Geometry: from Isometries to Special Relativity

This undergraduate textbook blends Euclidean and non-Euclidean geometry, using isometries as a unifying thread, and culminates in a geometric discussion of special relativity. The principal idea here is that defining different notions of “distance” on various sets (e.g., the Cartesian plane, the sphere in three-space, the upper half-plane, etc.) leads to different geometries, and (as Felix Klein noted in his Erlanger Program in 1872) studying the distance-preserving mappings in each case can shed some light on the geometry. 

 

The book opens with a chapter on the Euclidean plane and its isometries. Important results proved in this chapter include the fact that any such isometry can be written as a product of three or fewer reflections, and that any isometry of the plane must be a rotation, translation, reflection or glide reflection. Actually, the statement of the result in the book omits “reflections” because the text’s definition of glide reflection is broad enough to include reflection as a special case; this definition is not used in all textbooks, however, and it seems to me to be better to treat reflections and glides differently since in some respects—such as the number of fixed points—they behave quite differently.  

 

The next two chapters look at spherical geometry and stereographic projection. The unit sphere centered at the origin is defined in the usual way as a subspace of three-dimensional Euclidean space, the distance between points on this sphere is defined, and distance-preserving mappings from this sphere to itself are studied. Arguments that are similar to those used in the previous chapter are applied here to prove, for example, that any isometry of the sphere is a product of at most three reflections through planes. Ultimately, connections are also made between spherical isometries and orthogonal transformations of three-space. Our knowledge of the sphere from chapter 2 is then applied to study stereographic projection in chapter 3, which also covers inversion in the plane and sphere. 

 

Chapter 4 deals with hyperbolic geometry—or, more precisely, several specific models of the hyperbolic plane: the Euclidean upper half-plane, the Poincare disc and the Beltrami-Klein model. The upper half-plane model is discussed first and most extensively. The starting point is the definition of arc length in this geometry (as an integral), which allows for the definition of an isometry (as a mapping preserving arc length) and then a characterization of line segments as curves minimizing arc lengths. Isometries are characterized, and it is shown that inversion mappings play the role of reflections in that any isometry of the hyperbolic plane can be written as a product of three or fewer inversions. Area is defined in terms of a double integral and it is shown that the area of a triangle is the difference between the 180 degrees and the sum of the angles of the triangle. (This is typically called the defect of a triangle in other books, but that term is not used in this text.) All this takes several sections of the chapter, after which there are two shorter sections, one each on the Poincare disc and the Beltrami-Klein model. The chapter ends with a section titled “Euclid’s Fifth Postulate: The Parallel Postulate” that begins with a very brief discussion of Euclid’s Elements and the history of hyperbolic geometry, but quickly segues into a discussion (not hinted at in the title of the section) on tessellations in the Euclidean plane and hyperbolic geometry.  

 

The remaining two chapters, comprising about half the book, focus on the Lorentz-Minkowski plane and special relativity, discussed geometrically. In the first of these two chapters, the author defines the Lorentz-Minkowski plane and isometries of them, this time in terms of preserving the Lorentz-Minkowski distance, which defines a bilinear form. This chapter has a strong algebraic flavor to it, but geometric content appears at the end, where it is shown that the hyperbolic plane (specifically, the Klein model) is isometric to the hyperboloid in Lorentz-Minkowski three-dimensional space. In the final chapter, some physics enters the picture; the author discusses some aspects of relativity theory, but not in much detail (some of the famous paradoxes are mentioned in about one paragraph each). 

 

In addition to the foregoing textual material, there are a decent number of exercises in the book, and a 25-page section containing solutions to a number of them. In addition, the publisher’s webpage for the text contains a password-protected solutions manual for classroom adopters. 

 

I have mixed feelings about how successful this book would be as a text for a geometry course. On the one hand, it certainly has some things to recommend it. The collection of topics is interesting, and, although these topics are, individually, discussed elsewhere (see, for example, Ryan’s Euclidean and Non-Euclidean Geometry: An Analytic Approach, Greenberg’s Euclidean and Non-Euclidean Geometries, Anderson’s Hyperbolic Geometry, Callahan’s The Geometry of Spacetime, and Martin’s Transformation Geometry), I am not aware of any one other book that discusses them all in one place. The idea of proceeding from familiar, easily-visualized spaces to those that are more abstract is a good one. In addition, the author writes clearly and has taken pains to minimize the prerequisites necessary for reading the book. There are also a good number of illustrations, a few of them in color. 

 

Another positive feature of the book is a good bibliography, which lists a couple of articles as well as textbooks, and which describes, for each reference, the chapters of the book now under review to which that reference is relevant. 

 

On the other hand, however, the author’s writing style struck me as rather dry, very much in the Jack Webb “Just the facts, ma’am” tradition. Many of the explanations and proofs consist of little more than extended calculations. There were also times where, I thought, there was not very much in the way of motivational or background material. For example, by the end of the first chapter, the reader has learned some facts about isometries, but has not learned anything to demonstrate why they are important. There is no mention of how isometries can be used to prove interesting results in Euclidean geometry, and there is no hint of the role of isometries in geometric history (such as, for example, in the aforementioned Erlanger Program). 

 

This lack of motivation is particularly acute in the chapter on hyperbolic geometry. The discovery of non-Euclidean geometry is, to my mind anyway, one of the most fascinating chapters in the history of mathematics, but the book conveys little sense of just what a momentous event this really was. Basically, the reader is just shown several different models of hyperbolic geometry, and basically told “here’s a strange geometry”. The fact that the existence of these models of hyperbolic geometry demonstrates that the Euclidean parallel postulate cannot be proved from the other axioms of Euclidean geometry, thereby resolving an issue that had been studied for centuries, receives only brief mention at the end of chapter 4. (I also take exception to the author’s description of a “postulate” as being “something so self-evident that [it] must be accepted without proof”. Is the hyperbolic parallel postulate not a postulate, then?) 

 

I also thought that the final chapter on relativity theory could have used more historical discussion and motivation. The development of relativity theory struck me as rather rapid and terse, particularly since this is a subject that many people have heard of and find somewhat mysterious. There is some attempt to provide some motivation and historical background, but not much, which is puzzling, because this book is a rather slim one, with only about 200 pages of text, and could easily have absorbed another 50 or even 100 pages. 

 

In addition, I found some of the arguments used in the book to be a little too loose, sweeping things under the rug without telling the reader that this is being done. For example, it is proved that any isometry maps a line onto a line; immediately thereafter, it is said that by this theorem, an isometry maps a triangle onto a triangle. But that’s not what the theorem says at all, and the second result, it seems to me, only obviously follows from the first if we know that the inverse of an isometry is also an isometry—a result that has not, at this point, been established (or even stated). Also, I don’t believe I ever saw a precise definition of a reflection through an arbitrary line in the plane, only a picture illustrating the image of a point under reflection in a line. The author then states, referencing the picture (and only the picture), that it is easy to show that a reflection is an isometry. However, I don’t think it’s all that easy. The proof requires consideration of a number of special cases, including some technical arguments along the way that raise some fairly subtle points about whether certain points are collinear. I would have no objection to the author explicitly leaving this result to the reader to prove as an exercise (he doesn’t, by the way), but doing it the way the author does strikes me as giving the incorrect impression that the result is so obvious that it almost doesn’t require proof. 

 

To summarize and conclude: There are some useful things in this book, and an instructor who wants to structure a course around the idea of isometries of increasingly more abstract spaces will want to take a look at this text. However, such an instructor should be prepared to spend a fair amount of class time providing background and motivation, as well as filling in some details. Instructors not using this book as a text may find it a useful reference. 

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Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.