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A Revision of the Proof of the Kepler Conjecture
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  • Published: 17 March 2009

A Revision of the Proof of the Kepler Conjecture

  • Thomas C. Hales1,
  • John Harrison2,
  • Sean McLaughlin3,
  • Tobias Nipkow4,
  • Steven Obua4 &
  • …
  • Roland Zumkeller5 

Discrete & Computational Geometry volume 44, pages 1–34 (2010)Cite this article

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Abstract

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.

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References

  1. Appel, K., Haken, W.: The four color proof suffices. Math. Intell. 8(1), 10–20 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bauer, G., Nipkow, T.: Flyspeck I: Tame graphs. In: Klein, G., Nipkow, T., Paulson, L. (eds.) The Archive of Formal Proofs. http://afp.sf.net/entries/Flyspeck-Tame.shtml, May 2006

  3. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Automata Theory and Formal Languages, Second GI Conf., Kaiserslautern, 1975. Lecture Notes in Comput. Sci., vol. 33, pp. 134–183. Springer, Berlin (1975)

    Google Scholar 

  4. Denney, E.: A prototype proof translator from HOL to Coq. In: TPHOLs’00: Proceedings of the 13th International Conference on Theorem Proving in Higher Order Logics, London, UK, 2000, pp. 108–125. Springer, Berlin (2000)

    Chapter  Google Scholar 

  5. Fejes Tóth, L.: Lagerungen in der Ebene auf der Kugel und im Raum, 2nd edn. Springer, Berlin (1972)

    MATH  Google Scholar 

  6. Ferguson, S.P.: Sphere packings V. Pentahedral prisms. Discrete Comput. Geom. 36(1), 167–204 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Garloff, J.: Convergent bounds for the range of multivariate polynomials. In: Proceedings of the International Symposium on Interval Mathematics on Interval Mathematics 1985, London, UK, 1985, pp. 37–56. Springer, Berlin (1985)

    Google Scholar 

  8. Gonthier, G.: Formal proof—the four-colour theorem. Not. Am. Math. Soc. 55(11), 1382–1393 (2008)

    MATH  MathSciNet  Google Scholar 

  9. Hales, T.C.: Errata and revisions to the proof of the Kepler conjecture. http://code.google.com/p/flyspeck/

  10. Hales, T.C.: Sphere packings. I. Discrete Comput. Geom. 17, 1–51 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hales, T.C.: Kepler conjecture source code (1998). http://www.math.pitt.edu/~thales/kepler98/

  12. Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hales, T.C.: Introduction to the Flyspeck project. In: Coquand, T., Lombardi, H., Roy, M.-F. (eds.) Mathematics, Algorithms, Proofs, number 05021 in Dagstuhl Seminar Proceedings, Dagstuhl, Germany, 2006. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany. http://drops.dagstuhl.de/opus/volltexte/2006/432

  14. Hales, T.C.: Sphere packings. III. Extremal cases. Discrete Comput. Geom. 36(1), 71–110 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hales, T.C.: Sphere packings. IV. Detailed bounds. Discrete Comput. Geom. 36(1), 111–166 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hales, T.C.: Sphere packings. VI. Tame graphs and linear programs. Discrete Comput. Geom. 36(1), 205–265 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hales, T.C.: Some methods of problem solving in elementary geometry. In: LICS ’07: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science, Washington, DC, USA, 2007, pp. 35–40. IEEE Comput. Soc., Los Alamitos (2007)

    Google Scholar 

  18. Hales, T.C.: The Flyspeck Project (2007). http://code.google.com/p/flyspeck

  19. Hales, T.C.: The Jordan curve theorem, formally and informally. Am. Math. Mon. 114(10), 882–894 (2007)

    MATH  MathSciNet  Google Scholar 

  20. Hales, T.C.: Flyspeck: A blueprint for the formal proof of the Kepler conjecture (2008). Source files at http://code.google.com/p/flyspeck/source/browse/trunk/

  21. Hales, T.C.: Lemmas in elementary geometry (2008). Source files at http://code.google.com/p/flyspeck/source/browse/trunk/

  22. Hales, T.C.: Formal proof. Not. Am. Math. Soc. 55(11), 1370–1380 (2008)

    MATH  MathSciNet  Google Scholar 

  23. Hales, T.C., Ferguson, S.P.: The Kepler conjecture. Discrete Comput. Geom. 36(1), 1–269 (2006)

    Article  MathSciNet  Google Scholar 

  24. Hales, T.C., McLaughlin, S.: A proof of the Dodecahedral conjecture. J. Am. Math. Soc. (2009, to appear). math/9811079

  25. Harrison, J.: HOL Light: A tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) Proceedings of the First International Conference on Formal Methods in Computer-Aided Design (FMCAD’96). Lecture Notes in Computer Science, vol. 1166, pp. 265–269. Springer, Berlin (1996)

    Chapter  Google Scholar 

  26. Harrison, J.: A HOL theory of Euclidean space. In: Theorem Proving in Higher Order Logics. Lecture Notes in Comput. Sci., vol. 3603, pp. 114–129. Springer, Berlin (2005)

    Google Scholar 

  27. Harrison, J.: Verifying nonlinear real formulas via sums of squares. In: Schneider, K., Brandt, J. (eds.) Proceedings of the 20th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2007. Lecture Notes in Computer Science, vol. 4732, pp. 102–118. Springer, Kaiserslautern (2007)

