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Thinking Geometrically: A Survey of Geometries

Thomas Q. Sibley
MAA Press
Publication Date: 
Number of Pages: 
MAA Press Textbooks
[Reviewed by
Mark Hunacek
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This book is a revised and substantially expanded version of the author’s earlier text, The Geometric Viewpoint, originally published by Addison-Wesley in 1998. Like its predecessor, this book presents a potpourri of topics in both Euclidean and non-Euclidean geometry, arranged in a series of chapters that are largely independent of one another and therefore offer an instructor considerable flexibility in the book’s use. The intended target audience is upper-division undergraduates at roughly the junior level. Enough material is covered in this book to make it attractive not only to mathematics majors who plan to go on to graduate school, but also to those who plan on a career teaching secondary mathematics.

There are ten chapters, the last two of which are completely new. The first three discuss aspects of Euclidean geometry at a relatively elementary level. The text begins with a look at some of the basic Euclidean theorems, covered as in the Elements. Much of the chapter deals with plane geometry, but a section at the end introduces some higher-dimensional considerations, including a brief look at the geometry of the sphere. Although it is established that the sum of the angles of a spherical triangle is greater than 180 degrees, the idea of spherical geometry as constituting a non-Euclidean geometry is not pursued in detail at this point.

There is also no attempt in this chapter to put things on a rigorous axiomatic basis, but in chapter 2 the author looks at some of the logical defects in Euclid and discusses the axiomatic method. Several simple axiomatic systems are looked at for illustrative purposes; one of these offers an axiomatic foundation for the undefined notion of “betweenness”. Somewhat surprisingly, the simple axioms for an incidence geometry are not presented as an example; they often are in books at this level, such as Venema’s Foundations of Geometry and Greenberg’s Euclidean and Non-Euclidean Geometries: Development and History.

Later in the chapter the author looks at two approaches to providing a firm axiomatic development of Euclidean geometry — one due to David Hilbert, the other an amalgamation of Hilbert’s and the axiom system of Birkhoff, who got by with many fewer axioms than did Hilbert by invoking the power of the real numbers to measure distance and angles. The amalgamation of these systems was done by the School Mathematics Study Group (SMSG) in the 1950s; this system emphasizes pedagogical convenience at the expense of mathematical purity by deliberately taking as axioms certain “obvious” results that could, albeit with considerable effort, be proved as theorems. This recognizes and addresses a major pedagogical weakness of a rigorous axiomatic development, namely the need to convince students that it is necessary to spend a lot of time and effort proving results that they view as entirely self-evident.

Of course, a rigorous axiomatic development of Euclidean geometry can take an entire book by itself (see, e.g., Axiomatic Geometry by Lee) but the author at least provides a flavor of how such a development would look.

The next chapter is on analytic geometry, viewed here as a model of Euclidean geometry. While much of the material here (e.g., conics, polar coordinates) may be familiar from high school or early calculus courses, there are also some topics (Bézier curves, barycentric coordinates) that will almost surely be new to most students. The chapter ends with a look at analytic geometry in n-space and an introductory, but very brief, look at analytic geometry over fields other than the real numbers; only the rational numbers are considered, however, and in particular, finite fields are not exploited at this point. (This will, however, come later.)

In chapter 4, non-Euclidean geometry, specifically hyperbolic geometry, is introduced for the first time. The chapter begins with a quick look at various models of hyperbolic geometry (and I do mean “quick”: the Beltrami-Klein model receives four lines of explanation); the author defines what points and lines are in each model, and discusses incidence, but does not (in this section) address the more complicated questions such as distance measurement; a little further down the road, the author does give the definition of distance in the Klein and Poincaré models, but does not do much with it.

After looking at these models, the text discusses hyperbolic geometry from an axiomatic standpoint, plunging right into a discussion of the fairly subtle concept of sensed parallel lines and angles of parallelism. After a section on this topic, Saccheri quadrilaterals and their applications are discussed, as is the concept of area in hyperbolic geometry. (Saccheri’s book Euclid Vindicated from Every Blemish is mentioned briefly in the text, but does not appear in the “Suggested Readings” at the end of the chapter; I think it should be there, however, particularly in view of the recent Birkhäuser re-publication of it, complete with annotation and modern commentary by Vincenzo DeRisi.) There is also a short section introducing elliptic geometry. For reasons that I’ll go into later, I thought this was the least successful chapter of the book, but that conclusion may largely reflect my personal taste in the presentation of this material.

