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	Kepler conjecture - Wikipedia, the free encyclopedia
		Kepler had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606.
		Kepler did not have a proof of the conjecture, and the next step was taken by German mathematician Carl Friedrich Gauss, who published a partial solution in 1831.
		Gauss proved that the Kepler conjecture is true if the spheres have to be arranged in a regular lattice.
	en.wikipedia.org /wiki/Kepler_conjecture (1268 words)

	After 300 years, computers facilitate solution to Kepler Stacking Problem
		Kepler and Harriot had their differences--Harriot held that solid matter was made of closely packed atoms, and that this could explain properties such as the partial reflection of light falling on a transparent surface, but Kepler initially disagreed.
		To prove the Kepler conjecture requires a proof that the maximum possible density of spheres is that of face-centred cubic packing, 74.0 percent, a figure which may not seem much different than Muder's, but is far more difficult to achieve.
		To prove the Kepler conjecture, about 5,000 different packings (in which the centres of the spheres form triangles or four-sided shapes, as shown in the diagrams) were classified, and 100 were analysed in detail.
	www.wsws.org /articles/1999/jan1999/math-j06.shtml (2158 words)

	Kepler's conjecture
		In the nineteenth century, Carl Gauss proved that face-centered cubic packing is the densest arrangement in which the centers of the spheres form a regular lattice, but he left open the question of whether an irregular stacking of spheres might be still denser.
		In 1953, László Tóth reduced the Kepler conjecture to an enormous calculation that involved specific cases, and later suggested that computers might be helpful for solving the problem.
		Hales proof of Kepler's conjecture remains controversial simply because of the length of the computer calculations involved and the difficulty of verifying them.
	www.daviddarling.info /encyclopedia/K/Keplers_conjecture.html (227 words)

	Hales solves oldest problem in discrete geometry
		Known as the Kepler conjecture, the problem is named after the German astronomer Johannes Kepler who first proposed a solution in 1611.
		Kepler speculated that it would be most efficient to arrange the spheres in layers with each sphere resting in the small hollow between the three spheres beneath it.
		To the mathematically challenged, Kepler's solution seems so obvious, you can't help but wonder why generations of mathematicians have been wracking their brains ever since to determine whether or not he was right.
	www.umich.edu /~urecord/9899/Sep16_98/hales.htm (722 words)

	'Kepler's Conjecture'
		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.
		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.
	plus.maths.org /issue25/reviews/book4 (648 words)

	American Scientist Online - The Proof Is in the Packing (Site not responding. Last check: 2007-11-05)
		In 1611, the German physicist Johannes Kepler stated what he felt to be the obvious solution: You make a triangular array, then fit another layer into the interstices between the balls in the first layer, and so on.
		Kepler never even tried to prove that this was the densest packing.
		Like the proof of the Four-Color Conjecture, another notorious problem that was solved in the 1970s, his argument relies heavily on computer calculations: roughly 100,000 of them, virtually all of them too lengthy to do by hand.
	www.americanscientist.org /template/AssetDetail/assetid/15497 (1014 words)

	cass 1
		At any rate, this claim came to be known as Kepler's conjecture, and it turned out to be extremely difficult to verify.
		Kepler quite likely would have thought that the analogous assertion about the hexagonal packing in 2D was even more obvious.
		Kepler's assertions were possibly prompted by correspondence beginning in the year 1606 between him and the remarkable English mathematician Thomas Harriot.
	www.math.sunysb.edu /~tony/whatsnew/column/pennies-1200/cass1.html (551 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Kepler experimented and could not find any more efficient arrangement than the so-called face-centered cubic lattice, which has a packing efficiency of 74 percent.
		By the 20th century, generations of mathematicians had failed to demonstrate categorically that the Kepler conjecture was true, but no one had discovered a more efficient arrangement.
		Karl F. Gauss, the most brilliant mathematician of the 19th century, failed to prove the Kepler conjecture, but he solved the two-dimensional version of the problem, finding that the best way to arrange disks in two dimensions was to surround each disk by six others.
	www.math.gatech.edu /~mccuan/courses/archive/2401-fall2000/nyt (1393 words)

	Publisher description for Library of Congress control number 2002014422 (Site not responding. Last check: 2007-11-05)
		Kepler's Conjecture offers the nonspecialist genuine insights into the minds of research mathematicians when they are grappling with big, important questions.
		For the next four centuries, Kepler's conjecture became the figurative loose cannon in the mathematical world as some of the greatest intellects in history set out to prove his theory.
		Kepler's Conjecture provides a mesmerizing account of this 400-year quest for an answer that would satisfy even the most skeptical mathematical minds.
	www.loc.gov /catdir/description/wiley039/2002014422.html (233 words)

