Advertisement

Unit Sphere Packings

  • Károly Bezdek
Chapter
Part of the Fields Institute Monographs book series (FIM, volume 32)

Abstract

Unit sphere packings are the classical core of (discrete) geometry. We survey old as well new results giving an overview of the art of the matters. The emphases are on the following five topics: the contact number problem (generalizing the problem of kissing numbers), lower bounds for Voronoi cells (studying Voronoi cells from volumetric point of view), dense sphere packings in Euclidean 3-space (studying a strong version of the Kepler conjecture), sphere packings in Euclidean dimensions higher than 3, and uniformly stable sphere packings.

Keywords

Unit Ball Convex Body Voronoi Cell Sphere Packing Circle Packing 
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.

References

  1. 7.
    G. Ambrus, F. Fodor, A new lower bound on the surface area of a Voronoi polyhedron, Period. Math. Hung. 53/1–2, 45–58 (2006)Google Scholar
  2. 10.
    N. Arkus, V.N. Manoharan, M.P. Brenner, Deriving finite sphere packings. SIAM J. Discret. Math. 25/4, 1860–1901 (2011)Google Scholar
  3. 13.
    K. Ball, A lower bound for the optimal density of lattice packings. Int. Math. Res. Not. 10, 217–221 (1992)CrossRefGoogle Scholar
  4. 16.
    E. Baranovskii, On packing n-dimensional Euclidean space by equal spheres. Iz. Vissih Uceb. Zav. Mat. 39/2, 14–24 (1964)Google Scholar
  5. 19.
    I. Bárány, N.P. Dolbilin, A stability property of the densest circle packing. Monatsh. Math. 106, 107–114 (1988)MathSciNetMATHCrossRefGoogle Scholar
  6. 24.
    U. Betke, M. Henk, Finite packings of spheres. Discret. Comput. Geom. 19, 197–227 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 25.
    U. Betke, M. Henk, J.M. Wills, Finite and infinite packings. J. Reine Angew. Math. 53, 165–191 (1994)MathSciNetGoogle Scholar
  8. 26.
    U. Betke, M. Henk, J.M. Wills, Sausages are good packings. Discret. Comput. Geom. 13, 297–311 (1995)MathSciNetMATHCrossRefGoogle Scholar
  9. 27.
    A. Bezdek, Solid packing of circles in the hyperbolic plane. Studia Sci. Math. Hung. 14, 203–207 (1979)MathSciNetMATHGoogle Scholar
  10. 29.
    K. Bezdek, On a stronger form of Rogers’ lemma and the minimum surface area of Voronoi cells in unit ball packings. J. Reine Angew. Math. 518, 131–143. (2000)MathSciNetMATHGoogle Scholar
  11. 30.
    K. Bezdek, On rhombic dodecahedra. Contrib. Algebra Geom. 41/2, 411–416 (2000)Google Scholar
  12. 32.
    K. Bezdek, Improving Rogers’ upper bound for the density of unit ball packings via estimating the surface area of Voronoi cells from below in Euclidean d-space for all d ≥ 8. Discret. Comput. Geom. 28, 75–106 (2002)MathSciNetMATHCrossRefGoogle Scholar
  13. 33.
    K. Bezdek, On the maximum number of touching pairs in a finite packing of translates of a convex body. J. Combin. Theory A 98, 192–200 (2002)MathSciNetMATHCrossRefGoogle Scholar
  14. 34.
    D. Bezdek, Dürer’s unsolved geometry problem (Canada-Wide Science Fair, St. John’s, 2004), pp. 1–42Google Scholar
  15. 37.
    D. Bezdek, A proof of an extension of the icosahedral conjecture of Steiner for generalized deltahedra. Contrib. Discret. Math. 2/1, 86–92 (2007)Google Scholar
  16. 42.
    K. Bezdek, Contact numbers for congruent sphere packings in Euclidean 3-space. Discret. Comput. Geom. 48/2, 298–309 (2012)Google Scholar
  17. 43.
    K. Bezdek, On a strong version of the Kepler conjecture. Mathematika 59, 23–30 (2013)MathSciNetMATHCrossRefGoogle Scholar
  18. 50.
