A Proof of the Kepler Conjecture ( DCG Version ) Thomas C . Hales
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From Kepler to Hales , and Back to Hilbert
In layman’s terms the Kepler Conjecture from 1611 is often phrased like “There is no way to stack oranges better than greengrocers do at their fruit stands” and one might add: all over the world and for centuries already. While it is not far from the truth this is also an open invitation to a severe misunderstanding. The true Kepler Conjecture speaks about infinitely many oranges while most gro...
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The Hales program to prove the Kepler conjecture on sphere packings consists of five steps, which if completed, will jointly comprise a proof of the conjecture. We carry out step five of the program, a proof that the local density of a certain combinatorial arrangement, the pentahedral prism, is less than that of the face-centered cubic lattice packing. We prove various relations on the local d...
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This article gives an introduction to a long-term project called Flyspeck, whose purpose is to give a formal verification of the Kepler Conjecture. The Kepler Conjecture asserts that the density of a packing of equal radius balls in three dimensions cannot exceed π/ √ 18. The original proof of the Kepler Conjecture, from 1998, relies extensively on computer calculations. Because the proof relie...
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This paper formalizes the local density inequality approach to getting upper bounds for sphere packing densities in Rn. This approach was first suggested by L. Fejes-Tóth in 1954 as a method to prove the Kepler conjecture that the densest packing of unit spheres in R has density π √ 18 , which is attained by the “cannonball packing.” Local density inequalities give upper bounds for the sphere p...
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