	[No title]
		A 1998 paper which proved another long-standing conjecture using a computer, by Thomas Hales, of the University of Pittsburgh, has only recently been accepted by the Annals of Mathematics, perhaps the field's most prestigious journal, and is scheduled to be published later this year.
		Rather than argue by contradiction, he reduced what was a problem about an infinite number of things (the Kepler conjecture considers an infinite number of spheres in an infinitely large space) to a statement about a finite, but very large, number of mathematical objects.
		Loosely speaking, he reduced the Kepler conjecture to a problem of considering whether, given a set of cables, which have no minimum length, but can only be stretched to a certain extent, and struts, which have a limit on how much they can be compressed, one can build a sculpture of a certain type.
	www.lehigh.edu /~dmd1/sk413.txt (1693 words)

	Bees do it: Proving the Honeycomb Conjecture (Site not responding. Last check: 2007-11-05)
		After completing his proof of the Kepler Conjecture, Thomas Hales turned his attention to a related problem of even greater antiquity: What is the most efficient partition of a plane into equal areas?
		A few centuries later, Pappus of Alexandria presented an incomplete proof of the conjecture, based largely on the fact that only three regular polygons (the triangle, the square and the hexagon) fill out a plane, and the hexagon holds the most honey.
		Mathematicians from Kepler to Kelvin studied six-sided prisms, but the Honeycomb Conjecture resisted all efforts to prove it until Hales came along.
	www.pitt.edu /utimes/issues/34/011108/11.html (321 words)

	New Arrival of the Week (Site not responding. Last check: 2007-11-05)
		Unfortunately, though most mathematicians felt that Kepler's conjecture was correct, its justification proved to be quite stubborn.
		This book, using a delightful and engaging style, describes the 400-year effort to either justify or disprove Kepler's conjecture, with young mathematician Thomas Hales producing a "solution" in 1998.
		Though Kepler's conjecture and its resolution have not received as much press or attention as Wiles's solution to Fermat's problem, the story is equally interesting." -- Choice, July 1st, 2003
	www.uwec.edu /LIBRARY/news/kepler.htm (106 words)

	Thomas C. Hales - The Kepler Conjecture
		Hilbert: It seems `obvious' that Kepler's conjecture is correct.
		Harriot and Kepler: The genesis of Kepler's conjecture.
		The two-dimensional version of Kepler's conjecture asks for the densest packing of unit disks in the plane.
	pear.math.pitt.edu /PittMathZine/2001/fall/articles/cannonOverview.html (746 words)

	The New York Times > Science > In Math, Computers Don't Lie. Or Do They?
		Kepler concluded that the pyramid was most efficient.
		The Kepler Conjecture is also not the first proof to rely on computers.
		As for his 1998 proof of the Kepler Conjecture, Dr. Hales said that final publication, after a review process, originally expected to last a few months, would be almost anticlimactic.
	www.nytimes.com /2004/04/06/science/06MATH.html?ei=5007&en=4e26c69cb490ad90&ex=1396584000&partner=TECHDIRT&pagewanted=all&position= (1803 words)

	Kepler's Sphere Packing Problem Solved
		The general problem as considered by Kepler and subsequent mathematicians is formulated not in terms of the number of spheres that can be packed together but the density of the packing, i.e., the total volume of the spheres divided by the total volume of the container into which they are packed.
		Kepler believed that the face-centered cubic lattice was the most efficient of all arrangements (in terms of the density of the packing arrangement), but was unable to prove this.
		Most recently, in 1993, a highly-respected mathematician at the University of California at Berkeley produced a complicated proof of the Kepler Conjecture which, after many months of debate, most mathematicians decided was incorrect.
	www.maa.org /devlin/devlin_9_98.html (1088 words)

	BBC - Radio 4 - Another 5 Numbers - Kepler's Conjecture
		In 1606 this problem was presented to German astronomer, Johannes Kepler, who took it on but adapted it significantly.
		With Kepler's "face-centred cubic lattice" the first layer of oranges is formed in the same way you would spread penny coins on a desk to cover it leaving the least amount of gaps.
		Using this method, Kepler calculated that the packing efficiency rose to 74%, constituting the highest efficiency you could ever get.
	www.bbc.co.uk /radio4/science/another54.shtml (506 words)

	It's the way things stack up: Pitt professor solves long-standing math mystery
		In 1998, Thomas C. Hales, who was recruited to Pitt this fall as Mellon Professor in the mathematics department, astonished his colleagues by proving Kepler's Conjecture, one of the world's great math problems.
		Named after the German mathematician Johannes Kepler, who postulated it in 1611, Kepler's Conjecture held that the best possible stacking of balls is the "cannonball pyramid" familiar today to visitors of Civil War memorials.
		Hales began studying Kepler's Conjecture in 1988, when he was an assistant professor at Harvard.
	www.pitt.edu /utimes/issues/34/011108/10.html (915 words)

