2007;
544 pp;
Hardcover

MSC: Primary 51; 20;

Print ISBN: 978-0-8218-3900-3

Product Code: MBK/47

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-1197-8
Product Code: MBK/47.E**

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

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#### Supplemental Materials

# Continuous Symmetry: From Euclid to Klein

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*William Barker; Roger Howe*

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.

#### Readership

Undergraduates interested in geometry.

#### Reviews & Endorsements

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

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

# Table of Contents

## Continuous Symmetry: From Euclid to Klein

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Instructor Preface ix10 free
- Student Preface xiii14 free
- Acknowledgments xix20 free
- Chapter I. Foundations of Geometry in the Plane 124 free
- I.1. The Real Numbers 124
- I.2. The Incidence Axioms 629
- I.3. Distance and the Ruler Axiom 1740
- I.4. Betweenness 2245
- I.5. The Plane Separation Axiom 2750
- I.6. The Angular Measure Axioms 3457
- I.7. Triangles and the SAS Axiom 4669
- I.8. Geometric Inequalities 5679
- I.9. Parallelism 6285
- I.10. The Parallel Postulate 7093
- I.11. Directed Angle Measure and Ray Translation 84107
- I.12. Similarity 94117
- I.13. Circles 110133
- I.14. Bolzano's Theorem 115138
- I.15. Axioms for the Euclidean Plane 119142

- Chapter II. Isometries in the Plane: Products of Reflections 121144
- Chapter III. Isometries in the Plane: Classification and Structure 165188
- Chapter IV. Similarities in the Plane 217240
- Chapter V. Conjugacy and Geometric Equivalence 251274
- Chapter VI. Applications to Plane Geometry 287310
- VI.1. Symmetry in Early Geometry 287310
- VI.2. The Classical Coincidences 292315
- VI.3. Dilation by Minus Two around the Centroid 298321
- VI.4. Reflections, Light, and Distance 309332
- VI.5. Fagnano's Problem and the Orthic Triangle 315338
- VI.6. The Fermat Problem 322345
- VI.7. The Circle of Apollonius 340363

- Chapter VII. Symmetric Figures in the Plane 347370
- Chapter VIII. Frieze and Wallpaper Groups 375398
- Chapter IX. Area, Volume, and Scaling 459482
- References 531554
- Index 533556
- Back Cover Back Cover1570