Decomposition method related to saturated hyperball packings

In this paper we study the problem of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new procedure to get a decomposition of 3-dimensional hyperbolic space $\HYP$ into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices.

[1]  Top dense hyperbolic ball packings and coverings for complete Coxeter orthoscheme groups , 2016, 1612.04541.

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

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

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

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

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

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

[8]  L. Németh.  On the hyperbolic Pascal pyramid , 2015, 1511.02067.

[9]  J. Szirmai.  Packings with horo- and hyperballs generated by simple frustum orthoschemes , 2015, 1505.03338.

[10]  J. Szirmai.  The least dense hyperball covering to the regular prism tilings in the hyperbolic n-space , 2013 .

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

[12]  Robert Thijs Kozma,et al.  New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-Space , 2014, Discret. Comput. Geom..

[13]  A candidate for the densest packing with equal balls in Thurston geometries , 2012, 1210.2202.

[14]  The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3-space , 2006 .

[15]  Ausfüllungen der hyperbolischen Ebene durch kongruente Hyperzykelbereiche , 1979 .

[17]  J. Szirmai.  Hyperball packings in hyperbolic $3$-space , 2014, 1405.0248.

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

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

[20]  The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds ∗ , 2013, 1306.4221.

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