Acta Mathematica Hungarica
Volume/Issue:
Volume 136: Issue 1-2
Horoball packings and their densities by generalized simplicial density function in the hyperbolic space
Author: Jenő Szirmai 1
View More View Less
  • 1 Institute of Mathematics, Department of Geometry, Budapest University of Technology and Economics, H-1521 Budapest, Hungary
DOI:
https://doi.org/10.1007/s10474-012-0205-8
Pages:
39–55
Publication Date:
01 Jul 2012
Online Publication Date:
07 Mar 2012
Article Category:
Research Article
Restricted access

Abstract

The aim of this paper is to determine the locally densest horoball packing arrangements and their densities with respect to fully asymptotic tetrahedra with at least one plane of symmetry in hyperbolic 3-space extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a fully asymptotic tetrahedron. We allow horoballs of different types at the various vertices. Moreover, we generalize the notion of the simplicial density function in the extended hyperbolic space (n≧2), and prove that, in this sense, the well known Böröczky–Florian density upper bound for “congruent horoball” packings ofdoes not remain valid to the fully asymptotic tetrahedra.

The density of this locally densest packing is ≈0.874994, may be surprisingly larger than the Böröczky–Florian density upper bound ≈0.853276 but our local ball arrangement seems not to have extension to the whole hyperbolic space.

  • [1] Bezdek, K. 2006 Sphere packings revisited European Journal of Combinatorics 27 864–883 .

  • [2] Böhm, J. Hertel, E. 1981 Polyedergeometrie in n-dimensionalen Räumen konstanter Krümmung Birkhäuser Basel.

  • [3] Böröczky, K. 1978 Packing of spheres in spaces of constant curvature Acta Math. Acad. Sci. Hungar. 32 243–261 .

  • [4] Kellerhals, R. 1998 Ball packings in spaces of constant curvature and the simplicial density function Journal für reine und angewandte Mathematik 494 189–203 .

    • Crossref
    • Search Google Scholar
    • Export Citation

  • [5] Kozma, T. R. and Szirmai, J., Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types, Monatshefte für Mathematik, to appear.

    • Search Google Scholar
    • Export Citation

  • [6] Milnor, J. 1994 Geometry Collected Papers 1 Publish or Perish.

  • [7] Molnár, E. 1997 The projective interpretation of the eight 3-dimensional homogeneous geometries Beiträge Alg. Geom. (Contr. Alg. Geom.) 38 261–288.

    • Search Google Scholar
    • Export Citation

  • [8] Rogers, C. A. 1964 Packing and Covering Cambridge University Press.

  • [9] Szirmai, J. 2005 Horoball packings for the Lambert-cube tilings in the hyperbolic 3-space Beiträge Alg. Geom. (Contr. Alg. Geom.) 46 43–60.

    • Search Google Scholar
    • Export Citation

  • [10] Szirmai, J. 2007 The densest geodesic ball packing by a type of Nil lattices Beiträge Alg. Geom. (Contr. Alg. Geom.) 48 383–397.

    • Search Google Scholar
    • Export Citation

  • [11] Szirmai, J. 2010 The densest translation ball packing by fundamental lattices in Sol space Beiträge Alg. Geom. (Contr. Alg. Geom.) 51 353–373.

    • Search Google Scholar
    • Export Citation

  • [12] Zagier, D. 2007 The dilogarithm function Frontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization 3–65 Springer.

    • Search Google Scholar
    • Export Citation
Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632
Keywords:
hyperbolic geometry; horoball packing; optimal simplicial density; 52C17; 52C22; 52B15

Acta Mathematica Hungarica
P.O. Box 127
HU–1364 Budapest
Phone: (36 1) 483 8305
Fax: (36 1) 483 8333
E-mail: acta@renyi.mta.hu

  • Impact Factor (2019): 0.588
  • Scimago Journal Rank (2019): 0.489
  • SJR Hirsch-Index (2019): 38
  • SJR Quartile Score (2019): Q2 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.538
  • Scimago Journal Rank (2018): 0.488
  • SJR Hirsch-Index (2018): 36
  • SJR Quartile Score (2018): Q2 Mathematics (miscellaneous)

For subscription options, please visit the website of Springer Nature

Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation		 1950
Volumes
per Year		 3
Issues
per Year		 6
Founder Magyar Tudományos Akadémia
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Apr 2021 0 0 0
May 2021 1 0 0
Jun 2021 0 0 0
Jul 2021 0 0 0
Aug 2021 0 0 0
Sep 2021 0 0 0
Oct 2021 0 0 0