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The Kepler Conjecture and Its Proof

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The Kepler Conjecture

Abstract

This paper describes work on the Kepler conjecture starting from its statement in 1611 and culminating in the proof of Hales-Ferguson in 1998–2006. It discusses both the difficulty of the problem and of its solution.

This work was supported in part by NSF Grant DMS-0801029.

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References

  1. Annals of Mathematics Editors, Statement by the Editors, Annals of Mathematics 162 (2005), no. 3, un-numbered page preceding page 1165.

    Google Scholar 

  2. T.Aste and D.Weaire, The Pursuit of Perfect Packing, Institute of Physics Publishing, Bristol and Philadelphia 2000.

    MATH  Google Scholar 

  3. William Barlow, Probable nature of the internal symmetry of crystals, Nature 29 (1883), No. 738, 186–188.

    Article  Google Scholar 

  4. A. Bezdek, K. Bezdek and R. Connelly, Finite and uniform stability of sphere packings, Disc. & Comput. Geom. 20 (1998), no, 1, 111–130.

    Article  MathSciNet  Google Scholar 

  5. K. Böröczky, Jr, Finite Packing and Covering, Cambridge Math. Tracts No. 154, Cambridge Univ. Press: Cambridge 2004.

    Google Scholar 

  6. A. Bundy (Editor), Discussion Meeting Issue ’The nature of mathematical proof’ organized by A. Bundy, M. Atiyah, A. Macintyre and D. Mackenzie, Phil. Trans. of the Royal Society A Mathematical, Physical & Engineering Sciences 363 (2005), Issue 1835, October 15, 2005, pp. 2331–2461.

    Google Scholar 

  7. W. Casselman, The Difficulties of Kissing int Three Dimensions, Notices of the Amer. Math. Soc. 51, No. 8 (2004), 884–885.

    Google Scholar 

  8. J. H. Conway, C. Goodman-Strauss and N. J. A. Sloane, Recent progress in sphere packing, in: Current developments in mathematics, 1999 (Cambridge, MA), pp. 37–76, International Press, Somerville, MA1999.

    Article  MathSciNet  Google Scholar 

  9. J. H. Conway, T. C. Hales, D. J. Muder and N. J. A. Sloane, On the Kepler Conjecture, Letter to the Editor, Math. Intelligencer 16 (1994) No. 2, 5.

    Google Scholar 

  10. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, (Third Edition) Springer-Verlag: NewYork 1999.

    Google Scholar 

  11. H. S. M. Coxeter, Introduction to Geometry, Reprint of the 1969 Edition. JohnWiley and Sons: NewYork 1989.

    Google Scholar 

  12. B. Delaunay, Sur la sphère vide, Izv. Akad. Nauk. SSSR, 7 (1934), 793–800.

    MATH  Google Scholar 

  13. G. Fejes Tóth and J. C. Lagarias, Guest editor’s foreword, Discrete Comput. Geom. 36 (2006), 1–3.

    Article  MathSciNet  Google Scholar 

  14. L. Fejes Tóth, Über die dichteste Kugellagerung, Math. Zeitschrift 48 (1943), 676–684.

    Article  MathSciNet  Google Scholar 

  15. L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Springer-Verlag: Berlin 1953. (Second Edition 1972) [see pp. 171–181, both editions].

