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

Content Provider	Springer Nature Link
Author	Szirmai, Jenő
Copyright Year	2013
Abstract	The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a $$3$$ -dimensional space of constant curvature was settled by Böröczky and Florian for the hyperbolic space $$\mathbf H ^3$$ , and, with the proof of the famous Kepler conjecture, by Hales for the Euclidean space $$\mathbf E ^3$$ . The goal of this paper is to extend the problem of finding the densest geodesic ball (or sphere) packing for the other $$3$$ -dimensional homogeneous geometries (Thurston geometries) $$\mathbf S ^2\!\times \!\mathbf R , \mathbf H ^2\!\times \!\mathbf R , $$ $$\widetilde{\mathbf{S \mathbf L _2\mathbf R }}, \mathbf {Nil} , \mathbf {Sol} . $$ 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 $$\approx 0.85327613$$ that is not realized by a packing with equal balls of the hyperbolic space $$\mathbf H ^3$$ . However, that is attained, e.g., by a horoball packing of $$\overline{\mathbf{H }}^3$$ where the ideal centres of horoballs lie on the absolute figure of $$\overline{\mathbf{H }}^3$$ inducing the regular ideal simplex tiling $$(3,3,6)$$ by its Coxeter–Schläfli symbol. In this work we present a geodesic ball packing in the $$\mathbf S ^2\times \mathbf R $$ geometry whose density is $${\approx }0.87757183$$ . 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.
Starting Page	441
Ending Page	452
Page Count	12
File Format	PDF
ISSN	01384821
Journal	Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
Volume Number	55
Issue Number	2
e-ISSN	21910383
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-07-17
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Thurston geometries Ball packings Density The densest ball packing Packing and covering in $n$ dimensions Tilings in $n$ dimensions Non-Euclidean differential geometry Polyhedra and polytopes; regular figures, division of spaces Algebra Geometry Algebraic Geometry Convex and Discrete Geometry
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Geometry and Topology

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

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

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