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 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
).
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. http://arxiv.org/abs/math.MG/9811079.Muder,
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.
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
).
