Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types. (English) Zbl 1255.52015
Summary: The goal of this paper is to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space \(\mathbb H^3\). Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known Böröczky-Florian density upper bound for “congruent horoball” packings of \(\mathbb H^3\) remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.
MSC:
| 52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |
| 52C22 | Tilings in \(n\) dimensions (aspects of discrete geometry) |
| 52B15 | Symmetry properties of polytopes |
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