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A new bound on the local density of sphere packings
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  • Published: 01 December 1993

A new bound on the local density of sphere packings

  • Douglas J. Muder1 

Discrete & Computational Geometry volume 10, pages 351–375 (1993)Cite this article

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  • 16 Citations

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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.

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References

  1. J. H. Conway and N. J. A. Sloane,Sphere Packings, Lattices, and Groups, Springer-Verlag, New York, 1988.

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  2. Wu-Yi Hsiang, On the sphere packing problem and the proof of Kepler's conjecture, Preprint, 1992.

  3. J. H. Lindsey II, Sphere packing inR 3,Mathematika 33 (1986), 137–147.

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  4. J. H. Lindsey II, Sphere packing, Preprint, 1987.

  5. D. J. Muder, Putting the best face on a Voronoi Polyhedron,Proc. London Math. Soc. (3)56 (1988), 329–348.

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

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Author information

Authors and Affiliations

  1. The MITRE Corporation, 01730, Bedford, MA, USA

    Douglas J. Muder

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  1. Douglas J. Muder
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Cite this article

Muder, D.J. A new bound on the local density of sphere packings. Discrete Comput Geom 10, 351–375 (1993). https://doi.org/10.1007/BF02573984

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  • Received: 18 July 1991

  • Revised: 03 June 1993

  • Published: 01 December 1993

  • Issue Date: December 1993

  • DOI: https://doi.org/10.1007/BF02573984

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Keywords

  • Unit Sphere
  • Local Density
  • Solid Angle
  • Outer Radius
  • Discrete Comput Geom
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