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


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.


Unit Sphere Local Density Solid Angle Outer Radius Discrete Comput Geom 
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Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

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

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