	American Scientist Online - Foams and Honeycombs (Site not responding. Last check: 2007-11-05)
		Hales was led to the honeycomb problem in the wake of the excitement caused by his proof of the Kepler conjecture.
		With the expertise that he had built up in his years of work on the Kepler conjecture, it was only six months before he was able to announce a proof of the honeycomb problem.
		Unlike his proof of the Kepler conjecture, this new proof makes no use of computers and is only 20 pages long; comparatively speaking, it fell into his lap.
	www.americanscientist.org /template/AssetDetail/assetid/14718?fulltext=true (4543 words)

	The changing nature of proof
		Thomas C. Hales, the University of Pittsburgh Mellon Professor of Mathematics, famously and finally proved Johannes Kepler's 400-year-old conjecture that the most efficient way to pack spheres was in a pyramid shape.
		Though the Kepler conjecture may seem intuitive, no one was able to create a formal proof of it until Hales astonished mathematicians across the world by doing so in 1998.
		Hales estimates that his proof of Kepler's conjecture will take about 20 "work-years" to transcribe (meaning, for example, if 10 people worked on it, it would take two years).
	www.eurekalert.org /pub_releases/2006-02/uop-tcn021406.php (559 words)

	Do Unto Others Project-Church of the Science of God
		Examples include Fermat’s last theorem, Kepler’s conjecture and the four-color conjecture, all of which have been solved by mathematicians only in the past few decades.
		The four-color conjecture in particular attracted a lot of attention from recreational mathematicians, and it was in some ways a pity when it was finally proved, because a source of apparently endless fun had dried up.
		The (m,n) conjecture could very well turn out to be he false, which would explain why it has been so hard to prove for so long by so many..
	www.dountoothers.org /torusgrid.html (1326 words)

	The New York Times > Premium Archive > Mathematics 'Proves' What the Grocer Always Knew
		After four centuries of failure, Dr. Hales's announcement of a proof of the Kepler conjecture has been greeted with some surprise and a great deal of euphoria.
		Dr. Hales acknowledged that his work needed to be reviewed by other mathematicians before Kepler's conjecture officially turned into a theorem.
		Photo: Dr. Thomas Hales of the University of Michigan says he has solved the Kepler conjecture, one of the major problems of mathematics.
	crd.lbl.gov /~dhbailey/expmath/news/nyt-hales-1998.html (1513 words)

	Formalizing the Proof of the Kepler Conjecture (Site not responding. Last check: 2007-11-05)
		The Kepler Conjecture states that the densest packing of spheres in three dimensions is the familiar cannonball arrangement.
		Although this statement has been regarded as obvious by chemists, a rigorous mathematical proof of this fact was not obtained until 1998.
		The mathematical proof of the Kepler Conjecture runs 300 pages, and relies on extensive computer calculations.
	www.cs.utah.edu /tphols2004/hales.abstract.html (146 words)

	Mathematical mysteries: Kepler's conjecture
		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.
		A proof of the conjecture may also help physicists to understand the structure of crystals, something that Harriot himself urged Kepler to consider in his work on optics.
		Whether the holes in Hsiang's proof will be filled before Hales finishes his computation remains to be seen but it seems likely that Kepler's conjecture will, at least for the time being, remain a mathematical mystery.
	pass.maths.org.uk /issue3/xfile (572 words)

	Science News Online (7/24/99): The Honeycomb Conjecture (Site not responding. Last check: 2007-11-05)
		Last year, Hales proved Johannes Kepler's conjecture that the arrangement of the familiar piles of neatly stacked oranges at a supermarket represents the best way to pack identical spheres tightly (SN: 8/15/98, p.
		If Hales' proofs of the honeycomb and Kepler conjectures stand the test of time, "it's a remarkable double achievement," says physicist Denis Weaire of Trinity College Dublin in Ireland.
		The mathematicians' honeycomb conjecture therefore concerns a two-dimensional pattern-as if bees were creating a grid for laying out tiles to cover an infinitely wide bathroom floor.
	www.sciencenews.org /sn_arc99/7_24_99/bob2.htm (1749 words)

	Kepler, Johannes
		Kloster Maulbronn: Kepler, Hölderlin und Hesse lernten in Maulbronn (in German)
		Memorial plaque for Kepler in Zagan (Sagan), Poland (in Polish)
		Memorial plaque for Kepler at the Kepler High School, Zagan (Sagan), Poland (in Polish)
	www.astro.uni-bonn.de /~pbrosche/persons/pers_kepler.html (343 words)

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