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Discrete & Computational Geometry

, Volume 10, Issue 4, pp 351–375 | Cite as

A new bound on the local density of sphere packings

  • Douglas J. Muder
Article

Abstract

It is shown that a packing of unit spheres in three-dimensional Euclidean space can have density at most 0.773055..., and that a Voronoi polyhedron defined by such a packing must have volume at least 5.41848... These bounds are superior to the best bounds previously published [5] (0.77836 and 5.382, respectively), but are inferior to the tight bounds of 0.7404... and 5.550... claimed by Hsiang [2].

Our bounds are proved by cutting a Voronoi polyhedron into cones, one for each of its faces. A lower bound is established on the volume of each cone as a function of its solid angle. Convexity arguments then show that the sum of all the cone volume bounds is minimized when there are 13 faces each of solid angle 4π/13.

Keywords

Unit Sphere Local Density Solid Angle Outer Radius Discrete Comput Geom 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    J. H. Conway and N. J. A. Sloane,Sphere Packings, Lattices, and Groups, Springer-Verlag, New York, 1988.CrossRefMATHGoogle Scholar
  2. 2.
    Wu-Yi Hsiang, On the sphere packing problem and the proof of Kepler's conjecture, Preprint, 1992.Google Scholar
  3. 3.
    J. H. Lindsey II, Sphere packing inR 3,Mathematika 33 (1986), 137–147.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. H. Lindsey II, Sphere packing, Preprint, 1987.Google Scholar
  5. 5.
    D. J. Muder, Putting the best face on a Voronoi Polyhedron,Proc. London Math. Soc. (3)56 (1988), 329–348.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    C. A. Rogers,Packing and Covering, Cambridge University Press, Cambridge, 1964.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Douglas J. Muder
    • 1
  1. 1.The MITRE CorporationBedfordUSA

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