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Continuous Symmetry: From Euclid to Klein
William Barker, Bowdoin College, Brunswick, ME, and Roger Howe, Yale University, New Haven, CT

2007; 544 pp; hardcover
ISBN-10: 0-8218-3900-4
ISBN-13: 978-0-8218-3900-3
List Price: US$75
Member Price: US$60
Order Code: MBK/47
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The fundamental idea of geometry is that of symmetry. With that principle as the starting point, Barker and Howe begin an insightful and rewarding study of Euclidean geometry.

The primary focus of the book is on transformations of the plane. The transformational point of view provides both a path for deeper understanding of traditional synthetic geometry and tools for providing proofs that spring from a consistent point of view. As a result, proofs become more comprehensible, as techniques can be used and reused in similar settings.

The approach to the material is very concrete, with complete explanations of all the important ideas, including foundational background. The discussions of the nine-point circle and wallpaper groups are particular examples of how the strength of the transformational point of view and the care of the authors' exposition combine to give a remarkable presentation of topics in geometry.

This text is for a one-semester undergraduate course on geometry. It is richly illustrated and contains hundreds of exercises.

Request an examination or desk copy.


Undergraduates interested in geometry.


"This is a book about plane Euclidean geometry with special emphasis on the group of isometries. It includes the classification of plane isometries into reflections, translations, rotations, and glide reflections, and also the classification of frieze groups and the seventeen wallpaper groups with complete proofs. It offers unusual proofs of some standard theorems of plane geometry, making systematic use of the group of isometries. ... All in all, this is a substantial book with a lot of good material in it, well worth studying. The authors promise a volume 2, which should contain solid geometry and non-Euclidean geometry in the context of projective geometry."

-- Robin Hartshorne, MAA Monthly

"... I learned a lot by reading the book, mainly because the material is arranged in a manner that invites and inspires one to reflect about the connections among the ideas being discussed. It is thought-provoking throughout. If a textbook is meant to be a tool for learning, then the extent to which it makes one think in the manner of a mathematician is by far the most important feature -- much more important than any quibbles about the slickness of a proof. I am very much looking forward to the opportunity to use this book in my classes."

-- James Madden, MAA Reviews

"All in all, this is a very nice book (that is) worth reading. ... It should be in every library, and (would) be useful to students and teachers alike."

-- Hans Sachs, Mathematical Reviews

Teaching Experience:

"This book is demanding, but in all the right ways. The writing is exemplary in its attention to definitions and in making all logical steps in every argument explicit. It couples rigorous attention to detail with a towering understanding of role of symmetry in elementary Euclidean plane geometry, gradually and systematically building the same understanding in the mind of the student. It would be an excellent choice for a geometry class intended to explore the basic transformations of the plane deeply and in a mathematically mature way."

-- James Madden

"Continuous Symmetry is a marvelous text. Several things about the text stand out:

  • The writing is engaging without compromising rigor. The average student finds the book very readable.

  • The text can be used with the Geometer's Sketchpad software very effectively. In particular, students can construct similarity transformations on Sketchpad as 'macros,' i.e. hot keys. From such constructions, the symmetries of figures can be studied dynamically. The text reads even better because of this interface. A wonderful example of the 'hot key' method is the Euler line of a triangle. The Barker-Howe approach shows that the orthocenter H of a triangle arises as the image of the triangle's circumcenter O under dilation by a factor of -2 about the median G of the triangle. Students first build this dilation on Sketchpad. Next, they demonstrate using the dilation that the three perpendicular bisectors of the given triangle dilate to the three altitudes of the triangle. The student now sees the point of the argument, namely, that since the three perpendicular bisectors meet at the circumcenter, the three altitudes intersect. Hence, the points H, O, and G are collinear.

  • There is a nice range of exercises that support the philosophy of Klein's Erlanger Program, namely, that symmetry is the basis of geometry.

  • The book is flexible in that one can move around in the text without harming the logical flow of material. For example, working simultaneously with Chapters II and IV on isometries and similarities has some appeal.

  • The material of Chapter VIII on the analysis of the Wallpaper groups can be covered very well by using the Geometer's Sketchpad. In this regard, students can be assigned to build the generators of each of the 17 groups along with fundamental domains. By watching each domain tessellate, the split and non-split conditions become clear.

I have successfully used the Barker-Howe text for our Modern Geometry course at ISU for many years. Modern Geometry is a one-semester course required of secondary education majors. I have also used the text for a second semester course where the focus is placed on Chapters VII and VIII. In the second course, the classification of the wallpaper groups is given careful treatment."

-- Robert J. Fisher, Idaho State University

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