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January 2004

Archimedes Combinatorist.

 The Stomachion was also the name of this children's game, involving arrangements of the 14 shapes shown here, like a tangram puzzle. The vertices are all on the 12 x 12 grid. Archimedes asked the mathematics question: How many different ways can the pieces be arranged into a square? It turns out that the pieces separated by pale-shaded lines will always be joined as shown. The New York Times shows two solutions besides this one. "In Archimedes' Puzzle, a New Eureka Moment" was a front-page story by Gina Kolata in the December 14 2003 New York Times. The puzzle appears in the Archimedes Palimpsest, the 800-year-old Greek Orthodox holy book that had been overwritten on a 200-year-old scraped-off parchment copy of several of his works. Since the work's purchase by "an anonymous billionaire" at Christie's in 1998, the book has resided at the Walters Art Museum in Baltimore. There it has been examined by scholars using ultraviolet light and computer imaging techniques to tease out as much as possible of the Archimedean text still legible under the Christian prayers. One of the scholars is Reviel Netz of Stanford, who has recently deciphered parts of the Stomachion ("sto-MOCK-yon") among the pages of the dissassembled, erased and reassembled manuscript. The Stomachion, Kolata tells us, "has attracted the least attention" among all of Archimedes' works, "ignored or dismissed as unimportant or unintelligible." All that had been known was "a tiny fragment of the introduction," and it seemed to be about a children's puzzle, a third century B.C. greek tangram, involving 14 different geometric shapes. Netz has determined that what was really puzzling Archimedes was the computation of how many different ways those pieces may be assembled into a square. This makes him the first mathematical combinatorist, by a long shot. The answer is 17,152. Netz is sure that Archimedes had solved the problem ("or else he would not have stated it") but we do not know if he had the right answer.

Michael in India. Sir Michael Atiyah's visit to the subcontinent this fall has been noted in the local electronic press. ExpressIndia's Delhi Newsline for November 8, 2003 has a page one story by Amba Batra entitled: "Maths guru with Einstein's dream prefers chalk to mouse." Batra interviewed Atiyah, who refers to his current work linking geometry and physics as "a continuation of Einstein's dream," and states: "It will help us understand the forces of gravity, magnetism, and nuclear forces and give them geometric configurations." Batra quotes Delhi University Professor Dinesh Singh on Atiyah's reception: "The students have not left his side for a moment, their lunch today extended for more than four hours." (Atiyah: "I have never received the kind of audience that I have in India. I feel like a pop star.") On November 18 the Mumbai Newsline interviewed Sir Michael "in a quiet study at the Tata Institute." When asked about mathematics in 2050, he volunteered: "We're on the verge of a major revolution in biology, with big problems nobody really understands. Mathematicians will provide models on how the human brain works and to make sense of the human genome project. When any science gets complicated, you need mathematics."

Calculators vs. math skills. ABCNews.com has leapt into the controversy with a piece by Marc Levenson: Sci-Tech, December 11, 2003. The title is actually "Do High-Tech Calculators Take the Challenge Out of Learning Math?" and the focus is on CAS (Computer Algebra System) -equipped calculators, the kind that can solve an equation or calculate an indefinite integral. Levenson presents the pro (Henri Picciotto, math teacher at the Urban School in San Francisco: "The idea is, if you automate some of the more routine tasks, the kids can think about more challenging and interesting questions. It's as if I prevented you from writing if you don't know how to spell. Certainly, you should learn how to spell. But does that mean I should prevent you from using a spell checker when you may have interesting things to write about?") and the con (Berkeley's Hung-Hsi Wu: "I want them to learn how to think. That's the greatest virtue in having a good mathematics education. If you're a high school student, you're only beginning to learn the elementary thinking processes. The beginner needs the time, needs a little drudgery, even, to acquire that thinking skill.") He also interviews some students at the Urban School, like Aaron Brown, who remarks: "in some ways it's easier to know how to do it on paper than on the calculator."

Largest prime yet. "MSU student's prime number largest one yet" is a story by Sharon Terlep in the December 4 2003 Lansing State Journal. "Michael Shafer, a 26-year old chemical engineering student, made math history by discovering the largest prime number known." Shafer did it by running a program that "hooked up .., more than 200,000 computers world-wide." The program had been running for 19 days when "an alarm sounded letting him know his computers tagged a prime number." The number is a Mersenne prime (of the form 2p-1, where p is prime); in Schafer's case p = 20,966,011 and the number itself has over six million digits. According to Shafer, "The number itself really isn't useful. What's more important is what's gone into developing the server and that the programs can get all these computers to work together for a common goal." And: "There may come a time when there's more important research that can harness this technology and use it for something more relevant." Terlep's story is available online, as is Largest prime number ever is found on the New Scientist News Service.

Mathe-musical instrument. It's the tritare ("TREE-tar"), it's the invention of two Canadian mathematicians (Samuel Gaudet and Claude Gauthier, both at the University of Moncton), and it may revolutionize music. The story, by Karen Burchard, ran in the November 28, 2003 Chronicle of Higher Education. Gaudet and Gauthier are number theorists. In their research on "the odd-number portion of the 'p-series' problem" they came across a class of numbers with symmetries that seemed initially to have potential in engineering but ended up instantiated in a musical instrument "shaped like an inverted Y and equipped with six networks of strings that can produce a range of sounds, from guitarlike musical notes to percussive beats reminiscent of a church bell. If one string is plucked, it vibrates across all three of the fretboards." More details are available from various canadian sources. CBC - New Brunswick shows a photograph of the apparatus. Radio Canada has link to a 4-minute streaming video interview with the two inventors and their instrumentalist (en français. Gaudet: "Tout à coup on a pensé que ça pouvait faire un méchant instrument de musique.").Guitariste.com has some hints about the design (also en français). CBC Arts News has a link to a 1-minute broadcast where you can actually hear the tritare playing in the background.

-Tony Phillips
Stony Brook