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


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.


Hyperbolic geometry Horoball packings Honeycomb Tiling 

Mathematics Subject Classification (2000)

52C17 52C22 52B15 


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  1. 1.
    Bezdek K.: Sphere packings revisited. Eur. J. Combin. 27(6), 864–883 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bowen L., Radin C.: Optimally dense packings of hyperbolic space. Geom. Dedicata 104, 37–59 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Böhm J., Hertel E.: Polyedergeometrie in n-dimensionalen Räumen konstanter Krümmung. Birkhäuser, Basel (1981)zbMATHGoogle Scholar
  4. 4.
    Böröczky K.: Packing of spheres in spaces of constant curvature. Acta Math. Acad. Sci. Hungar. 32, 243–261 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Böröczky K., Florian A.: Über die dichteste Kugelpackung im hyperbolischen Raum. Acta Math. Acad. Sci. Hungar. 15, 237–245 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Coxeter H.S.M.: Regular honeycombs in hyperbolic space. Proc. Int. Congress Math. Amsterdam III, 155–169 (1954)Google Scholar
  7. 7.
    Dress A.W.M., Huson D.H., Molnár E.: The classification of face-transitive periodic three-dimensional tilings. Acta Crystallogr. A 49, 806–819 (1993)zbMATHCrossRefGoogle Scholar
  8. 8.
    Fejes Tóth G., Kuperberg G., Kuperberg W.: Highly saturated packings and reduced coverings. Monatsh. Math. 125(2), 127–145 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kellerhals R.: The dilogarithm and volumes of hyperbolic polytopes. AMS Math. Surveys Monographs 37, 301–336 (1991)MathSciNetGoogle Scholar
  10. 10.
    Kellerhals R.: Ball packings in spaces of constant curvature and the simplicial density function. Journal für reine und angewandte Mathematik 494, 189–203 (1998)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Molnár E.: Klassifikation der hyperbolischen Dodekaederpflasterungen von flächentransitiven Bewegungsgruppen. Math. Pannonica 4(1), 113–136 (1993)zbMATHGoogle Scholar
  12. 12.
    Molnár E.: The projective interpretation of the eight 3-dimensional homogeneous geometries. Beiträge zur algebra und Geometrie 38(2), 261–288 (1997)zbMATHGoogle Scholar
  13. 13.
    Marshall T.H.: Asymptotic volume formulae and hyperbolic ball packing. Annales Academiæ Scientiarum Fennicæ: Mathematica 24, 31–43 (1999)zbMATHGoogle Scholar
  14. 14.
    Prekopa A.: The Revolution of Janos Bolyai. In: Prekopa, A., Molnar, E. (eds) Non-eucledian geometries., pp. 3–60. Springer, Berlin (2006)CrossRefGoogle Scholar
  15. 15.
    Radin C.: The symmetry of optimally dense packings. In: Prekopa, A., Molnar, E. (eds) Non-eucledian geometries., pp. 197–207. Springer, Berlin (2006)CrossRefGoogle Scholar
  16. 16.
    Szirmai J.: Flächentransitiven Lambert-Würfel-Typen und ihre optimale Kugelpackungen. Acta Math. Hungarica 100, 101–116 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Szirmai J.: Horoball packings for the Lambert-cube tilings in the hyperbolic 3-space. Beiträge zur algebra und geometrie (contributions to algebra and geometry) 46(1), 43–60 (2005)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Szirmai J.: The optimal ball and horoball packings of the Coxeter tilings in the hyperbolic 3-space. Beiträge zur Algebra und Geometrie (contributions to algebra and geometry) 46(2), 545–558 (2005)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Szirmai J.: The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3-space. Acta Math. Hungarica 111(1–2), 65–76 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Szirmai J.: The regular prism tilings and their optimal hyperball packings in the hyperbolic n-space. Publ. Math. Debrecen Hungarica 69(1–2), 195–207 (2006)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Szirmai J.: The optimal ball and horoball packings to the Coxeter honeycombs in the hyperbolic d-space. Beiträge zur algebra und geometrie 48(1), 35–47 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Szirmai J.: The densest geodesic ball packing by a type of Nil lattices. Beiträge zur algebra und geometrie 48(2), 383–397 (2007)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Szirmai, J.: The densest translation ball packing by fundamental lattices in Sol space. Beiträge zur algebra und geometrie (Contributions to Algebra and Geometry) (2010, to appear)Google Scholar

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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