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december 14, 2003
Redan de gamla grekerna höll på med kombinatorik
I New York Times-artikeln In Archimedes' Puzzle, a New Eureka Moment berättas om en gammal matematisk gåta (Stomachion) skapad av Arkimedes, som nu fått sin lösning. Eller snarare, man har nu kommit på vilket problem det egentligen var. Det som överraskar historikerna är att man redan på den tiden höll på med kombinatoriska problem.
Twenty-two hundred years ago, the great Greek mathematician Archimedes wrote a treatise called the Stomachion. Unlike his other writings, it soon fell into obscurity. Little of it survived, and no one knew what to make of it.
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The Stomachion, concludes the historian, Dr. Reviel Netz, was far ahead of its time: a treatise on combinatorics, a field that did not come into its own until the rise of computer science.
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In fact, he has concluded, the prevailing wisdom was based on a misinterpretation. Archimedes was not trying to piece together strips of paper into different shapes; he was trying to see how many ways the 14 irregular strips could be put together to make a square.
The answer — 17,152 — required a careful and systematic counting of all possibilities. "It was hard," said Dr. Persi Diaconis, a Stanford statistician who worked on it along with a colleague, Dr. Susan Holmes, who is also his wife, and a second husband-and-wife team of combinatorial mathematicians, Dr. Ronald Graham and Dr. Fan Chung from the University of California, San Diego.
Independently, a computer scientist, Dr. William H. Cutler at Chicago Rawhide, a manufacturer of oil seals in Elgin, Ill., wrote a program that confirmed that the mathematicians' answer was correct.
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The diagram involved 14 pieces, and the word "multitude" seemed to be associated with it. Mr. Heiberg and those who followed him thought this meant that you could get many figures by rearranging the pieces.
"This is part of the reason people didn't see what it was about," Dr. Netz said. But the old interpretation seemed trivial, hardly worth Archimedes' time.
As he examined the manuscript pages, piecing together their text, he realized that what Archimedes was really asking seemed to be, "How many ways can you put the pieces together to make a square?" That question, Dr. Netz said, "has mathematical meaning."
"People assumed there wasn't any combinatorics in antiquity," he went on. "So it didn't trigger the observation when Archimedes says there are many arrangements and he will calculate them. But that's what Archimedes did; his introductions are always to the point."
But did Archimedes solve the problem? "I am sure he solved it or he would not have stated it," Dr. Netz said. "I do not know if he solved it correctly."
As for the name, derived from the Greek word for stomach, mathematicians are uncertain. But Dr. Diaconis has a hunch.
"It comes from `stomach turner,' " he said. "If you get involved with it, that's what happens."
Se även Mathworlds förklaringar:
Stomachion
Combinatorics
Posted by hakank at december 14, 2003 08:28 FM Posted to Matematik