2012/10/08 by J. Szirmai, Szirmai, Jen{\H}o, Szirmai, Jen{\\H}o · 1 citation
Engineering · Materials Science · Mathematics · #51M20 #52C17 #52C22 #53A35 #Advanced Materials and Mechanics #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.1210.2202
openalex publication_date 2012/10/08 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The ball (or sphere) packing problem with equal balls, without any symmetry\nassumption, in a 3-dimensional space of constant curvature was settled by\nB "or "oczky and Florian for the hyperbolic space HYP in citeBF64 and by\nproving the famous Kepler conjecture by Hales citeH for the Euclidean space\n EUC. The goal of this paper is to extend the problem of finding the densest\ngeodesic ball (or sphere) packing for the other 3-dimensional homogeneous\ngeometries (Thurston geometries)
SXR,~
HXR,~
SLR,~
NIL,~
SOL, where a\ntransitive symmetry group of the ball packing is assumed, one of the discrete\nisometry groups of the considered space.\n Moreover, we describe a candidate of the densest geodesic ball packing. The\ngreatest density until now is \≈ 0.85327613 that is not realized by\npacking with equal balls of the hyperbolic space HYP. However, it attains≠.g. at horoball packing of \ bH3 where the ideal centres of\nhoroballs lie on the absolute figure of \ bH3 inducing the regular\nideal simplex tiling (3,3,6) by its Coxeter-Schl "afli symbol. In this work\nwe present a geodesic ball packing in the SXR geometry whose density is\n\≈ 0.87499429. The extremal configuration is described in Theorem 2.8,\nOur conjecture and further remarks are summarized in Section 3.\n