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A monthly survey of math news
Math in The Lancet. The February 17, 2001 issue of the #1 medical journal features a review by Seamus Sweeney of ``The mystery of the aleph: mathematics, the Kabbalah, and the search for infinity" by Amir D Aczel (New York: Four Walls Eight Windows, 2000), a book about Georg Cantor's life and his discoveries in the theory of sets. Perhaps the profession's interest was sparked by Poincaré's judgment, quoted in the first sentence of the review, that Cantor's work was "a malady, a perverse illness from which someday mathematics will be cured." Sweeney gives the book a careful and sympathetic review, including a presentation of Cantor's famous diagonal argument establishing the uncountability of the real numbers. He does get a bit carried away: ``Reading of the dizzying orders of infinity that Cantor explored, one feels perhaps that maths and music are the closest humanity can get to any sense of the divine." This must sound odd to the M.D.s, many of whom confront transcendent problems on a daily basis. Sweeney concludes: ``Indeed the rarefied world of infinity and its relationship with the divine is perhaps the most beguiling seductress mathematics can rely on to persuade the reflex numerophobes conditioned to see mathematics as dry, soulless, and worst of all, boring." As other accessible introductions to the world of pure mathematics he recommends Paul Hoffman's ``The man who loved only numbers" (New York : Hyperion, 1998) and John D Barrow's ``Pi in the sky" (Oxford : Clarendon Press ; New York : Oxford University Press, 1992.)
Solitons in matter. Solitons, or solitary waves, were first discovered as surface waves in canals. They manifest solutions of the non-linear wave equation which have the remarkable property of maintaining their form unchanged as they propagate. Eran Sharon, Gil Cohen and Jay Fineberg, three members of the Racah Institute of Physics in Jerusalem, have a ``Letter to Nature" in the March 1, 2001 issue where they show how perturbations to a crack front in a brittle material result in long-lived and highly localized waves (`front waves') with many of the properties of solitons. They conclude that (presumably novel) nonlinear focussing processes, ``perhaps analogous to processes that occur in classical soliton formation, are at play."
Teenager finds new triangle theorem. The March 3, 2001 Atlanta Constitution ran a piece entitled "Theoretically, teen's a geometry whiz," by Kirk Kicklighter. It tells the story of Josh Klehr who on May 8, 1999, his last day of eighth grade, discovered a new concurrence theorem for triangles. In retrospect, it is the circumcenter theorem in the Minkowski metric ds2 = dx2 - dy2, but no one had ever noticed it before. Congratulations, Josh! The theorem is now known as the Klehr-Bliss Theorem. If we call the Minkowski perpendicular bisector of a segment with slope a the line through its midpoint with slope 1/a (the two lines are orthogonal in the Minkowski metric), the Klehr-Bliss theorem says that the Minkowski perpendicular bisectors of the three sides of a triangle are concurrent. The proof follows the usual proof: any point on the Minkowski perpendicular bisector of a segment is Minkowski equidistant from the two ends. For the proof of a more general statement, see Floor von Lamoen, Morley related triangles on the nine-point circle, Amer. Math. Monthly 107, 941-945.
God, Stephen Wolfram, etc. What has Stephen Wolfram, alumnus of Eton and Oxford, veteran of Argonne, CalTech, and the Institute for Advanced Study, MacArthur Fellow at age 21, been doing since his release of Mathematica (``the most popular scientific software ever made") in 1988? He has been planning the complete mathematization of science, and the overhaul of mathematics itself, through his work on cellular automata. This from a long essay by Michael S. Malone, in the online Forbes ASAP for November 27, 2000, entitled ``God, Stephen Wolfram, and Everything Else." Cellular automata go back to Von Neumann, but gained wide fame through John Conway's game ``Life" (try Alan Hensel's Java implementation online). How will Wolfram bring about his revolution? To a mathematician the article does not offer any useful clues. The one specific example given, the pattern of markings on a Textile Cone Shell, fits into perfectly conventional science, but it is not clear whether this example is to be taken literally or not, i.e. whether this remark is relevant. A piece apearing in Forbes, and containing statements like ``Everything from cars to cartoons, from farms to pharmaceuticals, may reflect the richness of the natural world as seen through Wolfram's cellular automata" and "Within 50 years, more pieces of technology will be created on the basis of my science than on the basis of traditional science," inevitably sounds more like the publicity for an IPO than the presentation of news about current scientific research. The beautiful and moving initial image (the 2-billion-tile rose generated from black and white squares laid according to "half a dozen ... abitrary rules") typifies the essay. We do not know if the rose is fact or metaphor. We have no way of judging if the tremendous technical developments hinted at are fact or science fiction. ``A New Kind of Science," Wolfram's magnum opus on the topic, is promised for sometime this year (see the website).
Calculus with Mother Hen. The March 30, 2001 edition of The Chronicle of Higher Education tells the story of ``Operation Mother Hen," a Web-based teaching tool for students who are ``chicken about calculus," and a new approach to the perennial problem of high drop-out and failure rates. Mother Hen is the nom de plume of Ann Piech, a Buffalo mathematics professor and calculus teacher. Each of her lectures is broken down into six concepts and accompanying problems, and each of the six segments is posted as a separate video clip on the Web: ``at any hour of the day or night a student who is having trouble with, say, polar coordinates or improper integrals can get a mini-tutorial in the subject with a mere click of the mouse." Check it out at http://motherhen.eng.buffalo.edu.