Way back when, as an undergraduate at Brooklyn College more than 40 years ago, I took a course called Projective Geometry, one of about four or five geometry courses offered at the time. Based on anecdotal evidence supplied by casual conversations with friends over the years, I suspect that I am one of comparatively few people who were lucky enough to take such a course. Especially today, it seems that most university math departments offer only one or two semesters of geometry, and very few offer a course devoted exclusively to projective geometry. Those undergraduate math majors who learn anything at all about the subject typically do so as part of a more general course in geometry, and I imagine that a fair number of mathematics majors graduate without ever having learned what a projective plane is.
This, I think, is a pity. Totally apart from the usefulness of projective geometry in more advanced areas of mathematics such as algebraic geometry or elliptic curves, and in applied areas like computer graphics, the subject is a beautiful one in its own right, particularly in its connections with algebra. The fact that a projective plane (defined axiomatically) turns out to satisfy Desargues’ theorem if and only if it can be coordinatized by a division ring, and, if so, this division ring is a field if only if the plane also satisfies Pappus’ theorem, is a wonderful piece of mathematics, right up there with Galois theory and the Killing-Cartan classification of simple complex Lie algebras.
Despite the paucity of courses in projective geometry, there must still be some demand for textbooks in the area. About five years ago, for example, Oxford University Press published Rey Casse’s Projective Geometry: An Introduction, and more recently there is Ueberberg’s Foundations of Incidence Geometry: Projective and Polar Spaces, a somewhat higher level account. We now also have Richter-Gebert’s text, which not only covers projective geometry but spends almost 600 pages doing so. Moreover, Richter-Gebert obviously also likes the connections between projective geometry and algebra, since algebraic aspects of the subject are emphasized throughout.
The book begins with an introductory chapter on Pappus’ theorem, providing a number of different proofs (nine of them!) in both the Euclidean and projective context, thereby providing an overview and preview of what is to come. (I write “Pappus”, by the way, because that’s the way I was brought up, but the book uses the Greek form: Pappos.)
After this introductory chapter (which, as the author points out, can be skipped), the book divides into three parts. The first (“Projective Geometry”) is an introduction to the basic ideas of the subject: after the (almost obligatory) brief discussion of perspective in painting, the axioms for a projective plane are given. The focus in part I and the rest of the text, however, is not on an axiomatic development of the subject but instead on one particular kind of projective plane, namely the one defined by a field K (the points in this plane being the one-dimensional subspaces of a three-dimensional vector space over K, or, in what amounts to the same thing, nonzero ordered triples of elements of K, two such triples being equated if one is a nonzero scalar multiple of the other). Other chapters in part I discuss standard projective geometry topics such as homogenous coordinates, cross-ratio, and projective transformations; the connection between commutativity of the underlying field and Pappus’ Theorem is stated but not proved (given the author’s emphasis on field planes, the result does not play a major role in the textual development of the subject).
The final two chapters of Part I discuss the bracket of a finite number of points in a projective space (defined as the determinant of the matrix whose columns are the coordinates of the points) as well as homogenous polynomials in these brackets. This turns out to be the key algebraic component utilized by the author in the rest of the book.
All in all, Part I of the text would likely (but for the total absence of exercises) make a good first course in projective geometry; most or all of the topics covered in the undergraduate course that I took appear here. The one exception is conics, but this omission is remedied in the next part of the book, entitled “Working and Playing with Geometry” and consisting of chapters 8 through 15, which is largely devoted to illustrating applications of projective geometry and which makes extensive use of the bracket formalism introduced earlier. Conics and their connection to quadratic forms are treated in considerable detail, but other, less well known topics are also covered, such as the tensor diagrams of chapter 13. Throughout part I of the book, the main emphasis was on the projective line and plane, but in chapter 12 of part II higher-dimensional projective spaces are introduced and discussed.
The final part of the book (“Measurement”) is about complex projective geometry. After an introductory chapter discussing complex numbers, the author begins the study of complex projective geometry by discussing the complex projective line and its connections with Möbius transformations.
Subsequent chapters engage in an interesting program: two points in the complex projective plane (which the author denotes I and J) are defined, and, over the course of several chapters, it is shown how formulas and concepts in Euclidean plane geometry can be expressed in terms of these points and the bracket notation previously introduced. As the author phrases it on page 349: “Euclidean geometry is projective geometry together with I and J.” While I was of course familiar with the fact that the Euclidean plane can be viewed as being contained in the projective plane, this particular formulation was not something I had ever seen in any book before.
Then, generalizing from the points I and J, the author is led to Cayley-Klein geometries, and finally to hyperbolic geometry, the subject of several chapters. While the material in parts II and III of the book may be a bit specialized for general coursework, it is interesting mathematics that is not readily found elsewhere in books. A final chapter of the book (“What We Did Not Touch”) consists of four expository sections, pointing the way to, and giving overviews of, other topics in projective geometry, including, for example, discrete mathematics (matroids and oriented matroids) and quantum theory.
Anybody reviewing a book is likely to have a few quibbles, and I am no exception. I thought that the notation was sometimes unnecessarily cumbersome and on occasion was downright perplexing; for example, on page 58 the author uses the symbol Ø, which is of course almost universally used to denote the null set, to represent an arbitrary space; the resulting definition of a “transformation” as a bijective function T: Ø Ø was a bit jarring, to say the least. Second, I would have preferred to see more of the standard textbooks in projective geometry appear in the bibliography. Finally, I noticed occasional (relatively minor) examples of mathematical sloppiness. For example, Lemma 5.7 (any additive and multiplicative function f from the field of real numbers to itself must be the identity function) omits the one other possibility that f is identically 0. Likewise, the author’s definition of the “bracket” of points as the determinant of a matrix whose columns are the homogenous coordinates of the points makes no mention of the fact that the definition depends on a particular set of homogenous coordinates and is therefore not well-defined. This doesn’t turn out to be an insurmountable problem, but it should, I think, have at least been mentioned.
These quibbles notwithstanding, there is much to admire in this book, which is clearly a labor of love on the part of the author, who mentions in the preface that it took an “eternity” to write. The time and effort shows: considerable care has been taken to make the book as accessible as possible (as illustrated by the inclusion of a chapter on complex numbers), but at the same time the book also contains discussion of a considerable amount of material that I have never before seen in any other text, thereby making it useful as a potential reference as well as teaching tool. And one has to admire the thoroughness of a nine-page index which lists “Calvin and Hobbes” as one of the entries (the reference is to a page on which a reprinted cartoon appears).
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University. He enjoyed having an excuse to dust off the textbook (Blattner’s Projective Plane Geometry) that he used as an undergraduate.