Densest Packing of Equal Spheres in Hyperbolic Space

Abstract. We propose a method to analyze the density of packings of spheres of fixed radius in the hyperbolic space of any dimension m≥ 2 , and prove that for all but countably many radii, optimally dense packings must have low symmetry.

[1]  Charles Radin,et al.  Space tilings and local isomorphism , 1992 .

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

[3]  Gábor Fejes Tóth,et al.  Packing and Covering , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[4]  A. Nevo.  Pointwise ergodic theorems for radial averages on simple Lie groups II , 1994 .

[5]  Shahar Mozes,et al.  Aperiodic tilings of the hyperbolic plane by convex polygons , 1998 .

[6]  I. Benjamini,et al.  Percolation in the hyperbolic plane , 1999, math/9912233.

[7]  E. Stein,et al.  Analogs of Wiener’s ergodic theorems for semisimple groups I , 1997 .

[8]  Michael Kapovich,et al.  Hyperbolic Manifolds and Discrete Groups , 2000 .

[9]  Shmuel Weinberger,et al.  Aperiodic tilings, positive scalar curvature, and amenability of spaces , 1992 .

[10]  F. Su.  The Banach-Tarski Paradox , 1990 .

[11]  J. Ratcliffe.  Foundations of Hyperbolic Manifolds , 2019, Graduate Texts in Mathematics.

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

[13]  Chaim Goodman-Strauss.  A strongly aperiodic set of tiles in the hyperbolic plane , 2005 .

[14]  Andrew S. Glassner,et al.  Aperiodic Tiling , 1998, IEEE Computer Graphics and Applications.

[15]  C. Radin.  Miles of tiles , 1999 .

[16]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

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

[18]  John G. Ratcliffe,et al.  Geometry of Discrete Groups , 2019, Foundations of Hyperbolic Manifolds.

[19]  C. Radin.  Aperiodic tilings in higher dimensions , 1995 .

[20]  Shahar Mozes,et al.  Aperiodic tilings , 1997 .