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The Strong Dodecahedral Conjecture and Fejes Tóth’s Conjecture on Sphere Packings with Kissing Number Twelve

Chapter
Part of the Fields Institute Communications book series (FIC, volume 69)

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).

Key words

Sphere packings Discrete geometry Voronoi cell 

Subject Classifications

52C17 

Notes

Acknowledgements

Research supported by NSF grant 0804189 and the Benter Foundation.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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