A candidate for the densest packing with equal balls in Thurston geometries

By Jeno Szirmai. 2012

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The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a $$3$$3-dimensional space of constant curvature was settled by Böröczky and Florian for the hyperbolic space $$\mathbf H ^3$$H3, and, with the proof of the famous Kepler conjecture, by Hales for the Euclidean space $$\mathbf E ^3$$E3. The goal of this paper is to extend the problem of finding the densest geodesic ball (or sphere) packing for the other $$3$$3-dimensional homogeneous geometries (Thurston geometries) $$\mathbf S ^2\!\times \!\mathbf R , \mathbf H ^2\!\times \!\mathbf R , $$S2×R,H2×R,$$\widetilde{\mathbf{S \mathbf L _2\mathbf R }}, \mathbf {Nil} , \mathbf {Sol} . $$SL2R~,Nil,Sol. In the following a transitive symmetry group of the ball packing is assumed, which is one of the discrete isometry groups of the considered space. Moreover, we describe a candidate of the densest geodesic ball packing. The greatest density until now is $$\approx 0.85327613$$≈0.85327613 that is not realized by a packing with equal balls of the hyperbolic space $$\mathbf H ^3$$H3. However, that is attained, e.g., by a horoball packing of $$\overline{\mathbf{H }}^3$$H¯3 where the ideal centres of horoballs lie on the absolute figure of $$\overline{\mathbf{H }}^3$$H¯3 inducing the regular ideal simplex tiling $$(3,3,6)$$(3,3,6) by its Coxeter–Schläfli symbol. In this work we present a geodesic ball packing in the $$\mathbf S ^2\times \mathbf R $$S2×R geometry whose density is $${\approx }0.87757183$$≈0.87757183. The extremal configuration is described in Theorem 2.6. A conjecture for the densest ball packing in Thurston geometries and further remarks are summarized in Sect. 1.1, 1.2 and 2.3.

Subfields: Applied Mathematics, Geometry and Topology, Materials Chemistry

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