musical symmetries: see Chapter 6, "The Geometry of Music", by Wilfrid Hodges. But most of these symmetries are only perceptible by optical inspection of the score. (As an experiment, I suggest playing the notes from the first two bars of a popular song in reverse order. It is rather unlikely that an untrained listener will recognize the original.) Not only group theory plays a role here. In Chapter 7, in the article by Der- mot Roaf and Arthur White on "Bells and mathematics", the emphasis is on combinatorics. "Ringing the changes" is the art of ringing a collection of n bells sequentially such that, at the end, all n! permutations have been heard. In ad- dition, certain conditions must be satis- fied, for example, from one round to the next, only transpositions between adjacent bells are admissible. This is so because, otherwise, it would be difficult to perform the sequence with really ex- isting heavy bells. It is interesting to see how this problem can be solved by rather simple algorithms and how the solutions are visualized graphically. In the last chapter of this part, "Com- posing with numbers" by Jonathan Cross, we are introduced to some math- ematical ideas which have found their way to being used as tools for com- posers. The story begins with the twelve- tone row of Arnold Sch6nberg; a num- ber of other examples are also discussed. The idea is always the same. First, one associates certain musical parameters, like pitch or duration, with numbers or more complicated mathematical objects, and then the structure of the mathemat- ical part is translated to a piece of mu- sic. For example, one could select a magic square and then use the rows (or the columns, or the diagonals) to define the pitches of the clarinet line or the du- rations of the bassoon line. Similar ideas are found in Part Four, "The Composer Speaks" (Carlton Gamer and Robin Wilson on "Microtones and projective planes" and Robert Sherlaw Johnson on "Composing with fractals"). The titles indicate the mathematical source of the compositions. Finite pro- jective planes are used to identify cer- tain subsets of tones. For example, if one wants to select three notes out of seven in such a way that the selection generates a "cyclic design", one finds everything that is needed in the geom- etry of the Fano plane. (A cyclic design in this case is a pattern such that trans- lations modulo 7 give rise to subsets of {0, . . . ,6} in which each pair of num- bers is contained in precisely one of the translations.) Dynamical systems are very com- mon in contemporary music. Here the well-known two-dimensional iterative patterns which lead to the Mandelbrot set generate the musical material. For example, if the channel of the synthe- sizer has to be determined where the next note will be generated, then a dis- cretization of the y-value of the pres- ent position of the system is important: e.g., if it lies in [7,8[, then choose chan- nel 5. It should be noted that the book is very carefully edited. It is a pleasure to read, and there are many interesting pictures and scores to illustrate the ma- terial. Readers who are particularly in- terested in the historical part of the sub- ject can consult the book Mathematics and Music- (edited by Gerard Assayag, Hans-Georg Feichtinger, and Jose Rod- riguez, Springer 2002; reviewed in The Mathematical Intelligencer, vol. 27, no. 3, p. 69). There is, surprisingly, only a small overlap in the content of these two books. The generation of scales by mathematical principles naturally plays a prominent role in both of them. For me, only two aspects are miss- ing. The first omission: I would have appreciated an article on Euler's work on music. He was one of the first to re- late mathematics to consonance, and it would be interesting to compare his work with that of Helmholtz. And I was surprised to see that one cannot find anything substantial on "probability and music". In the music of the last century there is an abundance of examples in which the building blocks of certain compositions are generated stochasti- cally, be it the pitches, the durations, or even the wave forms of the sounds. But these objections are not essen- tial. Let's praise the editors that they have presented an attractive volume that covers almost all of the important aspects of the interplay between math- ematics and music. Fachbereich Mathematik und Informatik Freie Universit&t Berlin D-14195 Berlin Germany e-maih behrends@math.fu-berlin.de The Pea and the Sun: A Mathematical Paradox by Leonard M. Wapner WELLESLEY, MA, A. K. PETERS, 2005, xiv + 218 PP., US $34.00, ISBN 1-56881-213-2 REVIEWED BY JOHN J. WATKINS T here is nothing quite like a good paradox. In their great comic opera, The Pirates of Penzance, W. S. Gilbert and Arthur Sullivan use a 'simple arithmetical process' to create "a paradox, a paradox, a most inge- nious paradooe' that renders the entire cast of pirates, maidens, and policemen helpless with laughter and amusement on the rocky seacoast of Cornwall. The paradox in this instance is extraordi- narily silly, but Gilbert and Sullivan lightly hang the entire plot of Pirates upon it. Leonard Wapner manages a similar sleight-of-hand with the Banach- Tarski paradox in his immensely en- gaging book The Pea and the Sun: A Mathematical Paradox. This book may not be quite as much fun as a Gilbert and Sullivan opera, but it is pretty close. To be fair, W. S. Gilbert and Arthur Sullivan should probably be ranked as geniuses, but Wapner does have one huge advantage over them. The rather flimsy basis of the Gilbert and Sullivan opera is the paradox that their naive young hero Frederic had the misfortune to be born on February 29, and so, by counting birthdays he is but "a little boy of five" and thus must continue on in his grossly unfair apprenticeship to a band of pirates until his twenty-first birthday. This is hardly a paradox wor- thy of the name. But Wapner has cho- sen as the basis for his book the Ba- nach-Tarski Paradox: quite simply, the finest paradox in all of mathematics. At first glance, the Banach-Tarski paradox seems so nonsensical that it might well belong in the world of Gilbert and Sullivan, for it says that it is possible to dissect a ball--that is, a solid sphere--into a finite number of pieces and then rearrange these pieces to form two balls exactly the same size as the 9 2006 SpringerScience+BusinessMedia, Inc., Volume28, Number3, 2006 71