Dodecahedral Conjecture

The kissing number of a sphere is 12. This led Fejes Tóth (1943) to conjecture that in any unit sphere packing, the volume of any Voronoi cell around any sphere is at least as large as a regular dodecahedron of inradius 1. This statement is now known as the dodecahedral conjecture. It implies a bound of eta<=0.754697... on the packing density for sphere packing, and thus provides a bound on the densest possible sphere packing. It is not, however, sufficient to establish the Kepler conjecture (which implies eta=0.74048).

This long-outstanding conjecture was proved by Hales and McLaughlin (2002) using techniques of interval arithmetic and linear programming.

See also

Kepler Conjecture, Kelvin's Conjecture, Kissing Number, Sphere Packing

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Bezdek, K. "Isoperimetric Inequalities and the Dodecahedral Conjecture." Int. J. Math. 6, 759-780, 1997.Fejes Tóth, L. "Über die dichteste Kugellagerung." Math. Z. 48, 676-684, 1943.Fejes Tóth, L. Regular Figures. Oxford, England: Pergamon Press, pp. 263-300, 1964.Hales, T. C. and McLaughlin, S. "A Proof of the Dodecahedral Conjecture." 5 Jun 2002., D. J. "Putting the Best Face on a Voronoi Polyhedron." Proc. London Math. Soc. 56, 329-348, 1988.Muder, D. J. "A New Bound on the Local Density of Sphere Packings." Disc. Comp. Geom. 10, 351-375, 1993.

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Dodecahedral Conjecture

Cite this as:

Weisstein, Eric W. "Dodecahedral Conjecture." From MathWorld--A Wolfram Web Resource.

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