A Revision of the Proof of the Kepler Conjecture

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

Abstract

The Kepler conjecture asserts that no packing of congruent balls in threedimensional 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.

Keywords

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Appel, K., Haken, W.: The four color proof suffices. Math. Intell. 8(1), 10–20 (1986)MathSciNetCrossRefGoogle Scholar
  2. 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. 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. 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)Google Scholar
  5. 5.
    Fejes Toth, L.: Lagerungen in der Ebene auf der Kugel und im Raum, 2nd edn. Springer, Berlin (1972)Google Scholar
  6. 6.
    Ferguson, S.P.: Sphere packings V. Pentahedral prisms. Discrete Comput. Geom. 36(1), 167–204 (2006)CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 8.
    Gonthier, G.: Formal proof—the four-colour theorem. Not. Am. Math. Soc. 55(11), 1382–1393 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Hales, T.C.: Errata and revisions to the proof of the Kepler conjecture. http://code.google.com/p/flyspeck/
  10. 10.
    Hales, T.C.: Sphere packings. I. Discrete Comput. Geom. 17, 1–51 (1997)CrossRefGoogle Scholar
  11. 11.
    Hales, T.C.: Kepler conjecture source code (1998). http://www.math.pitt.edu/~thales/kepler98/
  12. 12.
    Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005)MathSciNetCrossRefGoogle Scholar
  13. 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 fur Informatik (IBFI), Schloss Dagstuhl, Germany. http://drops.dagstuhl.de/opus/volltexte/2006/432
  14. 14.
    Hales, T.C.: Sphere packings. III. Extremal cases. Discrete Comput. Geom. 36(1), 71–110 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hales, T.C.: Sphere packings. IV. Detailed bounds. Discrete Comput. Geom. 36(1), 111–166 (2006)CrossRefGoogle Scholar
  16. 16.
    Hales, T.C.: Sphere packings. VI. Tame graphs and linear programs. Discrete Comput. Geom. 36(1), 205–265 (2006)CrossRefGoogle Scholar
  17. 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. 18.
    Hales, T.C.: The Flyspeck Project (2007). http://code.google.com/p/flyspeck
  19. 19.
    Hales, T.C.: The Jordan curve theorem, formally and informally. Am. Math. Mon. 114(10), 882–894 (2007)MathSciNetCrossRefGoogle Scholar
  20. 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. 21.
    Hales, T.C.: Lemmas in elementary geometry (2008). Source files at http://code.google.com/p/flyspeck/source/browse/trunk/
  22. 22.
    Hales, T.C.: Formal proof. Not. Am. Math. Soc. 55(11), 1370–1380 (2008)MathSciNetMATHGoogle Scholar
  23. 23.
    Hales, T.C., Ferguson, S.P.: The Kepler conjecture. Discrete Comput. Geom. 36(1), 1–269 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hales, T.C., McLaughlin, S.: A proof of the Dodecahedral conjecture. J. Am. Math. Soc. (2009, to appear). math/9811079Google Scholar
  25. 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)Google Scholar
  26. 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. 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)Google Scholar
  28. 28.
    Harrison, J.: Formal proof—theory and practice. Not. AMS 55(11), 1395–1406 (2008)MathSciNetMATHGoogle Scholar
  29. 29.
    Hormander, L.: The Analysis of Linear Partial Differential Operators. II. Classics in Mathematics. Springer, Berlin (2005). Differential Operators with Constant Coefficients; reprint of the 1983 originalGoogle Scholar
  30. 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)Google Scholar
  31. 31.
    Mahboubi, A., Pottier, L.: Elimination des quantificateurs sur les reels en Coq. In: Journees Francophones des Langages Applicatifs (JFLA) (2002). Available on the Web from http://www.lix.polytechnique.fr/~assia/Publi/jfla02.ps
  32. 32.
    McLaughlin, S.: KeplerCode: computer resources for the Kepler and Dodecahedral Conjectures. http://code.google.com/p/kepler-code/
  33. 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. 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)CrossRefGoogle Scholar
  35. 35.
    Milner, R., Tofte, M., Harper, R.: The Definition of Standard ML. MIT Press, Cambridge (1990)Google Scholar
  36. 36.
    Monniaux, D.: The pitfalls of verifying floating-point computations. TOPLAS 30(3), 12 (2008)CrossRefGoogle Scholar
  37. 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/
  38. 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)Google Scholar
  39. 39.
    Obua, S.: Flyspeck II: The basic linear programs. PhD thesis, Technische Universitat Munchen (2008)Google Scholar
  40. 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)CrossRefGoogle Scholar
  41. 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 optimizationMathSciNetCrossRefGoogle Scholar
  42. 42.
    Revol, N., Rouillier, F.: Motivations for an arbitrary precision interval arithmetic and the MPFI library. Reliab. Comput. 11(4), 275–290 (2005)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. J. Comb. Theory, Ser. B 70, 2–44 (1997)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Solovay, R.M., Arthan, R.D., Harrison, J.: Some new results on decidability for elementary algebra and geometry. APAL (2009, submitted)Google Scholar
  45. 45.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)MATHGoogle Scholar
  46. 46.
    Weekes, S.: MLton. http://mlton.org
  47. 47.
    Wiedijk, F.: Encoding the HOL Light logic in Coq. http://www.cs.ru.nl/~freek/notes/holl2coq.pdf
  48. 48.
    Wiedijk, F. (eds.): The Seventeen Provers of theWorld. Lecture Notes in Computer Science, vol. 3600. Springer, Berlin (2006). Foreword by Dana S. Scott, Lecture Notes in Artificial IntelligenceGoogle Scholar
  49. 49.
    Zumkeller, R.: Global optimization in type theory. PhD thesis, Ecole Polytechnique Paris (2008)Google Scholar

Copyright information

© T.C. Hales 2011

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 Corporation, JF1-13HillsboroUSA
  3. 3.Carnegie Mellon UniversityPittsburghUSA
  4. 4.Department for InformaticsTechnische Universität MünchenMunichGermany
  5. 5.École PolytechniqueParisFrance

Personalised recommendations