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Tony PhillipsTony Phillips' Take on Math in the Media
A monthly survey of math news

November 2000

* The tiniest dodecahedron. Scientists have achieved the synthesis of the C20 fullerene with the geometry of a dodecahedron. The C20 molecule has two other isomers, a "bowl" and a "ring;" all three were synthesized. The work is reported in the September 7 2000 Nature in a Letter by a US-German team led by Horst Prinzbach of Freiburg. The molecule was synthesized starting from C20H20 dodecahedrane by first replacing the hydrogen atoms with bromine, and then "debrominating." The substance existed in the gas phase for at least 0.4 milliseconds. This report was picked up by ScienceWeek in their October 13 e-issue.

This image shows the three isomers of C20 and the photoelectron spectra used to differentiate them. Horizontal coordinate is binding energy (eV); vertical coordinates are intensity in arbitrary units. Images reprinted by permission from Nature 407, 60 - 63 (2000) Macmillan Magazines Ltd.

* Strange attractors onstage. In the October 19 2000 New York Times, Jennifer Dunning reviewed a dance presentation at the Joyce Theater: "Strange Attractors," choreographed by Stephen Petronio. "The title ... comes from chaos theory and its probing of how apparently random behavior can occur in a world governed by deterministic laws. Program notes define a strange attractor as `a moving and magnetic focal point in a seemingly chaotic field.' Mr. Petronio has found the perfect analogy for his own approach to choreography." Ms. Dunning gives us an idea of how the mathematical abstraction works out on stage: "The protean empty space between ranks and ensembles of dancers seems the closest thing to a focal point ..., drawing and pushing out dancers as it shifts about."

* Incompressible is incomprehensible. Why are some things so hard to understand? Jacob Feldman of the Rutgers Psychology Department has an answer, reported in the October 5 1000 Nature. He found in a large set of experiments that for human learners, "the subjective difficulty of a concept is directly proportional to its Boolean complexity (the length of the shortest logically equivalent propositional formula)-that is, to its logical incompressibility." For example a concept which encodes as (A and B) or (A and not B) is equivalent to A and (B or not B), i.e. to A and so can be compressed to Boolean complexity 1. Whereas (A and B) or (not A and not B) cannot be compressed and has complexity 4. Subjects were asked to extract the concepts from sets of examples and non-examples. Main conclusion: "For each concept, learning is successful to the degree that the concept can be faithfully compressed." Feldman reflects on his result: "In a sense, this final conclusion may seem negative: human conceptual difficulty reflects intrinsic mathematical complexity after all, rather than some idiosyncratic and uniquely human bias. The positive corollary though is certainly more fundamental: subjective conceptual complexity can be numerically predicted and perhaps explained."

* A new Federal encryption algorithm was reported in the October 20, 2000 Chronicle of Higher Education. The article, by Florence Olsen, relates how the Commerce Department, after a 4-year search, has declared the new federal standard for protecting sensitive information to be Rijndael, an algorithm named after its inventors Vincent Rijmen and Joan Daemen. The two Belgians beat out 20 other entries, including teams from IBM and RSA. The new encryption algorithm, of which no mathematical details were given, can be made stronger as more powerful computer processors are developed. This was an entry requirement for the competition. According to Raymond G. Kammer of NIST, which managed the selection process, it should be good for about 30 years, "that is, if quantum computing doesn't manifest itself in five or six years."

*The geometry of a fender-bender. "Dynamics of singularities in a constrained elastic plate" is the title of a Letter in the October 12 2000 Nature, submitted by a French-Portuguese team led by Arezki Boudaoud of the ENS. But at the end the secret comes out: "The experiment we have performed is typically a controlled version of what happens when a car is bumped." The work involves geometry (surface curvature with cone singularities) numerics (simulation of solutions to the Föppl von Kármán equations) and careful experiment.

-Tony Phillips
SUNY at Stony Brook

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