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Continuous Symmetry: From Euclid to Klein

American Mathematical Society
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In a 1990 paper entitled “What Is Geometry?” Shiing-Shen Chern [3] identified six pivotal developments in the history of geometry:

  1. Euclid’s axiomatic treatment,
  2. the introduction of coordinates (Fermat and Descartes),
  3. the invention of the calculus (Newton and Leibniz),
  4. the recognition of the fundamental role of transformation groups (Klein and Lie),
  5. manifolds (Riemann), and
  6. fiber bundles (E. Cartan and Whitney).

The geometry presently taught in American high schools includes significant parts of (1) and (2), and some students get an introduction to (3). Transformations (4) make a cameo appearance in many contemporary American high school geometry textbooks, and play a featured role in a few. (5) and (6) have not influenced the high school curriculum.

The Common Core State Standards for Mathematics, which have now been adopted as a guide for the K–12 mathematics curriculum in 44 states, put transformations in center stage. Students in Grade 8 are introduced to geometry by experimenting with rotations, reflections and translations and they are expected to understand congruence in terms of rigid motions. High-school students develop these ideas more rigorously, learning standard terminology and fundamental facts about rigid motions and dilations of the plane as a foundation for the study of Euclidean geometry. In giving such prominence to transformations, the authors of the Common Core took into account both pedagogical evidence as well as the structure of the mathematics underlying the curriculum. Nonetheless, in view of the traditional organization of the curriculum, this is one of the bolder proposals of the Common Core.

The challenge will be implementation. We will need K–12 textbooks that are very different from those most widely used. At least as important will be assuring that teachers are intellectually equipped. Good scholarship presented in a form accessible to teachers (or those planning to become teachers) and focused on the meaning and role of transformations in geometry is needed. One would hope to find this in textbooks for junior/senior-level college geometry courses, but unfortunately few books fill the bill. There is no standard undergraduate course in geometry, and the available textbooks have less in common than the books for courses such as abstract algebra or basic analysis that have acquired a canonical form. Geometry textbooks tend either to present some kind of axiomatic treatment and then to branch off into topics or else to be, through and through, a sampling of approaches and ideas. Transformations tend to be treated in college in the same way that they are treated in high school, namely, as an interesting branch of a great tree.

The purpose of the book of Barker and Howe is to develop the main ideas about geometric transformations of the Euclidean plane and their applications. It could play a valuable role in introducing college students, especially future teachers, to this topic. However, it does not jump immediately to its main topic. Chapter I, which takes up the first 120 pages of the 529 pages of this text, is devoted to a careful development of Euclidean plane geometry based on an axiom system similar to SMSG, which features the Ruler and Protractor Postulates. This opening chapter sets the mathematical tone. It is careful, rigorous, thorough and explicit in its attention to detail. I expect that this would not only be a good book from which learn some geometry, but also a fine introduction to the habits of logical thinking and precise exposition that a math major needs to acquire.

Transformations make their first appearance in Chapter II, which culminates on pages 157–161 by classifying the isometries of the plane according to the number of their fixed points, demonstrating that every isometry is a composite of at most three reflections, and proving that, for any pair of congruent triangles, there is an isometry that takes one to the other. Readers who are familiar with these theorems might wonder if 160 pages of preparation are needed. The main ideas in the proofs are intuitively clear, and can be conveyed vividly by well-designed paper-folding activities. But what a student will take away from an informal presentation is very different from what he or she might acquire by following the course that the book lays out. Paper-folding, whatever legitimate pedagogical purposes it may serve, does not equip learners with a useful language for reasoning about transformations. This book will provide plenty of opportunity to learn good mathematical language and practice its use.

Chapter III begins with a discussion of compositions of reflections and the kinds of plane isometries that may be produced, leading up to the classification of isometries as reflections, rotations, translations or glide reflections on page 188. Numerous fine color illustrations make the twenty pages of preparation for this theorem a delightful visual experience. This is followed by a discussion of orientation based on the proposition that the parity of an isometry is well-defined. The chapter then introduces the idea of a group of transformations and in a sequence of exercises beginning on page 203 invites readers to investigate the structure of many examples. The remainder of the chapter deals with factorization in the plane isometry group.

Chapter IV concerns similarities. This chapter is especially important for future teachers because of the prominence of similarities in high-school mathematics. The treatment here parallels the treatment of isometries in the previous chapters, building up to the structure and classification theorems.

Chapter V studies the conjugacy relation in transformation groups and the decomposition of groups into conjugacy classes. This chapter ends with an interesting but sketchy discussion of the idea of developing geometry from the group of symmetries alone by means of the correspondences between lines and reflections and between points and 180-degree rotations. As a matter of fact, this is an idea that was developed extensively in the early 20th century by German-speaking mathematicians Gerhard Hessenberg, Johannes Hjelmslev and Gerhard Thomsen; see [4]. The standard reference is Friedrich Bachmann’s book [1]. Unfortunately, the works of Hessenberg, Hjelmslev and Thomsen have not been translated into English, nor has Bachmann’s book. However, a good synopsis in English is contained in chapter 5 of [2].

