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Kepler Conjecture - Wikipedia, The Free Encyclopedia Hsiang, WuYi (2001), Least action principle of crystal formation of dense packing type and Kepler's conjecture, Nankai Tracts in Mathematics, 3, River Edge, NJ World Scientific http://en.wikipedia.org/wiki/Kepler_conjecture
Extractions: From Wikipedia, the free encyclopedia Jump to: navigation search Face-centered cubic packing The Kepler conjecture , named after Johannes Kepler , is a mathematical conjecture about sphere packing in three-dimensional Euclidean space . It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing ( face-centered cubic ) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%. In 1998 Thomas Hales , following an approach suggested by Fejes Tóth (1953) , announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees have said that they are "99% certain" of the correctness of Hales' proof, so the Kepler conjecture is now very close to being accepted as a theorem Diagrams of cubic close packing (left) and hexagonal close packing (right). Imagine filling a large container with small equal-sized spheres. The density of the arrangement is the proportion of the volume of the container that is taken up by the spheres. In order to maximize the number of spheres in the container, you need to find an arrangement with the highest possible density, so that the spheres are packed together as closely as possible.
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Johannes Kepler - Wikipedia, The Free Encyclopedia His mother Katharina Guldenmann, an innkeeper s daughter, was a healer and herbalist who was later tried for witchcraft. Born prematurely, Johannes claimed http://en.wikipedia.org/wiki/Johannes_Kepler
Extractions: Johannes Kepler [ˈkɛplɐ] German mathematician astronomer and astrologer , and key figure in the 17th century scientific revolution . He is best known for his eponymous laws of planetary motion , codified by later astronomers, based on his works Astronomia nova Harmonices Mundi , and Epitome of Copernican Astronomy . These works also provided one of the foundations for Isaac Newton 's theory of universal gravitation During his career, Kepler was a mathematics teacher at a seminary school in Graz Austria , an assistant to astronomer Tycho Brahe , the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II, a mathematics teacher in
Kepler's Conjecture No packing of spheres of the same radius in three dimensions has a density greater than the facecentered (hexagonal) cubic packing. This claim was first published by Johannes http://www.daviddarling.info/encyclopedia/K/Keplers_conjecture.html
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Mathematical Mysteries: Kepler's Conjecture | Plus.maths.org Sir Walter Raleigh is perhaps best known for laying down his cloak in the mud for Queen Elizabeth I (though sadly this act of chivalry is probably a myth!) http://plus.maths.org/issue3/xfile/index.html
Extractions: Sir Walter Raleigh is perhaps best known for laying down his cloak in the mud for Queen Elizabeth I (though sadly this act of chivalry is probably a myth!) However, he also started a mathematical quest which to this day remains unsolved. Sir Walter posed a simple question to his mathematical assistant, Thomas Harriot. How can I calculate the number of cannon balls in a stack? Harriot solved this problem without difficulty but started to wonder about a more general problem. What arrangement of the balls takes up the least space? Harriot wrote about the problem to his colleague Johannes Kepler, best known for his work on planetary orbits. Kepler experimented with the problem and concluded that an arrangement known as the face centred cubic packing, a pattern favoured by fruit sellers, could not be bettered. This statement has become known as "Kepler's conjecture" or simply the sphere packing problem. Fruit stacked using the face centred cubic packing.
Extractions: ...that close packing (either cubic or hexagonal close packing , both of which have maximum densities of ) is the densest possible sphere packing In short, his proof involved large amounts of computer verification, and after three years of intensive effort, the Annals of Mathematics determined that it could not verify with certainty that the proof was correct (not because there were problems, but because the proof had many "low-level components" that lacked a more general intuition, and were thus hard to verify, especially given the degree of computer search involved). A strange arrangement was then made: the Annals of Mathematics published a briefer version of the proof, and made the code/data for the proof available unreviewed on its website. Discrete and Computational Geometry then undertook to publish detailed versions of the six preprints that Hales and his student S. P. Ferguson wrote in 1998.
