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
J. H. Conway and N. J. A. Sloane,Sphere Packings, Lattices, and Groups, Springer-Verlag, New York, 1988.
Wu-Yi Hsiang, On the sphere packing problem and the proof of Kepler's conjecture, Preprint, 1992.
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D. J. Muder, Putting the best face on a Voronoi Polyhedron,Proc. London Math. Soc. (3)56 (1988), 329–348.
C. A. Rogers,Packing and Covering, Cambridge University Press, Cambridge, 1964.
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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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DOI: https://doi.org/10.1007/BF02573984