The Strong Dodecahedral Conjecture and Fejes Toth's Contact Conjecture

Details The Strong Dodecahedral Conjecture and Fejes Toth's Contact Conjecture The Strong Dodecahedral Conjecture and Fejes Toth's Contact Conjecture

2011-10-03

arxiv.org/abs/1110.0402v1

Mathematics

Metric Geometry

Scientific paper

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 contact conjecture, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by twelve others consists of hexagonal layers. Both proofs are computer assisted. Complete proofs of these theorems appear in the author's forthcoming book "Dense Sphere Packings."

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