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Kepler Conjecture -- From Wolfram MathWorld On the Sphere Packing Problem and the Proof of Kepler's Conjecture. Int. J. Math. 4, 739831, 1993. Hsiang, W.-Y. A Rejoinder to Hales's Article. http://mathworld.wolfram.com/KeplerConjecture.html
Extractions: Kepler Conjecture In 1611, Kepler proposed that close packing (either cubic or hexagonal close packing , both of which have maximum densities of ) is the densest possible sphere packing , and this assertion is known as the Kepler conjecture. Finding the densest (not necessarily periodic) packing of spheres is known as the Kepler problem Buckminster Fuller (1975) claimed to have a proof, but it was really a description of face-centered cubic packing, not a proof of its optimality (Sloane 1998). A second putative proof of the Kepler conjecture was put forward by W.-Y. Hsiang (Cipra 1991, Hsiang 1992, 1993, Cipra 1993), but was subsequently determined to be flawed (Conway et al. 1994, Hales 1994, Sloane 1998). According to J. H. Conway, nobody who has read Hsiang's proof has any doubts about its validity: it is nonsense. Soon thereafter, Hales (1997a) published a detailed plan describing how the Kepler conjecture might be proved using a significantly different approach from earlier attempts and making extensive use of computer calculations. Hales subsequently completed a full proof, which appears in a series of papers totaling more than 250 pages (Cipra 1998). A broad outline of the proof in elementary terms appeared in Hales (2002). The proof relies extensively on methods from the theory of global optimization linear programming , and interval arithmetic . The computer files containing the computer code and data files for
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The Kepler Conjecture Information on the recent proof of Kepler conjecture on sphere packings. http://www.math.pitt.edu/~thales/kepler98/
Extractions: Faqs.org homepage Abstracts index Zoology and wildlife conservation Article Abstract: Proof that no denser packing is possible than in face-centred-cubic packing has taken 387 years to be found. This is due to technical difficulties arising from the fact that the density of packing is defined as the limit of the fraction of space occupied by the balls, the many different packings that are just as dense and the infinitely many ways to arrange 12 balls around another ball. Thomas C Hales has worked on the Kepler conjecture and proposed a five step attack on the problem, and the now completed proof depends heavily on computers. author: Sloane, Neil J.A. Publisher: Macmillan Publishing Ltd. Observations, Geometry, Mathematics problems, Sphere, Spheres (Geometry) User Contributions: Comment about this article or add new information about this topic: Comment: (50-4000 characters) Name: E-mail: Security Code: Display my email: Article Abstract: Mathematician Paul Erdos died of a heart attack on Sep. 20, 1996 at a conference in Warsaw, Poland. Born on Mar. 26, 1913 in Budapest, Hungary, Erdos made significant contributions to number theory, probability theory, real and complex analysis, geometry, approximation theory and combinatorics. He has also published about 1,500 papers and about 500-co-authorships throughout his career. A brief overview of Erdos' remarkable career is presented.
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Kepler's Conjecture On Huffduffer Episode four of Another Five Numbers, the BBC radio series presented by Simon Singh. Johannes Kepler experimented with different ways of stacking spheres. He concluded that the http://huffduffer.com/adactio/6613
Extractions: Search Go! Huffduffed by adactio on July 22nd, 2009 Episode four of Another Five Numbers, the BBC radio series presented by Simon Singh. Johannes Kepler experimented with different ways of stacking spheres. He concluded that the "face-centred cubic lattice" was best. Using this method, Kepler calculated that the packing efficiency rose to 74%, constituting the highest efficiency you could ever get. But, how to prove it? download Tagged with mathematics five numbers book:author=simon singh bbc ... boxman on September 11th, 2009 michele on July 22nd, 2009 iamdanw on October 26th, 2009 chrispederick on October 1st, 2009 srushe on February 25th, 2010 liqweed on February 23rd, 2010 tayles on June 15th, 2010 ninthart on June 23rd, 2010 jonkroll on September 11th, 2010 The Number Four Episode one of Another Five Numbers, the BBC radio series presented by Simon Singh. download Tagged with mathematics five numbers book:author=simon singh bbc ... adactio one year ago Episode one of Five Numbers, the BBC radio series presented by Simon Singh. download Tagged with mathematics five numbers book:author=simon singh bbc ... adactio one year ago Episode one of A Further Five Numbers, the BBC radio series presented by Simon Singh.
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'Kepler's Conjecture' | Plus.maths.org Kepler's Conjecture How some of the greatest minds in history helped solve one of the oldest math problems in the world By George Szpiro. George Szpiro has a most unusual day job for http://plus.maths.org/content/keplers-conjecture
Extractions: Skip to Navigation Search this site: review by Helen Joyce Issue 25 Submitted by plusadmin on April 30, 2003 in May 2003 George Szpiro has a most unusual day job for someone writing about the abstract world of pure mathematics. Although he first studied maths at university, he has been a political journalist now for a number of years, working as Israel correspondent for NZZ, a Swiss daily. He wrote this book at night, after the paper's deadline, and as it was being finished lost one of his closest friends in a suicide bombing. The contrast between sphere packings in three dimensions and his daytime subject material must often have struck him. The story of Kepler's Conjecture from the 1590s to the present day is told here via the stories of those who worked on it. Kepler - the great astronomer who discovered that the planets travel on elliptical paths - conjectured that the most efficient way to pack spheres in three-dimensional space (ie the way that wasted least space) was the familiar way that greengrocers stack oranges. But the world had to wait until the late 1990s for a proof, and not for lack of mathematicians trying. The eventual proof of Kepler's Conjecture in 1998 was computer-aided, as, notoriously, was the proof in 1976 of another famous longstanding problem; the Four Colour Theorem. (You can read the story of the Four Colour Theorem in "Four Colours Suffice"