The next two chapters are related. In the first, various kinds of geometric transformations are studied, including isometries, similarities and affine transformations in the plane and, and isometries of three-space and of the sphere. The chapter ends with a section on inversion in circles, with reference made to isometries of the Poincaré model of the hyperbolic plane. Since the concept of distance in this model is only briefly mentioned at the end of section 4.4, however, it is not possible to pursue this in great detail.

Isometries play a big role in the mathematical analysis of plane symmetry, and this is discussed in chapter 6: frieze groups are discussed and analyzed, for example, and the classification theorem for wallpaper groups is stated but not proved. The chapter also contains a look at three-dimensional symmetry, and applications to science and fractals are mentioned.

Chapters 7 and 8 are also related to one another. The former introduces projective planes, first from an intuitive historical point of view and then both axiomatically and analytically, and then discusses projective spaces and how other geometries can be viewed as “subgeometries” of projective geometry. The author’s approach here requires some discussion. The typical axiomatic definition of a projective plane is by means of simple incidence axioms that guarantee the existence of a sufficient number of points, and require that two distinct points determine a unique line and two distinct lines intersect in a unique point. (See, for example, Casse’s Projective Geometry and Borceux’s An Axiomatic Approach to Geometry.) The axioms given here, however, go beyond this; in addition to incidence axioms, he also gives axioms of harmonic sets and separation. These axioms are, therefore, for what might be called the real projective plane or the extended Euclidean plane; they are not axioms for what is typically defined as a “projective plane”. Indeed, many such planes are finite, and do not admit any kind of order relationship among the points. I’m not sure this distinction is made sufficiently clear in chapter 7, although it is clarified somewhat in the next chapter, where the term “projective plane” is given its usual, incidence-only, definition.

This next chapter (“Finite Geometries”) defines affine as well as projective planes from an incidence-axiom standpoint, and we see how analytic geometry, the subject of an earlier chapter, can be done with entries from a finite field rather than just the real numbers. Applications to such areas as design theory are referenced.

The next (and concluding) substantive chapters of the book are completely new to this revised edition; there was no analogue of them in the previous Geometric Viewpoint. The first of these is on classical differential geometry (curves and surfaces in the plane and space, no manifolds); here again, this is a subject that can easily fill an entire text (of which there are many, including Differential Geometry of Curves and Surfaces by Banchoff and Lovett), but the author has selected a few topics of a particularly geometric nature (notably curvature and geodesics) to discuss. The approach here is, of necessity, somewhat analytic, but Sibley works hard to minimize reliance on multivariable calculus, thereby increasing the accessibility of the discussion.

The final chapter of the book is on discrete geometry, where the analytic approach is replaced by a more combinatorial one. We see here a selection of interesting results, including the art-gallery problem, tilings of the plane and Voronoi diagrams.

After these ten chapters, there is a brief chapter (“Epilogue”) that gives a very cursory (three paragraph) discussion of topology, and a biographical sketch of Henri Poincaré; the discussion was so cursory, in fact, that I wondered why the author even bothered with it.

This is all followed by half a dozen appendices. The first of these lists the postulates, common notions, and propositions from book I of Euclid’s Elements. The second and third list the axioms used by the SMSG and Hilbert in their respective developments of Euclidean geometry. The fourth and fifth review, respectively, linear algebra and multivariable calculus, and the sixth spends a few pages discussing various proof techniques. I was pleased to see this, because, more and more, I am becoming convinced of the need for discussing proof ideas in upper-level courses; my only mild criticism of this appendix is that it would have been enhanced by a discussion of basic logic as applicable to proof techniques — i.e., negation, converse, contrapositive, etc.

Like most textbooks, this one has its strengths and weaknesses. On the plus side, it offers clear writing, lots of exercises (including projects), nice examples, historical and bibliographic sections, a good sense of how geometry interacts with other areas of mathematics (and areas outside mathematics), and a wide variety of topics from which to choose. On the other hand, however, precisely because so many different topics are covered, there is no opportunity to investigate any one of them very deeply. There were times, reading this book, when I felt like a tourist on one of those “If it’s Tuesday, this must be Belgium” tours of Europe — herded around from one cathedral to another at a breakneck pace, and not having the chance to really stop and savor the local culture.

