Abstract
This article sketches the proofs of two theorems about sphere packings in Euclidean 3-space. The first is K. Bezdek’s strong dodecahedral conjecture: the surface area of every bounded Voronoi cell in a packing of balls of radius 1 is at least that of a regular dodecahedron of inradius 1. The second theorem is L. Fejes Tóth’s conjecture on sphere packings with kissing number twelve, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by 12 others consists of hexagonal layers. Both proofs are computer assisted. Complete proofs of these theorems appear in Hales TC (Dense sphere packings: a blueprint for formal proofs. London mathematical society lecture note series, vol 400. Cambridge University Press, Cambridge/New York, 2012; A proof of Fejes Tóth’s conjecture on sphere packings with kissing number twelve. arXiv:1209.6043, 2012).
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Notes
- 1.
There are about 25,000 graphs that arise in the L 12 classification and only 8 graphs that arise in the contact graph. Because of the vast difference in complexity of these two classification problems, our discussion will focus on the L 12 classification.
- 2.
About 500 inequalities occur.
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Research supported by NSF grant 0804189 and the Benter Foundation.
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Hales, T.C. (2013). The Strong Dodecahedral Conjecture and Fejes Tóth’s Conjecture on Sphere Packings with Kissing Number Twelve. In: Bezdek, K., Deza, A., Ye, Y. (eds) Discrete Geometry and Optimization. Fields Institute Communications, vol 69. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00200-2_8
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