    Chapter  Google Scholar 

  28. Harrison, J.: Formal proof—theory and practice. Not. AMS 55(11), 1395–1406 (2008)

    MATH  MathSciNet  Google Scholar 

  29. Hörmander, L.: The Analysis of Linear Partial Differential Operators. II. Classics in Mathematics. Springer, Berlin (2005). Differential Operators with Constant Coefficients; reprint of the 1983 original

    MATH  Google Scholar 

  30. IEEE Standards Committee 754. IEEE Standard for binary floating-point arithmetic, ANSI/IEEE Standard 754-1985. Institute of Electrical and Electronics Engineers, New York (1985)

  31. Mahboubi, A., Pottier, L.: Elimination des quantificateurs sur les réels en Coq. In: Journées Francophones des Langages Applicatifs (JFLA) (2002). Available on the Web from http://www.lix.polytechnique.fr/~assia/Publi/jfla02.ps

  32. McLaughlin, S.: KeplerCode: computer resources for the Kepler and Dodecahedral Conjectures. http://code.google.com/p/kepler-code/

  33. McLaughlin, S.: An interpretation of Isabelle/HOL in HOL Light. In: Furbach, U., Shankar, N. (eds.) IJCAR. Lecture Notes in Computer Science, vol. 4130, pp. 192–204. Springer, Berlin (2006)

    Google Scholar 

  34. McLaughlin, S., Harrison, J.: A proof-producing decision procedure for real arithmetic. In: Automated deduction—CADE-20. Lecture Notes in Comput. Sci., vol. 3632, pp. 295–314. Springer, Berlin (2005)

    Google Scholar 

  35. Milner, R., Tofte, M., Harper, R.: The Definition of Standard ML. MIT Press, Cambridge (1990)

    Google Scholar 

  36. Monniaux, D.: The pitfalls of verifying floating-point computations. TOPLAS 30(3), 12 (2008)

    Article  Google Scholar 

  37. Nipkow, T., Paulson, L., Wenzel, M.: In: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Lect. Notes in Comp. Sci., vol. 2283. Springer, Berlin (2002). http://www.in.tum.de/~nipkow/LNCS2283/

    Google Scholar 

  38. Nipkow, T., Bauer, G., Schultz, P.: Flyspeck I: Tame graphs. In: Furbach, U., Shankar, N. (eds.) Automated Reasoning (IJCAR 2006). Lect. Notes in Comp. Sci., vol. 4130, pp. 21–35. Springer, Berlin (2006)

    Chapter  Google Scholar 

  39. Obua, S.: Flyspeck II: The basic linear programs. PhD thesis, Technische Universität München (2008)

  40. Obua, S., Skalberg, S.: Importing HOL into Isabelle/HOL. In: Automated Reasoning. Lecture Notes in Computer Science, vol. 4130, pp. 298–302. Springer, Berlin (2006)

    Chapter  Google Scholar 

  41. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program., Ser. B 96(2), 293–320 (2003). Algebraic and geometric methods in discrete optimization

    Article  MATH  MathSciNet  Google Scholar 

  42. Revol, N., Rouillier, F.: Motivations for an arbitrary precision interval arithmetic and the MPFI library. Reliab. Comput. 11(4), 275–290 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  43. Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. J. Comb. Theory, Ser. B 70, 2–44 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  44. Solovay, R.M., Arthan, R.D., Harrison, J.: Some new results on decidability for elementary algebra and geometry. APAL (2009, submitted)

  45. Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)

    MATH  Google Scholar 

  46. Weekes, S.: MLton. http://mlton.org

  47. Wiedijk, F.: Encoding the HOL Light logic in Coq. http://www.cs.ru.nl/~freek/notes/holl2coq.pdf

  48. Wiedijk, F. (eds.): The Seventeen Provers of the World. Lecture Notes in Computer Science, vol. 3600. Springer, Berlin (2006). Foreword by Dana S. Scott, Lecture Notes in Artificial Intelligence

    Google Scholar 

  49. Zumkeller, R.: Global optimization in type theory. PhD thesis, École Polytechnique Paris (2008)

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Author information

Authors and Affiliations

  1. Math Department, University of Pittsburgh, Pittsburgh, PA, USA

    Thomas C. Hales

  2. Intel Corporation, JF1-13, 2111 NE 25th Avenue, Hillsboro, OR, 97124, USA

    John Harrison

  3. Carnegie Mellon University, Pittsburgh, PA, USA

    Sean McLaughlin

  4. Department for Informatics, Technische Universität München, Munich, Germany

    Tobias Nipkow & Steven Obua

  5. École Polytechnique, Paris, France

    Roland Zumkeller

Authors
  1. Thomas C. Hales
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  2. John Harrison
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  3. Sean McLaughlin
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  4. Tobias Nipkow
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  5. Steven Obua
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  6. Roland Zumkeller
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Corresponding author

Correspondence to Thomas C. Hales.

Additional information

Research supported by NSF grant 0804189.

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Hales, T.C., Harrison, J., McLaughlin, S. et al. A Revision of the Proof of the Kepler Conjecture. Discrete Comput Geom 44, 1–34 (2010). https://doi.org/10.1007/s00454-009-9148-4

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  • Received: 10 December 2008

  • Revised: 31 January 2009

  • Accepted: 04 February 2009

  • Published: 17 March 2009

  • Issue Date: July 2010

  • DOI: https://doi.org/10.1007/s00454-009-9148-4

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Keywords

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