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


Emil Molnár, Jenő Szirmai




In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\geq2)$, 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 $\mathbf{H}^3$ 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 $\mathbf{H}^3$ with ``classical balls". We consider only periodic congruent ball arrangements (for simplicity) related to the generalized, so-called {ı 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 $0.77147\dots$ and the loosest ball covering with density $1.36893\dots$, 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.