    Google Scholar 

  16. T. C. Hales, The Sphere Packing Problem, J. Comp. App. Math. 44 (1992), 41–76.

    Article  MathSciNet  Google Scholar 

  17. T. C. Hales, Remarks on the density of sphere packings in three dimensions, Combinatorica, 13 (2), (1993), 181–197.

    Article  MathSciNet  Google Scholar 

  18. T. C. Hales, The status of the Kepler conjecture, Math. Intelligencer 16 (1994), no. 3, 47–58.

    Article  MathSciNet  Google Scholar 

  19. T. C. Hales, Sphere Packings I, Discrete Comput. Geom. 17 (1997), 1–51, eprint: math.MG/9811073.

    Google Scholar 

  20. T. C. Hales, Sphere Packings II, Discrete Comput. Geom. 18 (1997), 135–149, eprint: math.MG/9811074.

    Google Scholar 

  21. T. C. Hales, The honeycomb conjecture, Disc. Comput. Geom. 25 (2001), 1–22.

    Article  MathSciNet  Google Scholar 

  22. T. C. Hales, A proof of the Kepler conjecture, Annals of Math. 162 (2005), no. 3, 1065–1185.

    Article  MathSciNet  Google Scholar 

  23. T. C. Hales, Historical Overview of the Kepler Conjecture, Discrete Comput. Geom. 36 (2006), 5–20.

    Article  MathSciNet  Google Scholar 

  24. T. C. Hales and S. P. Ferguson, A Formulation of the Kepler Conjecture, Discrete Comput. Geom. 36 (2006), 21–69.

    Article  MathSciNet  Google Scholar 

  25. T. C. Hales, Sphere Packings III. Extreme cases, Discrete Comput. Geom. 36 (2006), 71–110.

    Google Scholar 

  26. T. C. Hales, Sphere Packings IV. Detailed bounds, Discrete Comput. Geom. 36 (2006), 111–166.

    Google Scholar 

  27. S. P. Ferguson, Sphere Packings V. Pentahedral prisms, Discrete Comput. Geom. 36 (2006), 167–204.

    Google Scholar 

  28. T. C. Hales, Sphere Packings VI. Tame graphs and linear programs, Discrete Comput. Geom. 36 (2006), 205–265.

    Google Scholar 

  29. T. C. Hales, Formal Proof, Notices of the American Math. Soc. 55, No. 11 (2008). 1370–1380.

    Google Scholar 

  30. T. C. Hales, J. Harrison, S. McLaughlin, T. Nipkow, S. Obua and R. Zumkeller,A Revision of the Kepler Conjecture, Discrete Comput. Geom. 44 (2010), 1–34.

    Article  MathSciNet  Google Scholar 

  31. T. C. Hales and S. McLaughlin, A proof of the dodecahedral conjecture, preprint, Nov. 1998, arXiv:math/9811079 v1, 54 pp.

    Google Scholar 

  32. T. C. Hales and S. McLaughlin, A proof of the dodecahedral conjecture, J. Amer. Math. Soc. 23 (2010), 299–344.

    Article  MathSciNet  Google Scholar 

  33. D. Hilbert, Mathematische Probleme, Nachrichten Kon. Ges. d.Wiss. Göttingen, Math.-Phys. Kl., 1900, pp. 253–297. English translation in: D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 8 (1902), 437–479. Reprinted in: Mathematical Developments Arising from Hilbert Problems, Proc. Symp. Pure Math XXVIII, American Math. Soc.: Providence 1976.

    Google Scholar 

  34. D. Hilbert, Grundlagen der Geometrie, 9th Edition, revised by Paul Bernays, Teubner: Stuttgart 1962. English Translation: D. Hilbert, Foundations of Geometry Authorized Translation by E. J. Townsend (1902) Open Court Publ. Co. 1959.

    Google Scholar 

  35. R. Hoppe, Berkeung der Redaktion, Archiv der Mathematik und Physik 56 (1874), 307–312. [Comment on paper: C. Bender, Bestimmung der grössten Anzahl gleich grosser Kugeln, welche sich auf eine Kugel von demselben Radius, wie die übrigen, auflegen lassen, Acrhiv der Mathematik und Physik 56 (1874), 302–306.]

    Google Scholar 

  36. W.-Y. Hsiang, On the sphere packing problem and the proof of Kepler’s conjecture, International J. Math. 4 (1993), No. 5, 739–831. (Math Reviews: 95g:52032)

    Article  MathSciNet  Google Scholar 

  37. W.-Y. Hsiang, On the sphere packing problem and the proof of Kepler’s conjecture, in Differential Geometry and Topology (Alghero, 1992),World Scientific: River Edge, NJ 1993, pp. 117–127. (Math Reviews: 96f:52028)

    Google Scholar 

  38. W.-Y. Hsiang, The geometry of spheres, in: Differential Geometry (Shanghai 1991), World Scientific : River Edge, NJ 1993, pp. 92–107.

