Dense Sphere Packings (London Mathematical Society Lecture Note Series, 400)

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He is particularly interested in geometric extremum problems, and equilibrium points of convex bodies. Account Options Connexion.

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Volumetric Discrete Geometry. Karoly Bezdek , Zsolt Langi. CRC Press , 24 avr.

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Provides a list of 30 open problems to promote research Features more than 60 research exercises Ideally suited for researchers and students of combinatorics, geometry and discrete mathematics. Droits d'auteur.

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Informations bibliographiques. Selected Proofs. Harriot published a study of various stacking patterns in , and went on to develop an early version of atomic theory. This meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one.

Kepler conjecture

But eliminating all possible irregular arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. In fact, there are irregular arrangements that are denser than the cubic close packing arrangement over a small enough volume, but any attempt to extend these arrangements to fill a larger volume is now known to always reduce their density.

After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century. In David Hilbert included it in his list of twenty three unsolved problems of mathematics —it forms part of Hilbert's eighteenth problem. This meant that a proof by exhaustion was, in principle, possible. Meanwhile, attempts were made to find an upper bound for the maximum density of any possible arrangement of spheres.

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In , Wu-Yi Hsiang claimed to have proven the Kepler conjecture. The current consensus is that Hsiang's proof is incomplete.


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In , assisted by his graduate student Samuel Ferguson, he embarked on a research program to systematically apply linear programming methods to find a lower bound on the value of this function for each one of a set of over 5, different configurations of spheres. If a lower bound for the function value could be found for every one of these configurations that was greater than the value of the function for the cubic close packing arrangement, then the Kepler conjecture would be proved.

To find lower bounds for all cases involved solving about , linear programming problems. When presenting the progress of his project in , Hales said that the end was in sight, but it might take "a year or two" to complete. In August Hales announced that the proof was complete.

Journal of Formalized Reasoning

At that stage, it consisted of pages of notes and 3 gigabytes of computer programs, data and results. Despite the unusual nature of the proof, the editors of the Annals of Mathematics agreed to publish it, provided it was accepted by a panel of twelve referees. Hales published a page paper describing the non-computer part of his proof in detail. Hales and Ferguson received the Fulkerson Prize for outstanding papers in the area of discrete mathematics for In January , Hales announced the start of a collaborative project to produce a complete formal proof of the Kepler conjecture.

An Introduction to the Theory of Surreal Numbers London Mathematical Society Lecture Note Series

The aim was to remove any remaining uncertainty about the validity of the proof by creating a formal proof that can be verified by automated proof checking software such as HOL Light and Isabelle. Hales estimated that producing a complete formal proof would take around 20 years of work. Hales first published a "blueprint" for the formal proof in ; [4] the project was announced completed on August 10, From Wikipedia, the free encyclopedia. Forum of Mathematics, Pi. June


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