This month's topics:
OMG! Math is cool!
"OMG! Actress and mathematician Danica McKellar wants girls to know that being good at numbers is cool." That's the subtitle from a piece Peg Tyler wrote for the for the August 6 2007 Newsweek. McKellar, who portrayed Winnie in the '90s TV serial "The Wonder Years," has written a book (entitled Math Doesn't Suck) "aimed at helping young girls survive--and even thrive--in math class." She "uses cutesy graphics and teen-magazine staples like personality quizzes, horoscopes and straight-from-the-mall examples to spell out often confusing concepts like reciprocal fractions and prime factorization." Her own story: "I was good at math in elementary school, and then around seventh grade the work got harder and I hit a rough patch." She pulled out of it thanks to an insightful and gifted teacher, went on to major in math at UCLA ("I thought it was just for nerdy white guys, but it's not"), and graduated with high honors. She even left her name on the Chayes-McKellar-Wynn Theorem (J. Phys. A: Math. Gen. 31(1998) 9055-9063) which makes her, according to Tyler, "the only television actress in America to coauthor a groundbreaking mathematical physics theorem." Among her recommendations to girls: "don't be afraid to sound dumb in class. Go ahead and ask."Electrical geometry: the Elephant-nose fish
Elephant-nose fish trained to discriminate by electrolocation between solid cube and pyramid were able to transfer their learning to wire models of the polyhedra, and even maintained the distinction when the vertical wires were interrupted. The modified models were presented in blocks of electrically transparent agar.
The September 1 2007 Journal of Experimental Biology has an article about the perceptual system of the Elephant-nose fish Gnathonemus petersii. These fish, typically 12-15 cm long, hunt at night in the murky waters of African rivers. Gnathonemus "sees" in the dark by detecting how external objects modify a weak electric field that it generates and that surrounds its body. Even though the images produced by this method (called electrolocation) are necessarily far out of focus, the fish can actually make very fine distinctions. Gerhard von der Emde and Steffen Fetz, the authors of "Distance, shape and more: recognition of object features during active electrolocation in a weakly electric fish" ran a series of tests of Elephant-nose fish perception in their laboratory in Bonn. In one of them, they showed that fish who had been trained to discriminate between a cube and a pyramid were able to maintain the distinction when the objects were replaced by wire models, and even when those models were modified by cutting segments out of all the vertical elements (the wires were embedded in blocks of electrically transparent agar to maintain proper relative position). As the authors say, "The fish were apparently able to complete the interrupted outlines, perceiving something similar to an illusionary contour."
Le Monde picked up this research news on August 28, with the headline: "Les calculs géometriques d'un poisson électrique." Their reporter contacted van der Emde, who remarked that Gnathonemus is "by far the most intelligent of the electric fish, and very nice to work with."Origami pinecones
Nature, on July 26, 2007, ran a "News and Views" piece by Ian Stewart about a new breed of mathematically inspired origami. Stewart begins by reminding us of the mathematical complexity hidden in this ancient Japanese art. "The basic problem of origami is the flat-folding problem: given a diagram of fold lines on a flat sheet of paper, can the paper be folded into a flat shape without introducing any further creases? ... [T]his question is ....an example of an NP-hard problem." Taketoshi Nojima (Department of Aeronautics and Astonautics, Kyoto) has recently published a series of papers where, among other things, he shows how to crease a sheet of paper so that it folds flat, but can also be uncompressed into a conical structure presenting equiangular spirals analogous to those produced by phyllotaxis. For example, the following fold diagram, with the dotted lines interpreted as "ridges" and the solid lines as "valleys," gives a flat object which, after stretching to bring the opposite vertical edges into coincidence, produces a cone:
Folding diagram for origami pinecone. The angles and lengths are carefully calculated so as to satisfy the local flat folding criterion (around each vertex, the sum of every other angle must be π), to ensure that the edges and the diagonals form piecewise equiangular spirals (with respect to an origin at the center of the circle implied by the lower dotted edge), and finally to ensure that the the free vertical edges match up properly. Image courtesy Taketoshi Nojima.
This cone was assembled from the diagram above, enlarged by a factor of 3.
Note that unlike the cones produced by phyllotaxis, this one has all three sets of equiangular spirals turning in the same direction. More "natural" configurations are also possible (see "Origami-Modellings of Foldable Conical Shells Consisting of Spiral Fold Lines," by Nojima and Takeuki Kamei, Trans. JSME 68 297-302 (2002) - in Japanese).The Myth. The Math. The Sex.
"Everyone knows men are more promiscuous by nature." That's how Gina Kolata starts her piece "The Myth. The Math. The Sex." in the New York Times for August 12, 2007. We even have darwinian explanations for the phenomenon, with woman being "genetically programmed to want just one man who will stick with her and help raise their children." Surveys bear this out: Kolata mentions a British study which "stated that men averaged 12.7 heterosexual partners in their lifetimes and women, 6.5." Whoa! It turns out this is mathematically impossible. Kolata refers to David Gale, who sanitizes the context and gives us the
If the numbers of men and women in the active heterosexual population are the same, as they approximately seem to be, the HSP Theorem does indeed imply that the average number of partners must be the same for both sexes. This should settle the matter. But Kolata makes the error of mentioning one study which reported an (almost identical) difference in the medians of the two distributions; this earns her a rebuke from Jordan Ellenberg, Slate's math guru: "Mean Girls: The New York Times slips up on sexual math" (August 13, 2007). "It's not every day I get to read a mathematical theorem in the New York Times, so I hate to complain. But Kolata isn't quite right here." Ellenberg goes on to give obvious examples of different medians with the same mean. Towards the end of the piece he acknowledges that some of Kolata's examples did in fact involve means; he changes tack and quotes serious studies of the problem of inaccurate self-reporting (unreliable memory plays a part). Kolata's essay is available online, thanks to the Dallas Morning News.
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