Packings with horo- and hyperballs generated by simple frustum orthoschemes

TL;DR

This paper investigates hyp-hor packings in 2- and 3-dimensional hyperbolic spaces generated by Coxeter tilings with frustum orthoschemes, determining their densities and optimal configurations, some exceeding known bounds but not extendable globally.

Contribution

It introduces a new class of packings in hyperbolic spaces based on simple frustum orthoschemes and analyzes their density properties and optimal configurations.

Findings

In 2D, packings approach the universal upper bound of 3/π.

In 3D, optimal packings have density approximately 0.83267.

Certain locally optimal packings exceed the B"or"oczky-Florian density upper bound.

Abstract

In this paper we deal with the packings derived by horo- and hyperballs (briefly hyp-hor packings) in the $n$-dimensional hyperbolic spaces $\HYN$ ($n=2,3$) which form a new class of the classical packing problems. We construct in the $2-$ and $3-$dimensional hyperbolic spaces hyp-hor packings that are generated by complete Coxeter tilings of degree $1$ i.e. the fundamental domains of these tilings are simple frustum orthoschemes and we determine their densest packing configurations and their densities. We prove that in the hyperbolic plane ($n=2$) the density of the above hyp-hor packings arbitrarily approximate the universal upper bound of the hypercycle or horocycle packing density $\frac{3}{\pi}$ and in $\HYP$ the optimal configuration belongs to the $[7,3,6]$ Coxeter tiling with density $\approx 0.83267$. Moreover, we study the hyp-hor packings in truncated orthosche\-mes $[p,3,6]$ $(6< p < 7, ~ p\in \bR)$ whose density function is attained its maximum for a parameter which lies in the interval $[6.05,6.06]$ and the densities for parameters lying in this interval are larger that $\approx 0.85397$. That means that these locally optimal hyp-hor configurations provide larger densities that the B\"or\"oczky-Florian density upper bound $(\approx 0.85328)$ for ball and horoball packings but these hyp-hor packing configurations can not be extended to the entirety of hyperbolic space $\mathbb{H}^3$.