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.
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Robert Thijs Kozma
Present address: Department of Mathematics, SUNY Stony Brook, Stony Brook, NY, 11794-3651, USA
Department of Mathematics and Statistics, Boston University, 111 Cummington St., Boston, MA, 02215, USA
Robert Thijs Kozma
Department of Geometry, Institute of Mathematics, Budapest University of Technology and Economics, 1521, Budapest, Hungary
Jenő Szirmai
Correspondence to Robert Thijs Kozma.
Additional informationCommunicated by A. Constantin.
About this article Cite this articleKozma, R.T., Szirmai, J. Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types. Monatsh Math 168, 27–47 (2012). https://doi.org/10.1007/s00605-012-0393-x
Received: 01 May 2011
Accepted: 02 February 2012
Published: 24 February 2012
Issue Date: October 2012
DOI: https://doi.org/10.1007/s00605-012-0393-x
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