Remarks on the density of sphere packings in three dimensions


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    J. H. Conway andN. J. A. Sloane:Sphere Packings, Lattices and Groups, Springer-Verlag, 1988.

  2. [2]

    L. Fejes Tóth:Reguläre Figuren, Akadémiai Kiadó, 1965.

  3. [3]

    W. Habicht andB. L. van der Waerden: Lagerung von Punkten auf der Kugel,Math. Ann. 123 (1951), 223–234.

    Google Scholar 

  4. [4]

    T. Hales: The Sphere Packing Problem,J. of Comp. and Appl. Math. 44 (1992), 41–76.

    Google Scholar 

  5. [5]

    J. H. Lindsey II: Sphere-Packing, II, preprint.

  6. [6]

    C. A. Rogers:Packing and Covering, Cambridge University Press, 1964.

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hales, T.C. Remarks on the density of sphere packings in three dimensions. Combinatorica 13, 181–197 (1993).

Download citation

AMS subject classification code (1991)

  • 05 B 40
  • 52 C 17