Dots and Lines by Richard J. Trudeau (review)

Books 333 and exclamation marks towards the end of the book, will encourage his readers to persist. Moreover, if readers remember to read at least in some degree 'as poets'-that is, responding to the striking imagery of black holes, wave packets, quarks, and baryon binding, rather than struggling with each and every equation-then they will find the book rewarding. Physics for Poets was first published in 1970. The 1978 revised version, which includes nearly 30% new material, seems to have been designed primarily to make it more accessible by cutting down on the mathematics that made the first edition harder going. It ends with a series of questions suitable for students embarking on their first serious course in physics. Dots and Lines. Richard J. Trudeau. Kent State Univ. Press, Kent, Ohio, U.S.A., 1978. 203 pp., illus. Paper, $6.50. Reviewed by Arthur L. Loeb" This is a book on graph theory, the mathematical treatment of configurations; the four-color problem and the problem of the Koenigsberg bridges, for example, fall within its scope. One of the principal difficulties facing one unfamiliar with the subject is the large number of special terms; unless one has used them or has visual analogs to which to tie them, it is difficult to grasp the theorems and proofs and the concrete meaning of results obtained. I found Trudeau's text a splendid and delightful introduction to graph theory. There are numerous illustrations for each topic discussed and he proceeds patiently from simpler to more complex abstractions that have considerable power. For example, in an exercise at the conclusion of Chapter 2, using only elementary properties of rather simple graphs, one is asked to prove that in any gathering of six people there are three people who are and three people who are not mutually acquainted. This example of the interplay between graph theory and its application to problems of logic convincingly presents the value of this branch of mathematics. Topics covered are Planar Graphs, Euler's Formula, Platonic Graphs, Coloring, Graphs of Genus Higher than Zero, Euler and Hamilton Walks. An Afterword deals with the solution of the four-color problem that had been announced just as the book was going to press. Rigor is stressed, and, in the rare instance when a theorem is not proven, the reason is pointed out explicitly as a demonstration of some of the idiosyncrasies of mathematics. One instance is Jordan's theorem: 'If C is a continuous simple closed curve in a plane, then C divides the rest of the plane into two regions having C as their common boundary. If a point P in one of these regions is joined to a point Q in the other by a continuous curve L in the plane, then L intersects c.' Trudeau points out that this theorem, seemingly so obvious, could not be proven by Jordan himself, and decades passed before a proof was found. Each chapter contains from 18 to 40 stimulating exercises, and they are frequently referred to in subsequent chapters. I found that some of the exercises were not adequately presented . For example, in Exercise 12 on page 94 the 'pigeon hole principle' is stated as follows: 'If m objects are distributed into n boxes and m is larger than n, then at least one box contains min ofthe objects.' Does Trudeau mean 'exactly min' or 'at least min'? Perhaps this uncertainty was left intentionally in view of an appended exercise, but I think the uncertainty should have been avoided. I also find it unfortunate that, in such an otherwise beautifully designed book, chapter numbers are not indicated at the head of at least every other page, in order to facilitate checking the numerous cross-references. My criticisms of the book are trivial, and I highly recommend it to those interested in design configurations as well as to those who delight in mathematical games. I hope that Trudeau will bring his same approach to group theory, which may be considered complementary to graph theory in the mathematics of configurations. (For discussions of visual art "Dept. of Visual and Environmental Studies, Harvard University , Cambridge, MA 02138, U.S.A. from the point...