While some find mathematics attractive simply because of its austere beauty, most of us, mere mortals, cope with its intellectual demands primarily because of its utility. The subject is so vast that, in order to cover it, most high school and college courses give only tiny glimpses of real-life applications. This book takes a more detailed view of mathematics in action, in several areas of technology; in doing so, it achieves two goals: it answers the question, “What is mathematics good for?” and it stimulates the development of mathematical modeling in current and interesting areas. Readers will also come to see the dynamic and evolving nature of mathematics, and perhaps appreciate the fact that they can contribute to its growth.

The required background is Euclidean geometry and the mathematical maturity than can be obtained in the first year of college; other material--linear algebra, multivariable calculus, and elementary probability theory--is covered as needed. The book emphasizes the following points: mathematical concepts and techniques can and do show up in different areas of interest; different mathematical concepts can lie behind an application; and, while mathematical sophistication need not be very high, technology is sometimes driven by a brilliant, yet fairly simple observation. In the authors’ words, “Ideas are a scientist’s most precious commodity.”

The book contains 15 chapters; each chapter is self-contained and can be read independently of the others, so that the material can be taught and studied in any order. Most chapters share a similar organization: an introduction to motivate the application; presentation of the underlying mathematics; cases, examples, and problems to illustrate the material; and, when needed, more elaborate and detailed examples and further mathematical development. All chapters end with references for further study. The last chapter, “Science Flashes,” contains ten small, self-contained subjects that require Euclidean geometry, high school algebra, and trigonometry. In a course, these could be introduced in class and left for study as exercises.

Links between chapters are given, so that the reader is alert to related applications; the many historical notes introduced throughout the book keep up the reader’s interest and enhance his or her well-rounded background and education. Numerous exercises are given, ranging in scope from simple ones to class projects. Those who are reading the book on their own will want to read all the exercises and do as many as time allows and interest dictates.

Conceptually, the book covers the following topics: lines and planes in all forms (regular and parametric equations, and subspaces) and in unexpected places, such as decoding messages; affine transformations in the plane and in space (robotic motion, fractals, and image compression); finite groups (mosaics and friezes, and primality tests in cryptography); finite fields (global positioning systems (GPS), error-correcting codes, and random number generation); modulo arithmetic (cryptography, error-correcting codes, large prime numbers construction, and random number generation); probability theory (random number generation and Google page ranking); Markov chains (Web page links); linear algebra (GPS, robotic motion, error-correcting codes, Google page ranking, and JPEG imaging); Fourier analysis; and calculus of variations (the best half pipe, the fastest tunnel, soap bubbles, and liquid mirrors). A detailed table of contents is downloadable from the publisher’s Web site (http://www.springer.com/math/book/978-0-387-69215-9?detailsPage=toc).

This is an excellent book for a varied audience. High school teachers and prospective teachers who want to be better prepared to instill in their students a vision of greater possibilities will certainly find in this book examples and techniques they can use; more importantly, they will also find tools they can use to develop their own examples, problems, and projects for their students. This book will also be attractive to college students who are eager to delve more deeply into serious and challenging applications, and to researchers in mathematics, computer science (CS), and technology, who want to acquire a more thorough understanding of the applications covered in the book. Finally, for those intrepid souls who wish to have more than a surface understanding of these applications and are willing to learn the underlying mathematics on their own, this presentation is accessible and the references are useful. The book could also serve as a textbook for an applications course; the authors give several pointers and suggestions to instructors.

While the emphasis is on the mathematical presentation of the material, the motivated programmer will find plenty of source material from which to take inspiration.

The book was originally written in French. The translation is smooth and excellent; except for the exuberant use of exclamation points, the reader cannot detect a Gallic accent.

I like this book and I recommend it.