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July/August 2004

Mathematical Origami.

 One of David Huffman's creations, made from a single, folded sheet of paper: four parabolic curved folds meet in a central square. Image from Paul Haeberli's Grafica Obscura website, used with permission.

"Cones, Curves, Shells, Towers: He Made Paper Jump to Life" is a piece by Margaret Wertheim in the June 22 2004 New York Times. She writes about David Huffman, a computer scientist who died in 1999, and his work on mathematically informed origami. As the image above exemplifies, Huffman's specialty was folds along curves. He wanted to be able "to calculate precisely what structures could be folded to avoid putting strain on the paper." Huffman, who is best known for the "Huffman codes" he discovered as an MIT graduate student, is also "a legend in the tiny world of origami sekkei," or computational origami. He published only one paper on the subject but his models and his notes are being carefully studied by today's mathematical paper-folders. Wertheim quotes Robert Lang: "he anticipated a great deal of what other people have since rediscovered or are only now discovering. At least half of what he did is unlike anything I've seen." And Michael Tanner, who says that what fascinated Huffman above all else "was how the mathematics could become manifest in the paper."

Local Boy Makes Good?

 The Zaman Daily Newspaper (Istanbul) online edition ran a dispatch dated May 18, 2004 from Omer Oruc in Izmir. "200 Year Old Math Problem Solved" is the headline; the story tells how Mustafa Tongemen, a retired mathematics teacher, has solved "a math problem brought forward by the Italian mathematician, Malfatti, in 1803," after working on the problem two hours a day for seven years. "Malfatti's problem aims to extract three circular vertical cylinders from a triangular vertical prism made of marble, with the least material loss." [In fact, there are two non-equivalent problems, initially confused by Malfatti. The problem just posed, which Conway calls problem (1), and the problem of inscribing three mutually tangent circles in a triangle, problem (2). In an equilateral triangle, in fact, the incircle and two smaller inscribed circles give a larger area than the three mutually tangent circles. Malfatti solved problem (2), which, as is clear from the image on the Zaman website, is the problem Mustafa Tongeman actually addressed. The much more difficult problem (1) was only solved in 1992. See the Historia Mathematica Mailing List Archive for a summary of the history and the source of the equilateral counterexample. -TP] The Purdue University Purdue News ran "Purdue mathematician claims proof for Riemann Hypothesis" on June 8, 2004. "Louis de Branges ... has posted a 124-page paper detailing his attempt at a proof on his university Web page. While mathematicians ordinarily announce their work at formal conferences or in scientific journals, the spirited competition to prove the hypothesis -which carries a \$1 million prize for whoever accomplishes it first- has encouraged de Branges to announce his work as soon as it was completed." The jury is out on this one. On June 22, 2004, the Daily Star (Beirut) ran May Habib's online story: Has local mathematician proven the `5th Postulate?' "Rachid Matta, a Lebanese mathematician and engineer, claims to have proven Euclid's parallel theorem - a theorem that has remained unproved since Euclid wrote it in 300 BC and one that many of the world's greatest minds have deemed improvable. If verified, Matta's work could have an enormous impact on mathematics because both elliptical and hyperbolic geometry - branches of geometry that violate the parallel theorem - would be discredited." We are told that Matta has spent 10 years working on this problem.
Mind, Music and Math. From the desk of New York Times cultural critic Edward Rothstein comes "Deciphering the Grammar of Mind, Music and Math" (June 19, 2004). The piece, under the Connections rubric, is a meditation on the nature of musical intelligence, in the light of recent work on the neural concomitants of musical perception and on the differences between the way the brain processes speech sounds and music. Rothstein emphasizes the "unique" power of music: how, even if completely unfamiliar music is heard in a locked room, where there is no reference to the outside world, "it can teach itself. Gradually, over repeated hearings ... music shows how it is to be understood. ... Sounds create their own context. They begin to make sense. ... Music may be the ultimate self-revealing code." He goes on: "This is one reason that connections with mathematics are so profound. Though math requires reference to the outside world, it too proceeds by noting similarities and variations in patterns, in contemplating the structure of abstract systems, in finding the ways its elements are manipulated, connected and transformed. Mathematics is done the way music is understood." The moral of the story: "This means that music can be fully understood only by maintaining access between the room and the world: neither can be closed off."

-Tony Phillips
Stony Brook