Math in the Media 0405

Math in the Media

April 2005

Differential geometry and the Venus Flytrap
Wrong, Wrong and Wrong
Amateur math in ancient Japan

$*$ Differential geometry and the Venus Flytrap.

"The rapid closure of the Venus flytrap (Dionaea muscipula) leaf in about 100ms is one of the fastest movements in the plant kingdom." Image from Nature 433, 422, used with permission.

An international team led by Yoël Forterre has used "high-speed video imaging, non-invasive microscopy techniques and a simple theoretical model" to investigate how the Venus flytrap can snap shut rapidly enough to catch its prey. The authors, reporting in Nature for January 27, 2005, argue that "the macroscopic mechanism of closure is determined solely by leaf geometry." The image on the left shows clearly that both in the open state and the closed the leaf has positive gaussian curvature, and that it starts curved outward but ends curved inward. Choosing curvilinear x and y coordinates on each half of the leaf, with x increasing in the direction of the spines, and y increasing perpendicularly to the right, they observe that the change in the principal curvature κ_x is the main actor in the phenomenon. "For a doubly-curved leaf ... bending and stretching modes of deformations are coupled, meaning that bending the leaf causes its mid-plane to be stretched. If the coupling is weak, the leaf can change its shape from open to closed by varying its gaussian curvature and stretch without a large energetic cost. In such a situation, the leaf deforms smoothly to accommodate the change in κ_x. If the coupling is strong, the leaf will not deform much (owing to the large energetic cost of stretching its mid-plane), until eventually the change in κ_x becomes so large that the leaf snaps shut rapidly." The authors derive ("poroelastic shell dynamics") a mathematical model which accurately mimics the detailed changes in leaf geometry. The Supplementary information posted on the Nature website includes a 1/4-speed video of the flytrap, tickled with a glass pipette as in the image above, snapping shut.

$*$ Wrong, Wrong and Wrong; Math Guides Are Recalled. That's the headline on an article by Susan Saulny in the March 25 2005 New York Times. The hapless New York math educators have done it again. This time, "City education officials were forced to recall test preparation materials for math exams late Wednesday after discovering that they were rife with errors, including basic arithmetic mistakes." Randi Weingarten, the head of the United Federation of Teachers, was reportedly outraged: "Tweed [the NYC Department of Education, located in the Tweed Courthouse] has no problem with excessively criticizing teachers for failing to meet its picayune mandates. But then it produces a test prep manual riddled with errors and misspellings. The hypocrisy is stunning." The Times printed two examples of questions with wrong answers and called on Alfred Posamentier, mathematician and dean of the City College School of Education, for the final word: "... in mathematics, where you have such an exact science, there is no room for error."

$*$ Amateur math in ancient Japan. Science magazine for March 18, 2005 ran a "News Focus" item by Dennis Normile, under the title "'Amateur' Proofs Blend Religion and Scholarship in Ancient Japan." Datelined Tokyo, the piece is prompted by an exhibition of Edo period sangaku (wooden tablets inscribed with geometric theorems) opening at the Nagoya City Science Museum next month. During that period (1603-1868) "when Japan was isolated from the rest of the world, a unique brand of mathematics flourished in the country's shrines and temples. Amateur mathematicians crafted geometric theorems on elegant wooden tablets ... and offered them to the gods." The exhibition is due largely to the efforts of Hidetoshi Fukagawa, a high school math teacher who stumbled upon sangaku while "looking for material to enliven his classes," and has spent decades tracking them down and deciphering their contents. Some of the theorems stated (notably Soddy's Hexlet - see Bob Allanson's animation) were published on a sangaku many years (in this case, 114 years) before their discovery in the West. And this was all the work of "amateurs." As Fukagawa puts it: "There was no academia as we know it. So samurai, farmers and merchants all felt free to study mathematics." The tablets contain theorems but, in fact, no proofs. Fukagawa again: "Ostensibly, the tablets were left as gifts to the gods. In reality, people were showing off and challenging others to work out the proof."

-Tony Phillips
Stony Brook

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