Recently, in connection with a review I wrote of Richter-Gebert’s Perspectives on Projective Geometry, I had occasion to look up Blattner’s Projective Plane Geometry, the textbook (long since out of print) that I used as a student for an undergraduate course in projective geometry, and, in the process, reminded myself how tastes change in mathematics education. Back in the 1970s and early 1980s, a person teaching a course in projective geometry had no real shortage of textbooks from which to choose (for example, the books by Blattner, Stevenson, Seidenberg, or Garner), but now, assuming that one could even find a college mathematics department offering such a course, the instructor looking for a suitable text would find the pickings to be much slimmer. Most modern texts which develop the subject do so as part of a more sophisticated program; the book by Richter-Gebert, for example, is clearly intended as a book for graduate students or mathematicians, and the upcoming book by Ueberberg (Foundations of Incidence Geometry) is advertised on the Springer website as a research monograph. Twenty-first century books devoted exclusively to developing the subject from scratch for study by undergraduates appeared to be heading towards extinction.
Which, of course, is one reason why Rey Casse’s book, intended as a text for an undergraduate course rather than as a monograph or encyclopedic treatment, is so welcome. Another, more important, reason, is that the book is quite a good one.
Casse develops the material in a way that I think is (with one minor exception that is really a matter of individual taste) very logical and sensible. After a chapter summarizing what knowledge is assumed on the part of the reader (some linear algebra and basic field theory), there is a chapter on the extended Euclidean plane EEP (obtained by attaching a “point at infinity” to every ordinary Euclidean line, with parallel lines having the same point at infinity; these extended lines, and the new line consisting of all the points at infinity, become the lines of the EEP) and the configurations of Desargues and Pappus in the context of the EEP. The next chapter introduces axioms for projective space (and is the one exception I referred to above — my inclination would be to introduce the considerably simpler axioms for the projective plane first, before tackling spaces of greater dimension) and discusses consequences of these axioms, including duality. It is proved that any n-dimensional projective space, with n > 2, satisfies Desargues’ theorem.
The next (fairly lengthy) chapter is a rather thorough discussion of planes and higher-dimensional projective spaces defined by fields. (In the case of planes, for example, the points are the ordered triples of points, not all zero, with two ordered triples identified if one is a nonzero scalar multiple of the other.) Basic concepts like cross ratio and harmonic points are introduced, it is proved that any field plane satisfies Desargues’ theorem, and an example is given to show that not all projective planes do. (Actually the author imposes the requirement that the field have at least three elements, since for the field with two elements the resulting projective plane has only seven points and a non-degenerate Desargues configuration does not exist; many authors avoid distinguishing this case by simply viewing the seven-point plane as vacuously satisfying Desargues’ theorem.) The author also discusses in this chapter the transformations of a projective plane, relating collineations of a field plane to the automorphisms of the underlying field (this is one version of the Fundamental Theorem of Projective Geometry), and proves theorems of Bruck and Singer that are not generally located in the undergraduate textbook literature.
Because the points in the projective planes discussed in this chapter are defined by homogenous coordinates, it is natural to ask whether, given any projective plane, it is possible to “coordinatize” it. This is the subject of the next chapter, in which Casse, starting with a general projective plane, introduces coordinates, the components of which define a set which he calls γ and on which are defined operations of addition and multiplication. The projective plane satisfies Desargues’ theorem if and only if γ turns out to be a division ring; the projective plane satisfies Pappus’ theorem if and only if γ turns out to be a field. (Combine this with Weddeburn’s theorem that any finite division ring is a field, and you have the interesting consequence that any finite projective plane which satisfies Desargues’ theorem also satisfies Pappus’; there is, I believe, no known purely geometric proof of this result.) For arbitrary projective planes γ doesn’t have this kind of nice algebraic structure; it is something called a ternary ring, a concept developed in this chapter.
The next chapter is concerned with non-Desarguesian planes, and primarily consists of the details of the construction of two classes of them, the Hughes planes and the Hall planes, based on a detailed analysis of the collineation group of a projective plane. The final two chapters of the book discuss, respectively, conics in a field plane and quadrics (the higher-dimensional analog) in projective three-dimensional field spaces. The discussion in these final two chapters is, as one would expect, heavily algebraic, and provides a nice glimpse of algebraic geometry.
This book seems to be a very viable candidate for a one-semester course in projective geometry (though there is almost surely more in the book than can be covered in that period of time). It is clearly addressed to the student rather than a researcher, and there are a reasonable number of worked-out examples and homework exercises of varying difficulty (none of which have answers in the back of the book). The bibliography, though, is poor; the author suggests in the preface that people looking for further reading consult the bibliography in Hirschfield’s Projective Geometry Over Finite Fields, but I think a student should not have to hunt up a different book (which his or her university library may not have; it is not in the Iowa State library) in which to find suggestions for further reading.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.