This month's topics:
"A Twist on the Möbius Band"
That's the title that Julie J. Rehnmayer used for her Science News Online report on recent answers to the question: when an inelastic rectangle (for example, a strip of paper) is twisted into a Möbius band in 3-dimensional space, what exactly is the resulting shape?
An rectangle with side ratio to 1 can be folded into a Möbius Band by reassembling it as a trapezoid (a), folding along the blue dotted line (b), and then folding along the green. The last fold (c) brings into congruence, with proper orientation, the sides that are to be identified. This configuration is the limiting case of the embeddings studied by Starostin and Van der Heijden.
When the side ratio is to 1, the strip can be folded into a configuration that respects the edge identification. For narrower strips the band assumes a "characteristic shape" minimizing the total bending energy; the exact determination of this shape has been an oustanding problem at least since 1930. Evgueni Starostin and Gert van der Heijden (University College London) recently nailed down the solution using "the invariant variational bicomplex formalism" and numerical methods. (The variational bicomplex is, according to Ian Anderson, a double complex of differential forms defined on the infinite jet bundle of any fibered manifold π : E --> M.) They report: "Solutions for increasing width show the formation of creases bounding nearly flat triangular regions..." .
Three of six characteristic shapes for length = 2π and various widths shown in Srarostin and Van der Heijden's article: width 0.2 (b), 0.8 (d) and 1.5 (f). "The colouring changes according to the local bending energy density, from violet for regions of low bending to red for regions of high bending." Image from Nature Materials Online (posted July 15, 2007), used with permission.Geometry and the Imagination
Bill Thurston (in Princeton, March 1990)
The 5-day conference with this title, held at Princeton on June 7-11 in honor of Bill Thurston's 60th birthday, was surveyed by Barry Cipra in the July 6 2007 Science. Cipra's 2-page spread covers four of the presentations.
Nature's Michael Hopkin interviewed The Simpsons' executive producer, Al Jean, for a News Feature item (July 27, 2007). Jean, it turns out, was an undergraduate math major at Harvard and has kept some of the ethos in his new job. "I look at comedy writing mathematically, it's sort of like a proof in which you're trying to find the ideal punchline for a setup, and when you get it it's a very elegant feeling." He and his fellow writers enjoy slipping obscure mathematical references into the Simpsons' storyboards. Contestants in a quiz will have to choose between various numbers "and each of the options is a different mathematical irregularity -- one's a perfect number, one's a sum of four squares. They're all in the thousands and they're numbers that nobody except a mathematician would, at face value, recognize as anything unusual, but if you're really sharp you'll pick it up." The closest he comes to what passes for humor at tea in math departments is the witticism referred to in the title: "...we did an episode where Apu was a witness in a courtroom and the lawyer asked if he had a good memory. He said, yes I do, I've memorized pi to one million decimal places, and Homer said "mmm... pi" and started drooling."
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