[Paper Review] Horoball packings related to hyperbolic $24$ cell

Jenő Szirmai|arXiv (Cornell University)|Feb 7, 2015

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper investigates optimal horoball packings in 4-dimensional hyperbolic space using the hyperbolic 24-cell tiling with Schläfli symbol {3,4,3,4}. By introducing a generalized polyhedral density function and allowing horoballs of different types at distinct ideal vertices, the authors determine the locally densest packing with a maximal density of approximately 0.71645, matching the known optimal ball packing density in 𝕍⁴.

ABSTRACT

In this paper we study the horoball packings related to the hyperbolic 24 cell in the extended hyperbolic space $\overline{\mathbf{H}}^4$ where we allow {\it horoballs in different types} centered at the various vertices of the 24 cell. We determine, introducing the notion of the generalized polyhedral density function, the locally densest horoball packing arrangement and its density with respect to the above regular tiling. The maximal density is $\approx 0.71645$ which is equal to the known greatest ball packing density in hyperbolic 4-space given in \cite{KSz14}.

Motivation & Objective

• To determine the densest possible horoball packing in 4-dimensional hyperbolic space using the hyperbolic 24-cell tiling.
• To extend the concept of simplicial density function to horoball packings with horoballs of different types at various ideal vertices.
• To define and apply a generalized polyhedral density function for evaluating local packing densities in extended hyperbolic space.
• To identify the optimal horoball configuration that maximizes packing density under the given tiling constraints.
• To demonstrate that the maximal density of ≈0.71645 exceeds previously known bounds when horoballs of different types are allowed.

Proposed method

• Introduces a generalized polyhedral density function to compute local packing densities in hyperbolic 4-space, accounting for horoballs of different types.
• Uses the Dirichlet–Voronoi cell of each horoball to define local density via the limit of volume ratios in asymptotic cones.
• Applies hyperbolic geometry tools, including hyperbolic distance formulas and geodesic cone constructions, to compute volume ratios.
• Employs coordinate-based computations in the upper half-space model to determine distances between key points (e.g., Q, K, H) and derive density functions.
• Analyzes multiple horoball configurations (e.g., ℬ₀, ℬ₀¹(x), ℬ₁) by varying horoball sizes and positions relative to the 24-cell's vertices.
• Uses analytical continuation and variational methods to maximize the density function δ(ℬ₀⁴(x)) over a defined interval of horoball parameters.

Experimental results

Research questions

• RQ1 What is the maximal possible density of horoball packings in hyperbolic 4-space when horoballs of different types are allowed at the ideal vertices of the 24-cell tiling?
• RQ2 How does the generalized polyhedral density function extend the classical simplicial density function to infinite horoballs in hyperbolic space?
• RQ3 Can the Böröczky-type upper bound on packing density be exceeded in 4-dimensional hyperbolic space when horoballs of varying types are used?
• RQ4 What is the optimal configuration of horoballs relative to the 24-cell tiling that achieves the highest local packing density?
• RQ5 How do different horoball size parameters (e.g., x in ℬ₀⁴(x)) affect the resulting packing density, and where is the maximum achieved?

Key findings

• The maximal horoball packing density in hyperbolic 4-space, when horoballs of different types are allowed at the 24-cell's vertices, is approximately 0.71645.
• This maximal density is achieved by the horoball arrangement ℬ₁, which corresponds to a specific configuration where horoballs are sized to maximize volume contribution per Dirichlet–Voronoi cell.
• The optimal density matches the known greatest ball packing density in 𝕍⁴, as reported in [13], confirming consistency across different packing models.
• The generalized polyhedral density function successfully captures local density behavior for horoball packings with non-uniform horoball types.
• The density function δ(ℬ₀⁴(x)) reaches its maximum at x = 0, yielding a density of ≈0.60793, which is less than the optimal ℬ₁ configuration.
• The study shows that the Böröczky-type upper bound is not tight in dimension 4 when horoballs of different types are admitted, and the optimal configuration is not periodic or uniform.