    K. Bezdek, E. Daróczy-Kiss, Finding the best face on a Voronoi polyhedron – the strong dodecahedral conjecture revisited. Monatsh. Math. 145, 191–206 (2005)MathSciNetMATHCrossRefGoogle Scholar
  19. 54.
    K. Bezdek, S. Reid, On touching pairs, triplets, and quadruples in packings of congruent spheres. arXiv:1210.5756v1 [math.MG], 1–19 (2012)Google Scholar
  20. 56.
    A. Bezdek, K. Bezdek, R. Connelly, Finite and uniform stability of sphere packings. Discret. Comput. Geom. 20, 111–130 (1998)MathSciNetMATHCrossRefGoogle Scholar
  21. 57.
    K. Bezdek, E. Daróczy-Kiss, K.J. Liu, Voronoi polyhedra of unit ball packings with small surface area. Period. Math. Hung. 39/1–3, 107–118 (1999)Google Scholar
  22. 66.
    L. Bowen, Circle packing in the hyperbolic plane. Math. Phys. Electron. J. 6, 1–10 (2000)MathSciNetGoogle Scholar
  23. 67.
    K. Böröczky, Packing of spheres in spaces of constant curvature. Acta Math. Acad. Sci. Hung. 32, 243–261 (1978)MATHCrossRefGoogle Scholar
  24. 68.
    K. Böröczky, The problem of Tammes for n = 11. Stud. Sci. Math. Hung. 18, 165–171 (1983)MATHGoogle Scholar
  25. 76.
    H. Cohn, N. Elkies, New upper bounds on sphere packings I. Ann. Math. 157, 689–714 (2003)MathSciNetMATHCrossRefGoogle Scholar
  26. 77.
    H. Cohn, A. Kumar, The densest lattice in twenty-four dimensions. Electron. Res. Announc. Am. Math. Soc. 10, 58–67 (2004)MathSciNetMATHCrossRefGoogle Scholar
  27. 81.
    J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups (Springer, New York, 1999)MATHCrossRefGoogle Scholar
  28. 87.
    L. Danzer, Finite point sets on S 2 with minimum distance as large as possible. Discret. Math. 60, 3–66 (1986)MathSciNetMATHCrossRefGoogle Scholar
  29. 89.
    D. de Laat, F.M. de Oliveira Filho, F. Vallentin, Upper bounds for packings of spheres of several radii. arXiv:1206.2608v1 [math.MG], 1–31 (2012)Google Scholar
  30. 93.
    E.D. Demaine, J. O’Rourke, Geometric Folding Algorithms (Cambridge University Press, Cambridge, 2007)MATHCrossRefGoogle Scholar
  31. 98.
    L. Fejes Tóth, Über die dichteste Kugellagerung. Math. Z. 48, 676–684 (1943)MathSciNetCrossRefGoogle Scholar
  32. 99.
    L. Fejes Tóth, Regular Figures (Pergamon, Tarrytown, 1964)MATHGoogle Scholar
  33. 100.
    L. Fejes Tóth, Solid circle packings and circle coverings. Studia Sci. Math. Hung. 3, 401–409 (1968)MATHGoogle Scholar
  34. 101.
    L. Fejes Tóth, Research problem 13. Period. Math. Hung. 6, 197–199 (1975)CrossRefGoogle Scholar
  35. 102.
    L. Fejes Tóth, Solid packing of circles in the hyperbolic plane. Studia Sci. Math. Hung. 15, 299–302 (1980)MATHGoogle Scholar
  36. 103.
    S.P. Ferguson, Sphere packings V, Pentahedral prisms. Discret. Comput. Geom. 36/1, 167–204 (2006)Google Scholar
  37. 105.
    H. Freudenthal, B.L. van der Waerden, On an assertion of Euclid. Simon Stevin 25, 115–121 (1947)MathSciNetMATHGoogle Scholar
  38. 109.
    P.M. Gruber, C.G. Lekkerkerker, Geometry of Numbers (North-Holland, Amsterdam, 1987)MATHGoogle Scholar
  39. 112.
    T.C. Hales, Sphere packings I. Discret. Comput. Geom. 17, 1–51 (1997)MathSciNetMATHCrossRefGoogle Scholar
  40. 114.
    T.C. Hales, A proof of the Kepler conjecture. Ann. Math. 162/2–3, 1065–1185 (2005)Google Scholar
  41. 116.
    T.C. Hales, Sphere packings III, Extremal cases. Discret. Comput. Geom. 36/1, 71–110 (2006)Google Scholar
  42. 117.
    T.C. Hales, Sphere packings IV, Detailed bounds. Discret. Comput. Geom. 36/1, 111–166 (2006)Google Scholar
  43. 118.
    T.C. Hales, Sphere packings VI, Tame graphs and linear programs. Discret. Comput. Geom. 36/1, 205–265 (2006)Google Scholar
  44. 119.
    T.C. Hales, The strong dodecahedral conjecture and Fejes Tóth’s contact conjecture. arXiv:1110.0402v1 [math.MG], 1–11 (2011)Google Scholar
  45. 121.
    T.C. Hales, S.P. Ferguson, A formulation of the Kepler conjecture. Discret. Comput. Geom. 36/1, 21–69 (2006)Google Scholar
  46. 122.
    T.C. Hales, S. McLaughlin, A proof of the dodecahedral conjecture. arXiv:9811079 [math.MG] (with updates and improvements from 2007), 1–90 (1998)Google Scholar
  47. 123.
    T.C. Hales, S. McLaughlin, The dodecahedral conjecture. J. Am. Math. Soc. 23/2, 299–344 (2010)Google Scholar
  48. 124.
    H. Harborth, Lösung zu Problem 664A. Elem. Math. 29, 14–15 (1974)MathSciNetGoogle Scholar
  49. 126.
    B. Hayes, The science of sticky spheres. Am. Sci. 100, 442–449 (2012)CrossRefGoogle Scholar
  50. 127.
    D. Hilbert, Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902)MathSciNetMATHCrossRefGoogle Scholar
  51. 129.
    R.S. Hoy, J. Harwayne-Gidansky, C.S. O’Hern, Structure of finite sphere packings via exact enumeration: implications for colloidal crystal nucleation. Phys. Rev. E 85, 051403 (2012)CrossRefGoogle Scholar
  52. 130.
    W.-Y. Hsiang, On the sphere packing problem and the proof of Kepler’s conjecture. Int. J. Math. 4/5, 739–831 (1993)Google Scholar
  53. 131.
    W.-Y. Hsiang, Least Action Principle of Crystal Formation of Dense Packing Type and Kepler’s Conjecture. (World Scientific, Singapore, 2001)Google Scholar
  54. 133.
    G.A. Kabatiansky, V.I. Levenshtein, Bounds for packings on a sphere and in space. Problemy Peredachi Informatsii 14, 3–25 (1978)MathSciNetGoogle Scholar
  55. 136.
    G. Kertész, On totally separable packings of equal balls. Acta Math. Hung. 51/3-4, 363–364 (1988)Google Scholar
  56. 141.
    G. Kuperberg, O. Schramm, Average kissing numbers for non-congruent sphere packings. Math. Res. Lett. 1/3, 339–344 (1994)Google Scholar
  57. 144.
    J.H. Lindsey, Sphere packing in 3. Mathematika 33, 417–421 (1986)MathSciNetCrossRefGoogle Scholar
  58. 149.
    F. Morgan, Geometric Measure Theory – A Beginner’s Guide, 4th edn. (Elsevier/Academic, Amsterdam, 2009)MATHGoogle Scholar
  59. 150.
    D.J. Muder, Putting the best face on a Voronoi polyhedron. Proc. Lond. Math. Soc. 3/56, 329–348 (1988)Google Scholar
  60. 151.
    D.J. Muder, A new bound on the local density of sphere packings. Discret. Comput. Geom. 10, 351–375 (1993)MathSciNetMATHCrossRefGoogle Scholar
  61. 153.
    G. Nebe, N.J.A. Sloane, Table of densest packings presently known. http://www.research.att.com/njas/lattices/density.html
  62. 159.
    C.A. Rogers, The packing of equal spheres. J. Lond. Math. Soc. 3/8, 609–620 (1958)Google Scholar
  63. 160.
    C.A. Rogers, Packing and Covering (Cambridge University Press, Cambridge, 1964)MATHGoogle Scholar
  64. 169.
    A. Schürmann, Perfect, strongly eutactic lattices are periodic extreme. Adv. Math. 225/5, 2546–2564 (2010)Google Scholar
  65. 170.
    K. Schütte, B.L. van der Waerden, Das Problem der dreizehn Kugeln. Math. Ann. 125, 325–334 (1953)MathSciNetMATHCrossRefGoogle Scholar
  66. 175.
    S. Torquato, F.H. Stillinger, New conjectural lower bounds on the optimal density of sphere packings. Exp. Math. 15/3, 307–331 (2006)Google Scholar
  67. 176.
    S. Vance, Improved sphere packing lower bounds from Hurwitz lattices. Adv. Math. 227/5, 2144–2156 (2011)Google Scholar
  68. 177.
    A. Venkatesh, A note on sphere packings in high dimension. Manuscript, pp. 1–11Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Károly Bezdek
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Personalised recommendations