    Google Scholar 

  39. W.-Y. Hsiang,Arejoinder toT. C. Hales’s article "The status of theKepler conjecture," Math. Intelligenger 17, No. 1 91995), 35–42. (Math Reviews: 97f:52029)

    Google Scholar 

  40. W.-Y. Hsiang, Least Action Principle of Crystal Formation of Dense Packing Type and the Proof of Kepler’s Conjecture,World Scientific: River Edge, NJ 2002. (Math Reviews: 2004g:52033)

    Google Scholar 

  41. R. H. Kargon, Atomism in England from Hariot to Newton, Oxford: Clarendon Press 1966.

    Google Scholar 

  42. Johannes Kepler, Strena Seu de Niue Sexangula, Published by Godfrey Tampach at Frankfort on Main, 1611. English Translation in: J. Kepler, The Six-Cornered Snowflake (Colin Hardie, Translator) Oxford University Press: Oxford 1966.

    Google Scholar 

  43. M. Kline, Mathematics: The Loss of Certainty, Oxford University Press 1982.

    Google Scholar 

  44. J. C. Lagarias, Local density bounds for sphere packings and the Kepler Conjecture, Disc. Comp. Geom. 27 (2002), 165–193

    Article  Google Scholar 

  45. I. Lakatos, Proofs and Refutations, The Logic of Mathematical Discovery, (ed. J.Worrall and E. Zahar), Cambridge University Press: Cambridge 1976.

    Google Scholar 

  46. I. Lakatos, What does a mathematical proof prove?, pp. 61–69 in: I. Lakatos, Mathematics, Science and Epistemology, Philosphical Papers, Volume 2, (J.Worrell and G. Currie, Eds), Cambridge Univ. Press; Cambridge 1978.

    Google Scholar 

  47. J. Milnor, Hilbert’s problem 18: On Crystalographic groups, fundamental domains, and on sphere packing, pp. 491–506 in: F.W. Browder, Ed., Mathematical Developments Arising from the Hilbert Problems, Proc. Symp. Pure Math. 28, American Math. Soc.: Providence, 1976.

    Google Scholar 

  48. F. Morgan,Kepler’s Conjecture and Hales’s Proof.ABook Review, Notices of theAmerican Math. Society 52 No. 1 (2005), 44–47.

    Google Scholar 

  49. I. Newton, The Correspondence of Isaac Newton, (9 volumes) H.W. Turnbull, F. R. S. ( Ed.), Cambridge University Press 1961.

    Google Scholar 

  50. J. Oesterlé, Densité maximale des empilements de sphères en dimension 3 [d’après Thomas C. Hales et Samuel P. Ferguson], Séminaire Bourbaki, Exp. No. 863, Juin 1999.

    Google Scholar 

  51. C. A. Rogers, The Packing of equal spheres, Proc. London Math. Soc. 8 (1958), 609–620.

    Article  MathSciNet  Google Scholar 

  52. C. A. Rogers, Packing and Covering, Cambridge Tracts on Mathematics and Physical Science No. 54, Cambridge Univ. Press, NewYork 1964.

    Google Scholar 

  53. K. Schütte and B. L. van der Waerden, Das Problem der dreizehn Kugeln, Math. Annalen 125 (1953), 325–334.

    Article  MathSciNet  Google Scholar 

  54. JohnW. Shirley, Thomas Hariot: A Biography, Clarendon Press: Oxford 1983.

    Google Scholar 

  55. George G. Szpiro, Kepler’s Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World,Wiley: NewYork 2003.

    MATH  Google Scholar 

  56. W. P. Thurston, On proof and progress in mathematics, Bull. Amer. Math. Soc. 30 (1994), no. 2, 161–177.

    Article  MathSciNet  Google Scholar 

  57. J. P. Troadec, A. Gervois and L. Oger, Statistics ofVoronoi cells of slightly perturbed face-centered cubic and hexagonal close-packed lattices, Europhysics Letters 42 no. 2 (1998), 167–172.

    Article  Google Scholar 

  58. D. F.Watson, Computing the n-dimensional Delaunay tesselation with applications toVoronoi polytopes, Computer J. 24 (1981), 167–172.

    Article  MathSciNet  Google Scholar 

  59. D. Zeilberger, Theorems for a price: tomorrow’s semi-rigorous mathematical culture, Notices of theAMS 40 (1993), no. 8, 978–981.

    MathSciNet  MATH  Google Scholar 

  60. C. Zong, Sphere Packings, Springer-Verlag: NewYork 1999.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Jeffrey C. Lagarias

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Jeffrey C. Lagarias

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Lagarias, J.C. (2011). The Kepler Conjecture and Its Proof. In: Lagarias, J.C. (eds) The Kepler Conjecture. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1129-1_1

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