A Formulation of the Kepler Conjecture

@article{Hales2006AFO,
  title={A Formulation of the Kepler Conjecture},
  author={Thomas C. Hales and Samuel P. Ferguson},
  journal={Discrete \& Computational Geometry},
  year={2006},
  volume={36},
  pages={21-69}
}
AbstractThis paper is the second in a series of six 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. The top level structure of the proof is described. A compact topological space is described. Each point of this space can be described as a finite cluster of balls with additional combinatorial markings. A continuous function on this compact space is defined. It is… 
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  • 2006
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Some Methods of Problem Solving in Elementary Geometry
  • T. Hales
  • Mathematics
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  • 2007
TLDR
The methods that were used in the original proof of the Kepler conjecture are investigated and a number of other methods that might be used to automate the proofs of these problems are described.
The dodecahedral conjecture
This article gives a summary of a proof of Fejes Toth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a three-dimensional packing of balls of unit radius is at least the volume of a
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References

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This short note describes the tentative form of a finite-dimensional optimization problem that may be of use in a second-generation proof of the Kepler conjecture. In the original 1998 proof of the
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A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.
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The second step of a program to prove the Kepler conjecture on sphere packings leads to a decomposition of R3 into polyhedra, which has density at most that of a regular tetrahedron.
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A series of nonlinear optimization algorithms are given and it is shown how a systematic application of these algorithms would bring substantial simplifications to the original proof of the Kepler conjecture.
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Using the Delaunay decomposition, a local notion of density for sphere packings in ℝ3 is defined and the face-centered-cubic and hexagonal-close-packings provide local maxima (in a strong sense defined below) to the function which associates to every saturated sphere packing in ™3 its density.
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