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Sphere Packings, VI. Tame Graphs and Linear Programs
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  • Published: 27 February 2006

Sphere Packings, VI. Tame Graphs and Linear Programs

  • Thomas C. Hales1 

Discrete & Computational Geometry volume 36, pages 205–265 (2006)Cite this article

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Abstract

This paper is the sixth and final part in a series of papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In a previous paper in this series, a continuous function f on a compact space is defined, certain points in the domain are conjectured to give the global maxima, and the relation between this conjecture and the Kepler conjecture is established. In this paper we consider the set of all points in the domain for which the value of f is at least the conjectured maximum. To each such point, we attach a planar graph. It is proved that each such graph must be isomorphic to a tame graph, of which there are only finitely many up to isomorphism. Linear programming methods are then used to eliminate all possibilities, except for three special cases treated in earlier papers: pentahedral prisms, the face-centered cubic packing, and the hexagonal-close packing. The results of this paper rely on long computer calculations.

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  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15217, USA

    Thomas C. Hales

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  1. Thomas C. Hales
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Correspondence to Thomas C. Hales.

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Hales, T. Sphere Packings, VI. Tame Graphs and Linear Programs. Discrete Comput Geom 36, 205–265 (2006). https://doi.org/10.1007/s00454-005-1215-x

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  • Received: 11 November 1998

  • Revised: 25 July 2005

  • Published: 27 February 2006

  • Issue Date: July 2006

  • DOI: https://doi.org/10.1007/s00454-005-1215-x

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Keywords

  • Continuous Function
  • Computational Mathematic
  • Planar Graph
  • Early Paper
  • Compact Space
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