demonstrating that extremal configurations are realized by several explicit Coxeter tilings and are tied to deep number-theoretic quantities.

Abstract

We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic -space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of packings that attain the universal simplicial density upper bound where denotes the Lobachevsky function. These results show that extremal packing densities in are realized by multiple explicit Coxeter tilings and are closely tied to special values of -functions and hyperbolic manifold volumes.
Paper Structure (15 sections, 10 theorems, 64 equations, 3 figures, 8 tables)

This paper contains 15 sections, 10 theorems, 64 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Statement of the Result
  3. Relation to Earlier Work
  4. Preliminaries
  5. The Projective Model of $\overline{\mathbb{H}}^3$
  6. Horospheres and Horoballs in $\overline{\mathbb{H}}^3$
  7. The Busemann function in $\overline{\mathbb{H}}^3$
  8. Packing Density in the Projective Model
  9. The Optimal Packing Densities of the Koszul Simplex Tilings
  10. Commensurability Class of $[3,3,6]$
  11. Commensurability Class of $[3,4,4]$
  12. Nonarithmetic Cases
  13. The commensurability class of $[5,3,6]$
  14. Commensurability classes of single simplices
  15. Tables

Key Result

Theorem 1

The optimal horoball packing density of Coxeter simplex tilings $\mathcal{T}_{\Gamma}$, $\Gamma \in \{ \overline{V}_3, \overline{Y}_3, \overline{VP}_3, \widehat{PP}_3, \overline{P}_3, \overline{Z}_3, \overline{DV}_3, \overline{DP}_3 \}$ is $\delta_{opt}(\Gamma) = \left( 2 \sqrt{3}\, \Lambda\!\left(\

Figures (3)

  • Figure 1: Subgroup relations of the 23 paracompact (Koszul simplex) Coxeter groups in $\overline{\mathbb{H}}^3$, 17 arithmetic and 6 nonarithmetic. The superscript indicates the number of ideal vertices.
  • Figure 2: Classification of the optimal horoball packings in the commensurability class $[3,3,6]$, organized by the ratio of the contributions of the individual horoballs as fractions of $\Theta$.
  • Figure 3: Classification of the optimal horoball packings in the commensurability class $[3,4,4]$, organized by the ratio of the contributions of the individual horoballs as fractions of $\Theta$.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Theorem 5: K. Böröczky
  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2: Local horoball density
  • ...and 9 more