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July 2001

Poor Richard's Magic Squares. Among Benjamin Franklin's many interests was a passion for magic squares. (An NxN magic square is a square array of the numbers from 1 to N2 with the property that each row, and each column, has the same sum, which must be N2(N2+1)/2N; so a 5x5 magic square has the numbers from 1 to 25 arranged so that each column and each row adds up to 65). An item (``Number Fun with Ben'') in the May 4, 2001 Science, expanded in a piece by Barry Cipra in the April 30, 2001 e-magazine Science now, reports on a recent discovery of new magic squares in Franklin's papers. Paul Pasles, a number theorist at Villanova, discovered three squares (4x4, 6x6 and 8x8) tucked away in Frankin's published collected papers. The 8x8 example is especially interesting in that it has ``wavy'' stripes that also sum to 260. These were illustrated in Cipra's article; his image (reproduced with permission) has been animated by Bill Casselman as a Java applet.

 An 8x8 magic square devised by Benjamin Franklin. Vertical, horizontal, diagonal and wavy columns all add to 260. The wavy columns also work after 90-degree rotation! Click on image for Java animation.

Two Tony's for Proof. The Broadway play about a mathematician, his daughter and his student won two Tony awards, as reported in an article by Clifford Ridley in thePhiladelphia Enquirer for June 4, 2001. Best Play, and Best Actress for Mary-Louise Parker. The article, which is mainly about The Producers, is available online. Before the awards, Melvin Rothstein had a piece in the New York Times (June 3, 2001) ``Getting `Proof' to Work is a Delicate Equation.'' Sounds promising, but the only hint of mathematics is in the title. Also available online.

``I prove a theorem and the house expands ..." is the beginning of a poem by Rita Dove, part of the syllabus for a new course Analogy, Mathematics, & Poetry taught at Rochester Institute of Technology in Spring 2001. The course, devised by RIT Professors Marcia Birken (Math & Stat) and Anne C. Coon (Language & Literature) is the focus of an article in the June 8, 2001 Chronicle of Higher Education. The Professors `` found analogy basic in forming the patterns inherent in both disciplines..'' The reading list: ``There are three required texts, The Rules for the Dance: A Handbook for Writing and Reading Metrical Verse (Houghton Mifflin, 1998), 101 Great American Poems (Dover, 1998), and To Infinity and Beyond: A Cultural History of the Infinite (Princeton University Press, 1991). The class also watched films like Pi and looked at `lots of Escher.' '' Prof. Birken's home page has other Math-Poetry links.
The rest of Dove's poem ``Geometry,'' as quoted in the article:

 I prove a theorem and the house expands: the windows jerk free to hover near the ceiling, the ceiling floats away with a sigh.

ex on the Op-Ed page. The June 4, 2001 New York Times Op-Ed page ran a piece by Evar D. Nering, professor emeritus of mathematics at Arizona State University. The title is `` The Mirage of a Growing Fuel Supply,'' but it's all about the exponential function. Nering shows with an elegant crescendo of examples how rapidly a constant growth rate in consumption can gobble up the world's fossil fuel resources, no matter how many more deposits are discovered. ``Exponential growth ... is inexorable.''

Geometric Quantum Computation is the topic of a ``Perspectives'' article in the June 1, 2001 issue of Science. The author, Seth Lloyd, of the MIT Mechanical Engineering Department, explains some recent work in quantum computation. The new research shows how holonomy, in particular the phase changes undergone by a particle moving through a tailored electromagnetic landscape, might be harnessed as the operations of quantum computation. Lloyd describes holonomy ``...imagine yourself walking over a gently curving landscape ... you wind up back where you started ... to your surprise you are now facing the opposite direction.'' Don't try this at home, unless you live on a very small asteroid.

``Surprisingly Square'' is the title of a piece by Ivars Peterson in the June 16, 2001 Science News. Peterson is reporting on recent developments in algebra that bear on the problem: how many ways can you express a number as a sum of n squares? Carl Jacobi in 1829 found simple formulas giving the number of different ways of doing it with two, four, six or eight squares, using elliptic function theory. And there the theory stood until 1996, when Stephen C. Milne of OSU came up with ``powerful new formulas'' to extend Jacobi's calculations to higher n. Powerful, but ``hard to fathom and use,'' according to Peterson, Milne's formulas spurred a search for alternate routes to the same information. Recently modular forms, the same tools that helped prove Fermat's Theorem, have been brought to bear on this problem, and with success. Don Zagier (Max Plank, Bonn) used them to re-do Milne's proof of a similar formula for triangular numbers, and Ken Ono (Wisconsin-Madison) extended Zagier's work to duplicate and clarify Milne's results on squares. Elliptic functions and modular forms are two different areas of mathematics, so their convergence on the sums-of-squares problem suggests hidden connections. As Milne puts it, ``Why do the two seemingly unrelated approaches give the same results?''

John Nash, and the movie. More news from Princeton in the June 3, 2001 New York Times. ``A Portrait of John Forbes Nash Jr.'s Shattered Brilliance'' is by Nina Darnton. It starts with an evocation of Nash today, gives a quick and lurid portrayal of ``The Phantom of Fine Hall'' and then goes on to talk about ``A Beautiful Mind,'' the movie. The news is that the film, although taking its title from Sylvia Nasar's biography, is going to be a true product of the times: infotainment. Here is the screenwriter, Akiva Goldsman, as quoted by Darnton: ``The film uses the architecture of Nash's life in the broadest possible sense. We hit landmarks - genius, marriage, Nobel prize, illness - and that became the frame on which to hang dramatic anecdotes that I'd like to believe are true to the spirit of John and Alicia's lives. It is certainly not factual and never pretends to be. Most of the things that happen in the movie didn't happen in John's life." After listening to exalted show-biz talk from the director (Ron Howard) and the producer (Brian Grazer) we move back to Nash. Darnton gives us a more complete and sympathetic sketch of Nash's life up to the present. The piece ends with a quote from Nasar about Nash's supreme self-confidence as a mathematician. ``He just went ahead and did it, and to have that kind of inner certainty is a little crazy. It's also a certain level of genius.'' The article, available online, includes a photograph of Nash in 1957, in his physical and intellectual prime.

-Tony Phillips
Stony Brook