Math in the Media

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Math in the Media 0599
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Tony PhillipsTony Phillips' Take on Math in the Media
A monthly survey of math news

May 1999

  • The music of the spheres
  • Learn inference and logical rigor from a computer game?
  • Ada and the Bernoulli Numbers
  • Plot of Internet traffic against time gives a curve with the self-similarity of a fractal!
  • * The music of the spheres: a paper in the Annals of Mathematics (January 1999) gives examples of one-parameter families of 10-dimensional drums which are all geometrically different but all "sound" the same. See "You Can't Always Hear the Shape of a Drum" in this series. What makes these new examples (due to Dorothee Schueth of Bonn) especially significant is that topologically the drums have no boundaries and no handles (the simplest examples are products of 3 and 4-dimensional spheres). These are the first examples of this type. The title is Continuous families of isospectral metrics on simply connected manifolds.

    * Learn inference and logical rigor from a computer game? This is Minesweeper, a game used by Patti Frazer Lock at St. Lawrence University to give students "a better sense of what proofs are all about," according to a Dana Mackenzie piece in the May-June American Scientist. And they love it, too. She got the idea from Allan Struthers of Michigan Tech, who observes: "Putting a flag on a square is a theorem -- you know there's a bomb there."

    * Ada and the Bernoulli Numbers. The May 1999 Scientific American has an article by Eugene Kim and Betty Toole on "Ada and the First Computer." Ada is Augusta Ada Byron (her father was the poet), later Lady Lovelace. The first computer is Charles Babbage's Analytical Engine (devised around 1840). In notes to her translation of a paper about this machine (which existed at that time only on paper) she lists the set of instructions that would make it generate the Bernoulli Numbers. Andrew Granville has information about these numbers, but beware of his notation: for Granville the Bernoulli numbers are the coefficients B_i occurring in the following infinite series:

    \frac{x}{e^x - 1}=  B_0 + B_1 x + B_2 \frac{x^2}{2!} + B_3 \frac{x^3}{3!} +  B_4 \frac{x^4}{4!} +\dots

    while Ada defines B_i by the series:

    \frac{x}{e^x - 1}=  1 - \frac{x}{ 2}  + B_1 \frac{x^2}{ 2!} + B_3 \frac{x^4}{ 4!} + B_5 \frac{x^6}{ 6!} + \dots

    (Since all Granville's B_{odd} beyond the first turn out to be 0, there is no contradiction, but a renumbering is necessary; the article uses both notations at once). These numbers turn up in many different mathematical contexts, some of which (homotopy theory and differential topology) Ada and her contemporaries could not have imagined.
    Ada explains how the internal symmetries of the calculation allow ``millions of these Numbers'' to be generated by the recursive use of a small set of instructions. In spirit, this is a completely modern computer program, and even explained in a kind of proto-Fortran.

    * The plot of Internet traffic against time gives a curve with the self-similarity of a fractal! This according to Herb Brody, reporting on work at AT&T Laboratories in the May 1999 Technology Review. ("The Fractal Net"). This is very different from the behavior of telephone traffic, and reflects the fact that "Lengths of Internet sessions range over a span of 6 to 7 orders of magnitude." The best known example of a fractal curve in mathematics is the Koch Snowflake (here in an image from Bellevue Community College); the self-similarity comes from the construction. Many other examples, with animations, are in the article Visualizing Space-Filling Curves with Fractals in Communications in Visual Mathematics for August 1998.

    -Tony Phillips
    SUNY at Stony Brook

    *Math in the Media Archive