Top Dense Hyperbolic Ball Packings and Coverings for Complete Coxeter Orthoscheme Groups

Emil Molnár, Jenő Szirmai

Abstract: In $n$-dimensional hyperbolic space ${𝐇}^n$ $(n\ge 2)$, there are three types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know a universal upper bound of the ball packing densities, where each ball’s volume is related to the volume of the corresponding Dirichlet–Voronoi (D-V) cell. E.g., in ${𝐇}^3$ a densest (not unique) horoball packing is derived from the $\{3,3,6\}$ Coxeter tiling consisting of ideal regular simplices $T_{\text{reg}}^{\infty }$ with dihedral angles $\frac{\pi }{3}$. The density of this packing is ${\delta }_3^{\infty }\approx 0·85328$ and this provides a very rough upper bound for the ball packing densities as well. However, there are no “essential" results regarding the “classical" ball packings with congruent balls, and for ball coverings either. The goal of this paper is to find the extremal ball arrangements in ${𝐇}^3$ with “classical balls". We consider only periodic congruent ball arrangements (for simplicity) related to the generalized, so-called complete Coxeter orthoschemes and their extended groups. In Theorems 1.1 and 1.2 we formulate also conjectures for the densest ball packing with density $0·77147\cdots$ and the loosest ball covering with density $1·36893\cdots$, respectively. Both are related with the extended Coxeter group $(5,3,5)$ and the so-called hyperbolic football manifold. These facts can have important relations with fullerenes in crystallography.

Keywords: hyperbolic geometry, ball packings and coverings

Classification (MSC2000): 52C17

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