Read This!

The MAA Online book review column

Briefly Noted

November 2000

VideoMath Festival is an eclectic mix, but one common theme of all the videos is the mathematically informative nature of the animation. This is accomplished in a number of different ways. In videos such as Fibonacci and the Golden Mean, The Story of Pi, On Archimedes' Path, The Shadows of Alexandria, and The Theorem of Pythagoras, ideas are presented in historical context with an emphasis on geometric verification and approximation. The mathematics is simple enough that formulas are given with justification or plausibility arguments. For example, in The Story of Pi, animation leads us through a simple limiting derivation the area formula for a circle by cutting the circle into n wedges, then rearranging the wedges into an approximately rectangular shape. More advanced geometric topics are discussed in the videos Geodesics and Waves and The Shape of Space. Here the emphasis is on introducing new mathematical ideas making full use of modern computer graphics. For example, in The Shape of Space we get to fly through a universe which is topologically a three-torus. This video gives a nice heuristic view of closed three-manifolds, reminiscent of other work done at The Geometry Center such as Not Knot. The computer animation is as equally stunning in the videos: Knot Energies, Touching Soap Films, The Optiverse, and The Topological House.

Of a slightly different nature are videos such as The Law of Large Numbers, Vehicle Dynamics Simulation, and Challenges in Fluid Dynamics. These videos provide excellent visualizations of applied mathematical systems produced by numerical methods. The emphasis here tends to be on the applications rather than the theory. For example, in Vehicle Dynamics Simulation we see the computer animated results of a numerical solution to a system of differential equations controlling the test drive of an yuppie-like SUV. Finally, in a category of its own is a very entertaining video called Evolved Virtual Creatures. In this video, we see the results of evolutions of virtual creatures grown form artificial genetic codes that have evolved interesting ways to accomplish (with varying success) different goals.

In summary, VideoMath Festival is wonderfully entertaining and informative collection. Most mathematical audiences will all find some part that is immensely enjoyable. There are also many possible uses for math educators. For example, videos related to numerical and graphical solutions to differential equations can be shown in class to either enhance topics covered in the classroom, to initiate ideas for student research projects, or just for pure entertainment. This video's wide range of topics, varied levels of mathematical sophistication, and stunning graphics make it a must have for any institution and a welcome addition to any personal video library.[Patrick D. Shanahan]

I wonder whether there shouldn't be more mathematics books like Michael C. Berg's The Fourier-Analytic Proof of Quadratic Reciprocity. Here is how Berg describes what he is up to. "This is not a book for experts. This is not a book for raw beginners. It is, instead, an exposition of and commentary on a handful of sources, most of them classical by now and at least one of them notoriously austere." Berg takes up an idea that goes back to Cauchy: one can use Fourier analysis to give a proof of the classical quadratic reciprocity theorem. Leaving the task of researching Cauchy's contribution to the reader, he starts with the treatment of this given by Hecke in his 1923 Lectures on the Theory of Algebraic Numbers (for some reason, Berg always quotes the original German, even though an English translation is available from Springer). After some preparation, he then turns to A. Weil's 1964 paper "Sur certains groupes d'opérateurs unitaires." This is the "notoriously austere" paper mentioned above, and Berg works hard at helping the reader through it. Finally, he looks at work of Kubota from the late 1960s that gives a general context for Weil's paper. The goal of all this is to consider whether these methods might yield a method for generalizing the proof, a challenge laid down by Hecke in the last paragraph of the 1923 book.

What I find most interesting in all this is that Berg has decided not to write his own account of these ideas. Instead, he produces a text that is more like a commentary on the original papers, a commentary that is not afraid to elaborate and to mention more recent points of view. It is not a "close reading" of the papers, and it does not attempt to read them in their historical context. Instead, it tries to help the reader grapple with the content of the original papers by giving an up-to-date account of what is in them.

Not having read the whole book, I can't say how well Berg succeeds in making this material accessible. I do feel, however, that this is something worth doing. The jump from textbooks to original papers is often very difficult; we should have more books that help students across the chasm. [Fernando Q. Gouvêa]

Publication Data

The Fourier-Analytic Proof of Quadratic Reciprocity, by Michael C. Berg. Wiley, 2000. Hardcover, 115pp, $79.96. ISBN 0-471-35830-4.

VideoMath Festival at ICM '98, edited by H.-C. Hege and K. Polthier. Springer-Verlag, 1998. Video, $39.95. ISBN: 3-540-92633-X.

Patrick D. Shanahan (pshanaha@popmail.lmu.edu) is Assistant Professor of Mathematics at Loyola Marymount University in Los Angeles. He researches in the field of geometric topology where his main focus is on geometric and algebraic invariants of three-manifolds.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvêa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.