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 Toth’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 Toth’s conjecture on sphere packings with kissing number twelve. arXiv:1209.6043, 2012).
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