Remarks on the density of sphere packings in three dimensions

Abstract

This paper shows how the density of sphere packings of spheres of equal radius may be studied using the Delaunay decomposition. Using this decomposition, a local notion of density for sphere packings in ℝ3 is defined. Conjecturally this approach should yield a bound of 0.740873... on sphere packings in ℝ3, and a small perturbation of this approach should yield the bound of\({\pi \mathord{\left/ {\vphantom {\pi {\sqrt {18} }}} \right. \kern-\nulldelimiterspace} {\sqrt {18} }}\). The face-centered-cubic and hexagonal-close-packings provide local maxima (in a strong sense defined below) to the function which associates to every saturated sphere packing in ℝ3 its density. The local measure of density coincides with the actual density for the face-centered cubic and hexagonal-close-packings.

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Hales, T.C. Remarks on the density of sphere packings in three dimensions. Combinatorica 13, 181–197 (1993). https://doi.org/10.1007/BF01303203

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AMS subject classification code (1991)

  • 05 B 40
  • 52 C 17