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Peter Hilton, Jean Pedersen

The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching. A joint venture of the Mathematical Association (in the United Kingdom) and the Mathematical Association of America. Edited by Chris Pritchard. xviii + 541 pp. Cambridge University Press, 2003. Paper, $40.

The increasing emphasis on standardized tests over the past 50 years has almost guaranteed that contemporary American students never come across any significant geometry instruction in school. The situation in the United Kingdom is not yet as dire but is undoubtedly deteriorating. Against this background, it seems incredible that a book such as The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching could have been published. But further consideration offers an explanation. Although it might seem that educators and bureaucrats have conspired to do away with the subject, geometry in fact survives, preserved by its own vitality. As one of us (J.P.) has written elsewhere, "geometry is the connection between the real world and the mathematics we use to solve problems about that real world." Indeed, the great French mathematician René Thom described geometry as the most fundamental abstract representation of the real world. This important property, together with the subject's intrinsic depth and subtlety, accounts for its remarkable resilience.

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Geometry enthusiasts will welcome this volume, in which the editor, Chris Pritchard, has gathered articles from the Mathematical Gazette and Mathematics in Schools as well as some from other sources, grouping them under six headings: "The Nature of Geometry," "The History of Geometry," "Pythagoras' Theorem," "The Golden Ratio," "Recreational Geometry" and "The Teaching of Geometry." The authors include some of the greatest mathematicians of the 20th century, such as G. H. Hardy, H. S. M. Coxeter and Michael Atiyah.

This book is a source of intellectual delight and can easily be dipped into during one's leisure moments. Of course, we do have a few criticisms, the most serious of which is the absence of an index, a curious omission for what is surely intended to be primarily a reference work. And we would have preferred that the content were less restricted; the articles from the Mathematical Gazette and Mathematics in Schools, which are journals of the Mathematical Association of the United Kingdom, largely reflect the intellectual appetites of teachers working at the secondary level. The somewhat grandiose title fails to indicate that postsecondary geometry is scarcely treated at all—topology and combinatorial geometry, for example, make only very occasional appearances. (We could have been more certain of this had there been an index!)

Certainly, we strongly recommend this book to all lovers of mathematics. The excellent foreword by Douglas Hofstadter makes a strong statement against the neglect of geometry in the current school and college curricula in the United States. And we especially liked the "Desert Island Theorems," which appear at the end of each of the first five sections of the book. "Imagine you are a mathematician," says the editor, "cast away on a desert island. You have no access to the modern world and little prospect of being reunited with the rest of humanity. Which geometrical theorem would you least wish to be without . . .?" Pritchard persuaded 30 mathematicians, physicists and educators to nominate a theorem and write 500 words about it. What genuine student of mathematics would not be intrigued by this material? The theorems are arranged in order of increasing difficulty, under the headings "Greek Geometry," "Elementary Euclidean Geometry," "Advanced Euclidean Geometry," "Non-Euclidean Geometry & Topology" and "Geometrical Physics."

Teachers can certainly use this and other material in The Changing Shape of Geometry to enrich their classes. The book should also be interesting and easily accessible to bright high school students working on their own. Allowed to browse, the curious young person will surely find much here to read with delight.—Peter Hilton, Mathematical Sciences, State University of New York, Binghamton, and Mathematics, University of Central Florida, Orlando; and Jean Pedersen, Mathematics and Computer Science, Santa Clara University, California

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