Periodica Mathematica Hungarica

, Volume 36, Issue 2–3, pp 171–180 | Cite as

Decomposing the 2-Sphere into Domains of Smallest Possible Diameter

  • A. Heppes


In the present paper the following sphere decomposition problem is discussed: For a given natural number n what is the smallest possible value σ(d, n) such that Sd can be decomposed into n parts each of (spherical) diameter ≤ σ(d, n)? The author investigates the problem for d = 2 and gives the answer for n < 7 as well as for n = 8 and n = 9. Partial results are given for n = 7, 10 and 12 and for the analogous problem in the plane.


Natural Number Partial Result Analogous Problem Decomposition Problem Sphere Decomposition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Bo33]
    K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fundamenta Math. 20 (1933), 177–190.Google Scholar
  2. [BS91]
    V. Boltjanski, and A. Soifer, Geometric etudes in combinatorial mathematics, Center for Excellence in Mathematical Education, Colorado Springs, 1991.Google Scholar
  3. [Eg54]
    H. G. Eggeleston, Covering a three-dimensional set with sets of smaller diameter, J. of the London Math. Soc. 30 (1955), 11–24.Google Scholar
  4. [Fe69]
    G. Fejes TÓth, Kresüberdeckungen der Sphäre, Studia Sci. Math. Hungar. 4 225–247.Google Scholar
  5. [Fe53]
    L. Fejes TÓth, Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, Berlin-Göttingen-Heidelberg, 1953.Google Scholar
  6. [Gr57]
    B. GrÜnbaum, A simple proof of Borsuk's conjecture in three dimensions, Proc. Cambridge Philos. Soc. 53 (1957), 776–778.Google Scholar
  7. [Ha54]
    H. Hadwiger, Von der Zerlegung der Kugel in kleinere Teile, Gaz. Mat. Lisboa 15 (1954), 1–3.Google Scholar
  8. [He57]
    A. Heppes, Decomposition of a 3-dimensional set into sets of smaller diameter, MTA III. oszt. Közl. 7/3–4 (1957), 413–416 (in Hungarian).Google Scholar
  9. [KK93]
    J. Kahn and G. Kalai, A counterexample to Borsuk's conjecture, Bull. (New) AMS 29/1 (1993), 60–62.Google Scholar
  10. [Sk45]
    D. Shkliarsky, On subdivisions of the two dimensional sphere, Math. Sbornik (New.Ser.) 16 (1945), 125–128 (in Russian).Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • A. Heppes

There are no affiliations available

Personalised recommendations