Advertisement

Discrete & Computational Geometry

, Volume 17, Issue 1, pp 1–51 | Cite as

Sphere packings, I

  • T. C. Hales
Article

Abstract

We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

Keywords

Convex Hull Dihedral Angle Unit Sphere Analytic Continuation Solid Angle 
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. [AH]
    G. Alefeld and J. Herzeberger,Introduction to Interval Computations, Academic Press, New York, 1983.MATHGoogle Scholar
  2. [FT]
    L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, Berlin, 1953.MATHGoogle Scholar
  3. [H1]
    T. C. Hales, Remarks on the density of sphere packings in three dimensions,Combinatorica,13(2) (1993), 181–197.MATHCrossRefMathSciNetGoogle Scholar
  4. [H2]
    T. C. Hales, The sphere packing problem,J. Comput. Appl. Math,44 (1992), 41–76.MATHCrossRefMathSciNetGoogle Scholar
  5. [H3]
    T. C. Hales, The status of the Kepler conjecture,Math. Intelligencer,16(3) (1994), 47–58.MATHMathSciNetCrossRefGoogle Scholar
  6. [H4]
    T. C. Hales, Sphere packings, II,Discrete Comput. Geom., to appear.Google Scholar
  7. [H5]
    T. C. Hales, Sphere packings, III, in preparation.Google Scholar
  8. [H6]
    T. C. Hales, Packings, http://www-personal.math.lsa.umich.edu/∼hales/packings.htmlGoogle Scholar
  9. [H7]
    J. F. Hartet al., Computer Approximations, Wiley, New York, 1968.MATHGoogle Scholar
  10. [IEEE]
    IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std. 754-1985, IEEE, New York.Google Scholar
  11. [P]
    W. H. Presset al., Numerical recipes in C,Less-Numerical Algorithms, second edition, Cambridge University Press, Cambridge, 1992. Chapter 20.Google Scholar
  12. [R]
    C. A. Rogers, The packing of equal spheres,Proc. London Math. Soc. (3)8 (1958), 609–620.CrossRefGoogle Scholar
  13. [W]
    What every computer scientist should know about floating-point arithmetic,Comput. Surveys, 23(1) (1991), 5–48.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1997

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations