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Study of the Kepler’s conjecture: the problem of the closest packing

Abstract

In the first and second parts the problem of the Kepler’s conjecture is reduced from a problem with an infinite number of parameters to a problem with a finite number of parameters. In the third part is it is shown that the latter problem can be solved by a numerical verification using only a finite number of computations. However that finite number remains large, even if modern computers can do it. The method of analysis is classical: to each sphere of an arbitrary packing is associated a corresponding large enough “associated domain”, disjoint from all other similar domains, in the hope of obtaining an interesting upper limit of the space occupation coefficient. Many methods of association have been tried in the past and the new method presented in this study is complex, but very well adapted to the problem of interest.

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References

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Correspondence to Christian Marchal.

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Marchal, C. Study of the Kepler’s conjecture: the problem of the closest packing. Math. Z. 267, 737–765 (2011). https://doi.org/10.1007/s00209-009-0644-2

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  • DOI: https://doi.org/10.1007/s00209-009-0644-2

Keywords

  • Close Packing
  • Absolute Minimum
  • Fourth Type
  • Neighboring Center
  • Space Angle