November/December 2012 (Vol. 14, No. 6) pp. 6-7
1521-9615/12/$31.00 © 2012 IEEE
Published by the IEEE Computer Society
Published by the IEEE Computer Society
Number-Crunching Could Have Crunched More
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If you're interested in a book on how to do the math and Matlab programming behind oscillators, three-body problems, and electrical circuits, then this is a fine book. You can stop reading this review now, and go buy the book. Really, it's pretty good, with those caveats.
If those caveats don't fit, keep reading. On p. 329, Paul Nahin writes, "This book has been an unabashed ode to the wonder of the modern electronic computer." My response to that is that this would have been a fine book 20 years ago. Almost all of the major examples in the book date before that. Even the 1992 versions of Matlab, Maple, and Mathematica could handle most of the computations in this book. Full disclosure: I work for Wolfram Research. Throughout, this didn't feel like a modern book, and didn't seem dedicated to modern results.
The author continually references his prior books. For example, on p. xi he talks about the "Mrs. Perkins's Quilt" problem that was in his previous book, Mrs. Perkins's Electric Quilt:
For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches. —Henry E. Dudeney
The electrical circuit method for handling this general problem was solved by Brooks, Smith, Stone, and Tutte in 1936–1938, and Tutte did a guest column about the problem for Martin Gardner in 1958.
Today, our computing power has multiplied by many trillions. Paul Nahin gave a pretty good explanation for the title problem in his previous book, but did he do any number crunching? He apparently didn't press "Run" on the program, but I did on my own version. Search for "Mrs. Perkins's Quilt" on the Web to see the solutions for the problem up to order 1,098. I used the 1936-era method on a modern system, and accidently trounced all known knowledge on the problem. This is easy to do. Most problems that haven't been solved recently can be examined again with millions to trillions of times as much power.
In the next few chapters, the author marvels at various hand computations, then shows Matlab solving the same today-it's-trivial problem. Someone greater than I slammed the pointlessness of some hand computations: Jacques Bernoulli. In Bernoulli's Ars Conjectandi, published in 1713, he says,
With the help of [Bernoulli numbers] it took me less than half of a quarter of an hour to find that the 10th powers of the first 1000 numbers being added together will yield the sum 91409924241424243424241924242500. From this it will become clear how useless was the work of Ismael Bullialdus spent on the compilation of his voluminous Arithmetica Infinitorum in which he did nothing more than compute with immense labor the sums of the first six powers, which is only a part of what we have accomplished in the space of a single page.
Nahin continues on this path when on p. 329 he writes, "You'll recall that Riemann then made a huge conceptual leap and conjectured that every last one of the infinity of complex zeros would have a real part of 1/2. Trillions of complex zeros have been calculated since 1859, and every last one does indeed have real part 1/2." In 2004, the Zetagrid.net project calculated 935.7 billion nontrivial zeros. I was one of the people waiting for the distributed project to reach a trillion. Then, suddenly, Xavier Gourdon used an Odlyzko-Schönhage algorithm to calculate the first 10 trillion (10,000,000,000,000) nontrivial zeros (see http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf). It's a really incredible paper—not a solution, but Gourdon provides an unexpectedly powerful number-crunching method. A single person trouncing a large distributed computer project was worth a mention.
I've been harsh so far, but the bulk of the Nahin's book stresses the link between differential equations and visualizations, a vital 21st century skill.
In the middle section, the author discusses chaos in pendulums, the three-body problem, and electrical circuits. Those examples comprise more than 200 pages of the book. Do an online search for "damped spherical spring pendulum," "planar three-body problem," or "three-branch electrical circuit" for some further examples of these topics. The mathematical explanations are excellent, and he includes plots, code, explanations, background stories, and equations for all. For the three-body problem, he could have mentioned the Sun-Earth-Cruithne triple system as a modern example. The Christopher Moore figure-eight, three-body solution gets a thorough explanation, though.
Nahin writes out numerical experiments and programs with great enthusiasm, and encourages readers to do the same throughout the book. With that vibe of "go enjoy math and coding," I did enjoy the bulk of the book. So despite the drawbacks I've noted, it's pretty good, especially for Matlab fans.
Ed Pegg Jr. runs the Wolfram Demonstrations Project at Wolfram Research and mathpuzzle.com. He served as a consultant for the TV show Numb3rs. He researches recreational mathematics, combinatorics, and all forms of mathematical elegance. While serving at the North American Aerospace Defense Command (NORAD), he completed an MS in mathematics at the University of Colorado at Colorado Springs. Contact him at firstname.lastname@example.org.