Discrete & Computational Geometry

, Volume 48, Issue 1, pp 128–141 | Cite as

The Strong Thirteen Spheres Problem

  • Oleg R. Musin
  • Alexey S. Tarasov


The thirteen spheres problem asks if 13 equal-size non-overlapping spheres in three dimensions can simultaneously touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schütte and van der Waerden only in 1953.

A natural extension of this problem is the strong thirteen-sphere problem (or the Tammes problem for 13 points), which calls for finding the maximum radius of and an arrangement for 13 equal-size non-overlapping spheres touching the unit sphere. In this paper, we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on an enumeration of irreducible graphs.


Planar Graph Discrete Comput Geom Regular Triangle Contact Graph Sphere Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aigner, M., Ziegler, G.M.: Proofs from THE BOOK. Springer, Berlin (1998) 1st edn. and (2002) 2nd edn. zbMATHGoogle Scholar
  2. 2.
    Anstreicher, K.: The thirteen spheres: A new proof. Discrete Comput. Geom. 31, 613–625 (2004) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bachoc, C., Vallentin, F.: New upper bounds for kissing numbers from semidefinite programming. J. Am. Math. Soc. 21, 909–924 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Böröczky, K.: The problem of Tammes for n=11. Studia Sci. Math. Hung. 18, 165–171 (1983) zbMATHGoogle Scholar
  5. 5.
    Böröczky, K.: The Newton-Gregory problem revisited. In: Bezdek, A. (ed.) Discrete Geometry, pp. 103–110. Dekker, New York (2003) CrossRefGoogle Scholar
  6. 6.
    Böröczky, K., Szabó, L.: Arrangements of 13 points on a sphere. In: Bezdek, A. (ed.) Discrete Geometry, pp. 111–184. Dekker, New York (2003) Google Scholar
  7. 7.
    Böröczky, K., Szabó, L.: Arrangements of 14, 15, 16 and 17 points on a sphere. Studia Sci. Math. Hung. 40, 407–421 (2003) zbMATHGoogle Scholar
  8. 8.
    Brass, P., Moser, W.O.J., Pach, J.: Research Problems in Discrete Geometry. Springer, Berlin (2005) zbMATHGoogle Scholar
  9. 9.
    Brinkmann, G., McKay, B.D.: Fast generation of planar graphs (expanded edition).
  10. 10.
    Casselman, B.: The difficulties of kissing in three dimensions. Not. Am. Math. Soc. 51, 884–885 (2004) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Danzer, L.: Finite point-sets on \(\textbf{S}^{2}\) with minimum distance as large as possible. Discrete Math. 60, 3–66 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fejes Tóth, L.: Über die Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems. Jber. Deutch. Math. Verein. 53, 66–68 (1943) Google Scholar
  13. 13.
    Fejes Tóth, L.: Lagerungen in der Ebene, auf der Kugel und in Raum. Springer, Berlin (1953). Russian translation, Moscow, 1958 zbMATHGoogle Scholar
  14. 14.
    Hales, T.: The status of the Kepler conjecture. Math. Intell. 16, 47–58 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hoppe, R.: Bemerkung der Redaktion. Archiv Math. Phys. (Grunet) 56, 307–312 (1874) Google Scholar
  16. 16.
    Hsiang, W.-Y.: Least Action Principle of Crystal Formation of Dense Packing Type and Kepler’s Conjecture. World Scientific, Singapore (2001) zbMATHCrossRefGoogle Scholar
  17. 17.
    Leech, J.: The problem of the thirteen spheres. Math. Gaz. 41, 22–23 (1956) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Levenshtein, V.I.: On bounds for packing in n-dimensional Euclidean space. Sov. Math. Dokl. 20(2), 417–421 (1979) Google Scholar
  19. 19.
    Maehara, H.: Isoperimetric theorem for spherical polygons and the problem of 13 spheres. Ryukyu Math. J. 14, 41–57 (2001) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Maehara, H.: The problem of thirteen spheres—a proof for undergraduates. Eur. J. Comb. 28, 1770–1778 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Musin, O.R.: The kissing problem in three dimensions. Discrete Comput. Geom. 35, 375–384 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Musin, O.R.: The kissing number in four dimensions. Ann. Math. 168, 1–32 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Odlyzko, A.M., Sloane, N.J.A.: New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. J. Comb. Theory, Ser. A 26, 210–214 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
  25. 25.
    Pfender, F., Ziegler, G.M.: Kissing numbers, sphere packings, and some unexpected proofs. Not. Am. Math. Soc. 51, 873–883 (2004) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Robinson, R.M.: Arrangement of 24 circles on a sphere. Math. Ann. 144, 17–48 (1961) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Schütte, K., van der Waerden, B.L.: Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand 1 Platz? Math. Ann. 123, 96–124 (1951) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Schütte, K., van der Waerden, B.L.: Das Problem der dreizehn Kugeln. Math. Ann. 125, 325–334 (1953) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Szpiro, G.G.: Kepler’s Conjecture. Wiley, New York (2002) Google Scholar
  30. 30.
    Tammes, R.M.L.: On the origin number and arrangement of the places of exits on the surface of pollengrains. Rec. Trv. Bot. Neerl. 27, 1–84 (1930) Google Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at BrownsvilleBrownsvilleUSA
  2. 2.Institute for System AnalysisRussian Academy of ScienceMoscowRussia

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