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Archimedes und Kombinatorik

Hipparchus study that leads to numbers known today as the Schröder numbers lets us assume that the ancient Greeks considered not only geometrical but also combinatorial problems. Another example is another discovery from the Archimedes Palimpsest. A study revealed that Archimedes considered combinatorial problems for the Stomachion.

The Stomachion was mentioned by Caesius Bassus, a Roman poet, in his work "De metris". The combinatorial Game was played by children as it was supporting their memory. (H. Keil, Grammatici Latini, Volume VI Hildesheim 1961)

Stomachion in Greek Textus (misere corruptus): Archimedis opera omnia, vol. 2, p. 416 sqq. ed. J. L. Heiberg, Lipsiae 1881/1915

The Stomachion is a puzzle game considered to be from Archimedes. A square with a 12x12 grid is divided in 14 pieces of 3,4 and 5 sided polygons. As the vertices of the polygons are on the grid points their areas, due to Picks theorem, can be expressed as integer ratios to the area of the square.

Pick's Theorem (Java Applet) , Proof of Pick's Theorem

With the area of the square A =12*12 =144 units, the area of the pieces and its relation to the A are:

No of pieces

Area units

Area of part/A

2

3

1/48

4

6

1/24

1

9

1/16

5

12

1/12

1

21

7/48

1

24

1/6

Rearranging the pieces it is possible to form a very large number of figures.

Was Archimedes using the Stomachion to form such figures? The historian Dr. Reviel Netz 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 used to form a square.

The answer of 17152 combinations by Archimedes requires a careful, systematic counting of all possibilities. "It was hard," said Persi Diaconis, a Stanford statistician who worked on it with his wife Susan Holmes, and a second husband-and-wife team of combinatorial mathematicians, Ronald Graham and Fan Chung from the University of California, San Diego. Independently, a computer scientist, William Cutler at Chicago Rawhide wrote a program that confirmed that the mathematicians' answer was correct. The Stomachion, according to 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.

Only In November 2003, Bill Cutler found that 536 possible distinct arrangements of the Stomachion's 14 pieces into the shape of a square exist, where solutions that are equivalent by rotation and reflection are considered identical and thus counted only once.

"People assumed there wasn't any combinatorics in antiquity," Netz said. "So it didn't trigger the observation when Archimedes says there are many arrangements and he will calculate them."

As for the name, derived from the Greek word for stomach, mathematicians are uncertain but Persi Diaconis has a funny explanation: "It comes from 'stomach turner,' " he said. "If you get involved with it, that's what happens."

Testimonia , Ludus «loculus Archimedius»

http://www.mcs.drexel.edu/~crorres/Archimedes/Stomachion/intro.html

http://www.mcs.drexel.edu/~crorres/Archimedes/Stomachion/Pick.html

Proof of Pick's theorem

Pegg, E. Jr. "Math Games: The Loculus of Archimedes, Solved." Nov.17, 2003,http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html.

Pegg, E. Jr. "The Loculus of Archimedes, Solved." http://library.wolfram.com/infocenter/MathSource/5108/.

Eric W. Weisstein. "Stomachion." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Stomachion.html