Projective geometry is one of those subjects that should be, but rarely is, taught to American undergraduate mathematics majors. It is an elegant subject that can be understood by people with relatively modest background in college mathematics, yet is also a very useful one, finding applications not only in art and science but also in other branches of mathematics such as algebraic geometry and elliptic curves. Notwithstanding this, very few undergraduate colleges currently offer courses in projective geometry, and my guess is that quite a few people who graduate from these colleges with degrees in mathematics do so without knowing, for example, what homogenous coordinates are, or what Desargues’ Theorem states.
It wasn’t always this way. Back when I was an undergraduate, more than 40 years ago, my college offered, as I recall, five different upper-level courses in geometry: one each on “college geometry”, “higher analytic geometry”, foundations of geometry, projective geometry and geometric transformations. The university I currently teach at, by contrast, only offers two semesters of geometry (the precise contents of which vary to some extent with the interests of the person teaching it), and in comparison to other institutions that’s on the high end; many colleges only offer one semester of geometry. Back then, undergraduate-level books devoted entirely to projective geometry were not uncommon: for example, I remember reading books on the subject by Blattner, Seidenberg, Stevenson and Garner. Nowadays it seems somewhat harder for a novice undergraduate to find an elementary introduction to this subject.
The book under review attempts to help fill that gap. It is intended for beginners, and the prerequisites (essentially some background in beginning linear algebra, although a semester of abstract algebra wouldn’t hurt either) are fairly minimal. However, even assuming that there are universities around these days that offer courses on projective geometry, this book, as the author himself states in the preface, “is not a ‘textbook’ ”. There are no exercises, and although there are proofs presented here, the book does not generally follow the traditional theorem/proof format usually employed by more conventional mathematics texts; it is mostly organized around examples.
Also, the selection of topics (discussed in more detail in the next few paragraphs) is a bit eclectic: the author wanted, in his words, to “present a collection of ideas that have appealed to me”. As a result, there are things covered here that one might not expect to find in a beginning course on the subject (specifically, a great deal of attention is paid to configurations of points and lines which exhibit a lot of symmetry), but there are also some topics that are not covered as early, or in as much detail, as one might expect: finite projective planes and spaces do not appear, for example, until late in the book, and although the author does mention that there are examples of projective planes in which Desargues’ theorem fails, no examples of such non-Desarguesian planes are given.
The first two chapters in the book address, respectively, the synthetic (i.e., axiomatic) and analytic (homogenous coordinates) aspects of the subject. In chapter 1, the author briefly refers to the first five axioms of Euclid’s Elements, points out how Bolyai and Lobachevsky replaced the fifth postulate with an equally consistent one, and then proposes going the other route and eliminating parallel lines altogether. After a brief discussion of perspective drawing to illustrate this idea, the author (bypassing projective planes) gives a set of axioms for projective three-space (and, later in the chapter, for projective n-space). He also mentions how Euclid’s first and second postulates, together with Playfair’s postulate (through a given point P not on a given line l, there is one and only one line parallel to l) can be used to define affine geometry (really an affine plane, since none of the axioms he quotes mentions more than one plane). The theorems of Desargues and Pappus are discussed and there is a rapid summary of how coordinates can be introduced into a projective space. Even in this early chapter, the reader sees how certain configurations of points and lines will play an important role in what follows. Chapter 2 introduces (real) homogenous coordinates; among the applications made of them are proofs of the theorems of Desargues and Pappus. Plucker coordinates are also mentioned at the end of the chapter.
The next four chapters discuss, respectively, sets of points defined by homogenous polynomials of degree one, two, three and four; these chapters are titled “Linear Figures”, “Quadratic Figures”, “Cubic Figures” and “Quartic Figures”. The emphasis in all these chapters is on interesting and symmetric configurations of points. The chapter on linear figures, for example, discusses not only familiar concepts such as the projective line and cross ratio, but also configurations of points such as “Richmond’s hexastigm” and “Schlaffli’s double-six”, neither of which I had ever heard of. It is in this chapter that the author extends the idea of homogenous coordinates to those with complex entries, and provides a discussion of the complex projective line. The chapter on quadratic figures is, not at all surprisingly, focused on conics and related topics such as Pascal’s theorem. The chapters on cubic and quartic figures address topics that would not be out of place in an introductory algebraic geometry text, including, for example, plane and twisted cubic curves and cubic surfaces (and the canonical forms of these objects) and various quartic surfaces such as Kummer surfaces.
The final chapter is on finite projective geometries and their associated groups. The author begins this chapter by pointing out that when dealing with homogenous coordinates, one can take the coordinate entries from any field rather than just from the field of real or complex numbers, and, in particular, one can take them from finite fields. (He provides a rapid overview of finite fields, but students who are not already somewhat familiar with the technique of constructing them by taking quotients in a polynomial ring by ideals generated by irreducible polynomials might find the discussion a bit hard to follow.) Following this, the author then surveys various two- and three-dimensional projective geometries, emphasizing configurations in each and mentioning some interesting connections to such other areas as error-correcting codes.
I have a few quibbles. First, the notation occasionally struck me as nonstandard and likely to discourage “drop in” readers of the book. For example, the author uses a “box notation” (listing numbers in a square array enclosed by a box) which I have not seen previously; a casual reader picking up the book and seeing a certain configuration described by 16 numbers described in a 4 × 4 boxed array might be understandably somewhat confused.
Second, there were occasions when I thought things could have been more precisely stated: of the four axioms that the author gives for projective three-space, for example, none of them actually assert the existence of any points, lines or planes; in addition, it is never explicitly stated that a plane containing two points also contains the line determined by these points. As another example from later in the book (page 133), the author defines an algebraic variety to be a homogenous polynomial (rather than the set of solutions to a homogenous polynomial), and then, having done so, proceeds to state in the very next sentence that a variety is a “hypersurface in N-dimensional projective space”, which it certainly is not if it is a polynomial.
Finally, and I freely admit that this reflects my own biases, the author makes use of the hated (at least by me, and surely by others!) Einstein Summation Convention. Since one of the author’s research interests is relativity theory, it is not surprising that he would be more kindly disposed towards the Summation Convention than I am. While I have gotten used to having to deal with this in books on differential geometry, I would prefer not to have to encounter it elsewhere.
Notwithstanding these quibbles, I do think there is a lot of interesting material in this book that is not readily available elsewhere, and while I doubt (for reasons expressed above) that it would function well as a text for a course on projective geometry (for that, I would look to books like Casse’s Projective Geometry: An Introduction, or Bennett’s Affine and Projective Geometry), it is certainly something that should be kept close at hand, as a potential source of interesting examples, by anybody interested in the material.
Mark Hunacek (email@example.com) teaches mathematics at Iowa State University.