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Tony PhillipsTony Phillips' Take on Math in the Media
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Ant Geometry


Exits on the Pheromone Highway branch on average 53o from the "away" direction, so each intersection reads like an arrow pointing home.
Image © Duncan Jackson, used with permission.

"Pheromone trails are used by many ants to guide foragers between nest and food. But how does a forager that has become displaced from a trail know which way to go on rejoining the trail?" Richard Feynman (in "Surely you're joking ...") considered this problem and speculated that a direction might be written into the pheromone trail (e.g. A-B-space-A-B-space). In fact, the ants use information encoded in the geometry of the (plentiful) bifurcations along the trail. This has been conclusively shown by Duncan Jackson, Mike Holcombe and Francis Ratnieks, a computer science-biology team at the University of Sheffield; work reported in the December 16 2004 Nature. The Sheffield team worked with colonies of Pharaoh's ant (Monomorium pharaonis). In one of their reported experiments, they allowed individual ants to walk along experimental, straight trails: "any reorientations occurring were as likely to be correct as incorrect in relation to the polarity of the trail when it was originally formed." In contrast, when meeting a trail bifurcation, "43% of fed ants [who would presumably be heading home] made U-turns upon meeting the bifurcation point when walking in the 'wrong' direction: that is, away from the nest. Conversely only 8% of fed ants walking the 'correct' way made ... corrections that led them to heading incorrectly away from the nest." On the other hand, of the unfed ants, who presumably would be heading out for food, "47% made course corrections at the bifurcation point when moving the 'wrong' way" while "only 8% walking the 'correct' way made incorrect course changes." The authors add: "Note that although only 45% of the ants corrected their orientation at a single bifurcation, real networks contain many bifurcations and many opportunities for course correction." [Interpreted as Markov chains, these numbers say that in the steady state 84% of the fed ants and 85.4% of the unfed will be moving in the appropriate direction. -TP]

The Magic of Math, in Queens

On November 24, 2004 the New York Times ran "From Internet Arm Wrestling to the Magic of Math," Edward Rothstein's review of the new wing of the New York Hall of Science, in Queens. Rothstein glances at the high-tech baubles of the new installations, but saves most of his admiration for the Mathematica exhibition, which the Hall of Science recently acquired from the California Science Center. Mathematica was created for IBM in 1961 by the celebrated design team of Charles and Ray Eames. Rothstein remembers seeing it as a child: "I still recall wired structures rising out of soapy liquid, their swirling surfaces demonstrating solutions of mathematical problems; the cubic array of bulbs that translated simple multiplication into three-dimensional patterns of light; the suspended Moebius strip - a surface with only one side and one edge - on which a train continuously ran." And he ponders the difference between this exhibition, assembled at the apex of the post-sputnik wave of enthusiasm for science, and the flashier but shallower productions of today. "Mathematica samples varied branches of mathematics, not blanching from explaining functions or projective geometry; contemporary exhibitions set their sights lower, restricting each display's focus. Mathematica knows you won't fully understand it all ... . Contemporary displays are more concerned that you grasp a single concept. They are play stations in a science lesson."

Father of fractals

That's the title of Jim Giles' News Feature in the November 18 2004 Nature. Benoit Mandelbrot is the Father; the article is illustrated with a large and spooky image of part of "the set that bears his name." Giles gives us a capsule intellectual history of Mandelbrot, taking him from his 1963 paper on self-similarity in graphs of cotton prices, through his years as an "academic wanderer" and the 1982 publication of The Fractal Geometry of Nature, when "the worlds of math and physics took notice." After a brief and completely non-technical digression on fractals, we come to the main point of the paper: Mandelbrot's attitude. Apparently, he has not been very nice. "As so often happens in academia, questions of precedence were central." We hear reports of aggressivity and misbehavior at conferences. Then Giles focuses on Mandelbrot and Vilfredo Pareto (1848-1923), who had published "similar studies on power laws in enonomics" many years before. Giles claims that in the most recent reprinting of Mandelbrot's 1963 paper on cotton, "many references to Pareto have been removed." And that one paper by a third author, Mandelbrot and the stable paretian hypothesis, appears in the same collection with a new title: Mandelbrot on price variation. Are we supposed to be horrified? In fact Giles ends up fairly conciliatory: "Even researchers who have been the subject of his attacks praise his contributions to maths." [A deeper analysis would examine the divisions in in post-war french society, politics and science (even mathematics!), and how they played themselves out in Mandelbrot's career. -TP]

Tony Phillips
Stony Brook University
tony at math.sunysb.edu