Chapter VI describes applications of transformations to Euclidean plane geometry. Running from page 287 to 346, it includes numerous interesting results. Transformations appear sometimes as essential tools, sometimes as useful aids, sometimes as a way to interpret a construction and sometimes merely as a point of view. Section 2 describes some theorems on the concurrence of special lines in triangles.

The first theorem on the circumcenter (the point of concurrence of the perpendicular bisectors of the sides of triangle ABC) helps to illustrate my remark about the varying role of transformations. The proof is essentially as follows: Let w be the perpendicular bisector of segment AB and let u be the perpendicular bisector of segment BC. Let P be the point of intersection of w and u. Then P is equidistant from A and B because it’s on w and also P is equidistant from B and C because it’s on u. Therefore, P is equidistant from A and C, so P lies on the perpendicular bisector of segment AC. Transformations illuminate the argument — w and u are axes of reflections and it is these reflections that show the congruence of segments AP, BP and CP — but certainly it is not necessary to know this to follow the proof.

The incenter, centroid and orthocenter are treated in a similar fashion, using arguments in which the role of transformations is clear, illuminating but not essential. Section 3 uses dilation about the centroid by a factor of –2 to unify a discussion of the Euler line and the nine point circle. Here, the transformation does some real work. Section 5 of this chapter concerns the orthocenter (the point of concurrence of the altitudes). The discussion hinges on the analysis of the composition of the reflections whose axes are the sides of a triangle. Here, transformations are not only hard at work, but are leading the development.

Chapters VI (pages 347–375) concerns the symmetries of bounded figures in the plane, leading up to a discussion of dihedral groups. Chapter VII (pages 376–458) is a careful treatment of frieze and wallpaper groups. The last chapter concerns area, volume and scaling.

I had hoped to use this book in a one-semester course in geometry that I teach periodically, but up till now the scheduling has not worked out. The course is populated primarily by math majors who are seeking certification as secondary teachers. Most of these people take a proof-based course in real analysis as well as other junior/senior courses, including probability, abstract algebra, number theory and other advanced topics. The typical student in this group will likely struggle at first with the level of exposition in Continuous Symmetry, particularly if he or she has not previously taken the analysis course, but surely will be capable of handling it. Because the text is detailed and methodical, patience and persistence is needed. However, it clear and explicit enough that it will never leave students helpless or befuddled, provided they are serious and spend the time to read carefully.

I plan to use the text next time I teach the class. I expect to have to pick and choose carefully, especially in the first chapter. I find the level of rigor of Euclid to be a very workable goal in this class. It does not pay off to be overly concerned with facts, such as the Crossbar Theorem, that are evident from the topological properties of diagrams. I think the geometry of the number line is an essential topic, and a good opening for the course. This is consistent with the layout of the book.

My goal will be to spend only enough time in chapter I to prepare for chapters II, III and IV. These, I want to treat carefully because of the tasks these future high-school teachers will face in the classrooms where they will eventually teach. I shall need to include some material on area and volume, and Chapter IX can support this. Some discussion of coordinates, particularly linear change of coordinates, will be desirable. For this, I shall need to supplement the book. R. Hartshorne has made some observations about conventions related to the Ruler Postulate in a review of this book that appeared recently in the American Mathematical Monthly. Anyone who plans to use the book should be aware of his remarks.

Although I have mentioned them only once, the abundant, high-quality illustrations throughout the text are one of the most attractive features.

The book does not always choose the quickest or most elegant route to a result. For example, Proposition IX.2.14 on the area of a parallelogram could have been proved more elegantly by repeating Euclid’s proof of Proposition 35 of Book 1 of the Elements. Nevertheless, 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.


  1. [1] Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edition. Berlin, Springer, 1973.
  2. [2] Behnke, H., Bachmann, H., Fladt, K. & Kunle, H., Fundamentals of Mathematics, Volume 2 (translated by S. H. Gould). Cambridge MA, MIT Press, 1974.
  3. [3] Chern, S-S., What Is Geometry? The American Mathematical Monthly, Vol. 97, No. 8, Special Geometry Issue (Oct., 1990), pp. 679–686.
  4. [4] Thomsen, G., Grundlagen der Elementargeometrie in gruppenalgebraischer Behandlung. Leipzig, Berlin, B. G. Teubner, 1933.

James Madden is Professor of Mathematics at Louisiana State University.

Date Received: 
Friday, December 7, 2007
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William Barker and Roger Howe
Publication Date: 
James J. Madden
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Wednesday, February 29, 2012


akirak's picture

A full review of Continuous Symmetry was written by Robin Hartshorne for the American Mathematical Monthly, Volume 118, Number 6, June 2011, pp. 565-568. Subscribers can see this review online by going to