Discrete & Computational Geometry

, Volume 44, Issue 1, pp 1–34 | Cite as

A Revision of the Proof of the Kepler Conjecture

  • Thomas C. Hales
  • John Harrison
  • Sean McLaughlin
  • Tobias Nipkow
  • Steven Obua
  • Roland Zumkeller


The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic packing. The original proof, announced in 1998 and published in 2006, is long and complex. The process of revision and review did not end with the publication of the proof. This article summarizes the current status of a long-term initiative to reorganize the original proof into a more transparent form and to provide a greater level of certification of the correctness of the computer code and other details of the proof. A final part of this article lists errata in the original proof of the Kepler conjecture.


Formal proof Sphere packings Linear programming Interval analysis Higher order logic Hypermap 


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Thomas C. Hales
    • 1
  • John Harrison
    • 2
  • Sean McLaughlin
    • 3
  • Tobias Nipkow
    • 4
  • Steven Obua
    • 4
  • Roland Zumkeller
    • 5
  1. 1.Math DepartmentUniversity of PittsburghPittsburghUSA
  2. 2.Intel CorporationHillsboroUSA
  3. 3.Carnegie Mellon UniversityPittsburghUSA
  4. 4.Department for InformaticsTechnische Universität MünchenMunichGermany
  5. 5.École PolytechniqueParisFrance

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