I felt this most keenly in the chapter on hyperbolic geometry, perhaps because I have been teaching this material for the past several years and have developed a routine about how I like to present it. I have always thought that the way to teach the development of non-Euclidean geometry is to treat it, as Greenberg says in the preface to his geometry book, “as a suspense story”. I (following Greenberg, as well as the much earlier book from which I first learned this material, Basic Concepts of Geometry by Prenowitz and Jordan) begin by discussing some of the many attempts over thousands of years to prove the Fifth Postulate (or some equivalent formulation), which in turn leads to a consideration of neutral geometry (i.e., Euclidean geometry minus any assumption of a parallel postulate at all). This gives students a deeper appreciation of what specific role the Fifth Postulate plays in Euclidean geometry.

At this point, one can depart from neutrality and actually assume the negation of the Fifth Postulate, thereby churning out lots of results that seem strange to a student brought up on Euclid: the sum of the angles of a triangle is less than 180 degrees, parallel lines are not equidistant, similar triangles are congruent, etc. In fact, one non-obvious fact here concerns the very negation of the parallel postulate. If we negate the phrase “through a given point P not on line \(\ell\), there exists a unique line parallel to \(\ell\)” we of course get “there exists a point P and a line \(\ell\) not containing P, such that through P there are either no lines, or more than one line, parallel to \(\ell\).” However, it can be proved in neutral geometry that there is always at least one line parallel to \(\ell\), and that, moreover, this is an “all or nothing” proposition: if there is one point-line pair that negates the Euclidean parallel postulate, then they all do. So, in hyperbolic geometry, the negation of the Euclidean parallel postulate is “through a given point P not on line \(\ell\), there exist more than one line parallel to \(\ell\)”. This is the hyperbolic parallel postulate, which was called the Lobachevskian parallel postulate when I was a student.

By now the students, naturally enough, are wondering whether this “strange new universe” (János Bolyai’s term), with its results that are so different than the Euclidean ones they grew up on, has any real mathematical validity, and they are therefore ripe for a discussion of one or more models of hyperbolic geometry, the existence of which show that hyperbolic geometry is in fact at least as consistent as Euclidean. This development, it seems, never fails to intrigue the students, and seems to me to be a wonderful way to cap about a month’s worth of work. About this time, I ask the students to read the short story Euclid Alone by Orr, about the consequences of a mathematician’s discovery that Euclidean geometry really is inconsistent after all; I discovered this story in a footnote in Greenberg’s book.

In Sibley’s text, however, models of hyperbolic geometry are introduced early, without first giving the student a chance to be led up to just how momentous their significance really is, and without first discussing the strange results of hyperbolic geometry; it is only after models have been discussed that some of the theorems of hyperbolic geometry are established, and even here, the first ones to be discussed are rather difficult ones concerning “omega triangles” rather than the easier results concerning angle sums and the non-existence of rectangles. There is nothing logically wrong with how Sibley does things, of course, but somehow, to my mind at least, this approach seems to take something of the suspense and excitement out of the development, and makes things more complicated and less accessible than necessary.

But this is, of course, largely a question of opinion, and it also may be that I am complaining here about something that is largely impossible to fix; if one wants to cover all the topics that Sibley does, the discussions must of necessity be rather brief (though I still think the arrangement of topics is not always optimal, as in chapter 4). Any individual professor will, therefore, have to decide for himself or herself what the best way to teach geometry is: concentrate on one or two topics and really pursue them, or provide a quick “if it’s Tuesday” tour of a wide variety of topics. Instructors who prefer the latter approach will certainly want to obtain a copy of this book and examine it in detail. The large selection of topics covered also make this book a valuable reference.

Mark Hunacek ( teaches mathematics at Iowa State University. 

1. Euclidean Geometry
2. Axiomatic Systems
3. Analytic Geometry
4. Non-Euclidean Geometries
5. Transformational Geometry
6. Symmetry
7. Projective Geometry
8. Finite Geometries
9. Differential Geometry
10. Discrete Geometry
11. Epilogue
A. Definitions, Postulates, Common Notions, and Propositions from Book I of Euclid’s Elements
B. SMSG Axioms for Euclidean Geometry
C. Hilbert’s Axioms for Euclidean Plane Geometry
D. Linear Algebra Summary
E. Multivariable Calculus Summary
F. Elements of Proofs
Answers to Selected Exercises