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

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


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




Research supported by NSF grant 0804189 and the Benter Foundation.


  1. 1.
    Bachoc, C., Vallentin, F.: New upper bounds for kissing numbers from semidefinite programming. J. Am. Math. Soc. 21, 909–924 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bezdek, K.: On a stronger form of Rogers’ lemma and the minimum surface area of Voronoi cells in unit ball packings. J. Reine Angew. Math. 518, 131–143 (2000)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Fejes Tóth, L.: über die dichteste Kugellagerung. Mathematische Zeitschrift 48, 676–684 (1943)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fejes Tóth, L.: Remarks on a theorem of R.M. Robinson. Studia Scientiarum Hungarica 4, 441–445 (1969)zbMATHGoogle Scholar
  5. 5.
    Fejes Tóth, L.: Lagerungen in der Ebene auf der Kugel und im Raum, 2nd edn. Springer, Berlin/New York (1972)zbMATHCrossRefGoogle Scholar
  6. 6.
    Fejes Tóth, L.: Research problems. Periodica Mathematica Hungarica 29, 89–91 (1989)CrossRefGoogle Scholar
  7. 7.
    Hales, TC.: Linear programs for the Kepler conjecture. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) Mathematical Software – ICMS 2010, Springer, Berlin/New York (2010)Google Scholar
  8. 8.
    Hales, T.C.: Dense sphere packings: a blueprint for formal proofs. London Mathematical Society Lecture Note Series, vol. 400. Cambridge University Press, Cambridge/New York (2012)Google Scholar
  9. 9.
    Hales, T.C.: A proof of Fejes Tóth’s conjecture on sphere packings with kissing number twelve. arXiv:1209.6043 (2012)Google Scholar
  10. 10.
    Hales, TC., Ferguson, S.P.: The Kepler conjecture. Discret. Comput. Geom. 36(1), 1–269 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hales, T.C., McLaughlin, S.: A proof of the dodecahedral conjecture. J. AMS 23, 299–344 (2010). Google Scholar
  12. 12.
    Marchal, C.: Study of the Kepler’s conjecture: the problem of the closest packing. Mathematische Zeitschrift 267, 737–765 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Nipkow, T., Bauer, G., Schultz, P.: Flyspeck I: tame graphs. In: Furbach, U., Shankar, N. (eds.) International Joint Conference on Automated Reasoning, Seattle. Lecture Notes in Computer Science, vol. 4130, pp. 21–35. Springer (2006)Google Scholar
  14. 14.
    Obua, S.: Flyspeck II: the basic linear programs. Ph.D. thesis, Technische Universität München (2008)Google Scholar
  15. 15.
    Rogers, C.A.: The packing of equal spheres. J. Lond. Math. Soc. 3/8, 609–620 (1958)Google Scholar
  16. 16.
    Solovyev, A., Hales, T.C.: Efficient formal verification of bounds of linear programs. LNCS, vol. 6824, pp. 123–132. Springer, Berlin (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

Personalised recommendations