New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-Space

In this paper we consider ball packings in $$4$$4-dimensional hyperbolic space. We show that it is possible to exceed the conjectured $$4$$4-dimensional realizable packing density upper bound due to L. Fejes-Tóth (Regular Figures, Macmillian, New York, 1964). We give seven examples of horoball packing configurations that yield higher densities of $$0.71644896\dots $$0.71644896⋯, where horoballs are centered at ideal vertices of certain Coxeter simplices, and are invariant under the actions of their respective Coxeter groups.

[1]  The Densest Translation Ball Packing by Fundamental Lattices in Sol Space To the Memory of Professor , 2010 .

[2]  Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types , 2010, 1007.0722.

[3]  The Densest Geodesic Ball Packing by a Type of Nil Lattices , 2007 .

[4]  Emil Molnár,et al.  The projective interpretation of the eight 3-dimensional homogeneous geometries. , 1997 .

[5]  Matthieu Jacquemet,et al.  The Inradius of a Hyperbolic Truncated $$n$$n-Simplex , 2014, Discret. Comput. Geom..

[6]  K. Böröczky.  Packing of spheres in spaces of constant curvature , 1978 .

[7]  Über die dichteste Kugelpackung im hyperbolischen Raum , 1964 .

[8]  Symmetries in the 8 homogeneous 3-geometries* , 2010 .

[9]  Horoball packings and their densities by generalized simplicial density function in the hyperbolic space , 2011, 1105.4315.

[10]  G. Tóth,et al.  Packing and Covering with Convex Sets , 1993 .

[11]  N. Johnson,et al.  The size of a hyperbolic Coxeter simplex , 1999 .

[12]  C. Radin.  The Symmetry of Optimally Dense Packings , 2006 .

[13]  Ruth Kellerhals,et al.  The dilogarithm and volumes of hyperbolic polytopes , 1991 .

[14]  The Optimal Ball and Horoball Packings of the Coxeter Tilings in the Hyperbolic 3-space , 2005 .

[15]  Asymptotic Volume Formulae and Hyperbolic Ball Packing , 1997 .

[16]  Simply transitive geodesic ball packings to $$\mathbf{S}^2\!\times \!\mathbf{R}$$ space groups generated by glide reflections , 2014 .

[17]  N. Johnson,et al.  Commensurability classes of hyperbolic Coxeter groups , 2002 .

[18]  Károly Bezdek,et al.  Sphere packings revisited , 2006, Eur. J. Comb..

[19]  J. Szirmai.  Simply transitive geodesic ball packings to $S^2 \times R$ space groups generated by glide reflections , 2012, 1206.0566.

[20]  The Optimal Ball and Horoball Packings to the Coxeter Honeycombs in the Hyperbolic dspace , 2007 .

[21]  Annali DI Matematica,et al.  Annali di Matematica pura ed applicata , 1892 .

[22]  Charles Radin,et al.  Densest Packing of Equal Spheres in Hyperbolic Space , 2002, Discret. Comput. Geom..

[23]  G. Kuperberg,et al.  Highly saturated packings and reduced coverings , 1995, math/9511225.

[24]  N. Johnson,et al.  Quaternionic modular groups , 1999 .

[25]  Horoball packings to the totally asymptotic regular simplex in the hyperbolic n-space , 2011, 1112.1969.

[26]  Thomas C. Hales,et al.  Historical Overview of the Kepler Conjecture , 2006, Discret. Comput. Geom..

[27]  Regular prism tilings in $${{\widetilde{\bf SL_{2}R}}}$$SL2R~ space , 2014 .