In n-dimensional hyperbolic space Hn (n > 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 H3 a densest (not unique) horoball packing is derived from
the {3,3,6} Coxeter tiling consisting of ideal regular simplices T? reg
with dihedral angles ?/3. The density of this packing is ??3 ? 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 H3 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... and
the loosest ball covering with density 1.36893..., 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.
Lorentzian Coxeter systems and Boyd–Maxwell ball packings
