arXiv subjects

Robert T. Kozma

Publications and source records attributed to Robert T. Kozma.

3 recordsLinked to original sources

arXiv2025

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

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.

math.MG↗

arXiv2021

New Lower Bounds for Optimal Horoball Packing Density in Hyperbolic $n$-space for $6 \leq n \leq 9$

Koszul type Coxeter simplex tilings exist in hyperbolic $n$-space $\mathbb{H}^n$ up to $ n = 9$, and their horoball packings achieve the highest known regular ball packing densities for $n = 3, 4, 5$. In this paper we determine the optimal horoball packing densities of Koszul simplex tilings in dimensions $6 \leq n \leq 9$, which give new lower bounds for optimal packing density in each dimension. The symmetries of the packings are given by Coxeter simplex groups, and a parameter related to the Busemann function gives an isometry invariant description of different optimal horoball packing configurations.

math.MG↗

arXiv2016

Structure and Visualization of Optimal Horoball Packings in $3$-dimensional Hyperbolic Space

Four packings of hyperbolic 3-space are known to yield the optimal packing density of $0.85328\dots$. They are realized in the regular tetrahedral and cubic Coxeter honeycombs with Schläfli symbols $\{3,3,6 \}$ and $\{4,3,6\}$. These honeycombs are totally asymptotic, and the packings consist of horoballs (of different types) centered at the ideal vertices. We describe a method to visualize regular horoball packings of extended hyperbolic 3-space $\bar{\mathbb{H}}^3$ using the Beltrami-Klein model and the Coxeter group of the packing. We produce the first known images of these four optimal horoball packings.

math.MG↗