Tony Phillips' Take Blog on Math Blogs

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February 2001

 Plimpton 322, George A. PlimptonCollection, Rare Book and Manuscript Library, Columbia University.Click for larger image.
Rereading Plimpton 322. Plimpton 322, the tiny dark star of Columbia University's Rare Book Library, is probably the world's best known piece of Mesopotamian mathematics. A clay tablet, about 5 inches long and 3.5 inches wide, with 4 columns and 15 rows of cuneiform hexadecimal numbers, it is usually dated 1900-1600 BC. Sir Christopher Zeeman lectured on it in San Antonio in 1995 AD, calls it the oldest preserved document on number theory," and gives a mathematical argument for its interpretation as a set of pythagorean triples. Lectures available online. The January 27 2001 Science News has a piece by Ivars Peterson describing a recent reevaluation of the mathematical content of the tablet. The work is due to Eleanor Robson (Oriental Institute, Oxford) and is presented as an example of new scholarly approaches to Mesopotamian mathematics" which combine historical, linguistic and mathematical techniques." Dr Robson pins down the date of the tablet to 1800 +/- 40 BC, and gives an alternative explanation of the tablet's purpose: it served as a guide for a teacher preparing exercises involving squares and reciprocals." Despite this downgrading from number theory to arithmetic, Robson considers Plimpton 322 the epitome of Mesopotamian mathematical culture at its best. ... It's a well-organized, well-executed, beautiful piece of mathematics."

"Loving Math Infinitely" is the title of a piece by Josephina Alvarez, professor of mathematics at New Mexico State University at Las Cruces, that ran in the Chronicle Review section of the January 19, 2001 Chronicle of Higher Education. Prof. Alvarez describes her experiences teaching a course on mathematics appreciation for nonscience majors." This is a rich piece, with many valuable ideas for mathematics teachers at any level. She finds that in order to reach her students, she has to look into their training and interests. I need to learn to use their language. My challenge is to extract the mathematical ideas in music, philosophy, art, and other seemingly nonmathematical fields." She elaborates on As soon as I knew what I should be looking for, I started to see mathematics in unexpected places." with many examples, and concludes: Interestingly enough, once we saw mathematics as firmly rooted in human need and experience, it seemed easy to depart on a tour of increasing abstraction." To summarize her article in two sentences: Mathematics appreciation is more than a course. It is an attitude that we should cultivate in every mathematics course."

How do fish swim? The dynamics of swimming fish and flapping flags involves a complicated interaction of their deformable shapes with the surrounding fluid flow." This is the beginning of a letter to Nature" (14 December 2000) from a Courant Institute/Rockefeller University team headed by Jun Zhang. Their research used flexible filaments in a flowing soap film 3-4 microns thick. In particular they report that beyond a certain critical length the system becomes bi-stable, with both a stretched-straight state" and a stable flapping state" possible.

 A snapshot of the flapping state. Note that theend of the filament is near the center of the image. Image from NYUWetlab, used with permission. Click for larger image.
The stable flapping state has an especially simple mathematicalform: Unlikea simple pendulum, the undulation is well fitted by a travelling harmonic wave with aspatially varying envelope: . Here, y(x, t) is the horizontal displacement of the filament from the centre-line,measured at a vertical distance x from the fixed point for time t. f(x) is a spatialenvelope function (increasing monotonically from the fixed point), nu is the flapping frequency and lambda the wavelength."

Aninteresting final point: Swimming offers alternatives comparable to the bistability of our filament. Thestretched-straight state is the analogue of a glide, whereas the flapping state isanalogous to swimming. Efficient propulsion uses the natural oscillations of theswimmer, which in the filament is a property mediated by stiffness."A web presentation of this research is available.

Digital Archimedes. The thousand-year-old palimpsest that turned up on the front pages in 1998 is undergoing a thoroughly modern analysis. A report entitled Through the Layers, a Glimmer of Archimedes" by Lawrence Biemiller (The Chronicle of Higher Education, January 26, 2001) describes this treasure: a precious euchologion, a 10th century Greek prayer book, was written on parchment that had previously been used for a collection of works by the great Archimedes. Even though the pages had been scraped clear of the original text, enough remained for the Danish scholar Johan Ludvig Heiberg to recognize. The discovery (published in 1907) was sensational: among other works the manuscript contained a copy of The Method," an important work of Archimedes that had been thought completely lost. Biemiller tells the story of how this item came to be sold at Christies' in 1998 (for \$2 million) and how the present owner has agreed to pay for the Walters Art Gallery in Baltimore to take over the conservation of the manuscript and the computer-aided digital imaging of its pages. Two teams, one from Johns Hopkins and one from the Rochester Institute of Technology, are collaborating with the Walters. They predict a much more complete transcription of the text and, quite important, access to the original drawings. The project should take four years.

Math teachers are nerds ... not. John Dunford, a math teacher himself (and general secretary of the somewhat ominously named Secondary Heads Association) takes issue with a survey of 12-year-old schoolchildren's impressions of their maths teachers." In a piece in the January 4, 2001 Guardian he defends the profession from the merciless gaze of the young: Fat, bald nerds with glasses and beards apparently predominate at the front of maths classrooms. These sad figures are apparently seen as unmarried and unstylish, wrinkled from all the hard thinking that they have had to do in order to solve maths problems throughout their lives." Not so, says Dunford. Speaking of his colleagues over the years: I cannot really recall anyone who might have been described as a nerd. ... family people mostly, with a good range of interests from swimming and badminton to reading and cooking." But wait! There could be an age factor too. I may not find my colleagues nerdish, but I do recall my own maths teachers at school as coming into that category." The piece is available online.

Math and Fiction. Incompatible careers? Not if you ask Manil Suri, expert in partial differential equations, Professor of Mathematics at the University of Maryland Baltimore County, and author of a New Yorker short story (The Seven Circles," February 14, 2000) and a novel The Death of Vishnu" (Norton, January 2001). Suri was interviewed by Ivar Stakgold in the January-February 2001 SIAM News. The interview ranges over many topics, but keeps returning to the similarities and differences between doing math and writing fiction. Suri: What's a little eerie is how similar the thought processes can be for the two activities. Suppose I am trying to decide whether a mathematical entity X is bounded. I might try to think of varous ways that X might try to misbehave ... . Fiction presents a similar scenario -- perhaps X is now a character in a certain situation. To find out what happens next, I would try to put myself in X's place, looking for all the ways I could proceed, maneuvering around any of the story's imposed constraints, and choosing the most interesting path." He mentions a reading at MSRI after which a colleague told him It's all about Fourier series," i.e. that all fiction is composed of the same Fourier modes: love, death, marriage, sex and so on, and one gets different stories by assembling different combinations of these basis functions." Prof. Suri knows how to milk a metaphor: Certainly a key strategy to getting fiction published is to orthogonalize to what has been written before."

-Tony Phillips
Stony Brook