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Monatshefte für Mathematik

, Volume 168, Issue 1, pp 27–47 | Cite as

Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types

  • Robert Thijs KozmaEmail author
  • Jenő Szirmai
Article

Abstract

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.

Keywords

Hyperbolic geometry Horoball packings Honeycomb Tiling 

Mathematics Subject Classification (2000)

52C17 52C22 52B15 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  2. 2.Department of MathematicsSUNY Stony BrookStony BrookUSA
  3. 3.Department of Geometry, Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary

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