Once again, Dover Publications has done a service to the mathematical community by saving from extinction a classic, decades-old, text. The rescued object this time is a slim (and very inexpensive, about 10 dollars as I write this) little book, first published in 1968, that gives a good introduction to some of the very interesting mathematics surrounding the theory of projective planes, particularly the finite ones. This Dover edition is an unchanged reproduction of the original, which, as best as I can tell, was one of the earliest texts that attempted to make this material accessible to a broad undergraduate audience. (Artin’s *Geometric Algebra*, published before this text, discusses these ideas, but is not undergraduate-accessible).

Abraham Adrian Albert was of course primarily known as an algebraist, but it is not surprising to see his name on a book about finite projective planes, because this subject offers a beautiful combination of geometry, algebra and combinatorics. Because the basic ideas can be easily summarized and because the mathematics is so pretty, let us first survey some of the main ideas before looking at the book in more detail.

A projective plane can be defined axiomatically by taking as undefined terms “point”, “line” and an incidence relation (which I’ll describe here by using informal language like “lies on” or “passes through”), and postulating the following: there exist four points, no three of which lie on a line; any two distinct points lie on a unique line; any two distinct lines pass through a unique point. So, in a projective plane, there are no parallel lines; any two lines intersect.

The classic example of a projective plane is the “extended Euclidean plane”, obtained by affixing an “ideal point” to every Euclidean line (the same ideal point to all parallel lines) and then creating an “ideal line” whose points are precisely the ideal points. Another family of examples of a projective plane is obtained by taking a division ring \(D\), defining a “point” to be an ordered triple \([x, y, z]\) of elements of \(D\), subject to the rules that not all three components are zero and that two triples are considered equal if one is a nonzero scalar multiple of the other. Lines can be defined the same way, though let’s (following the textbook’s notation) use columns \(\begin{bmatrix}a\\b\\c\end{bmatrix}\) to denote lines. Then, by definition, the point \([x, y, z]\) lies on the line \(\begin{bmatrix}a\\b\\c\end{bmatrix}\) if and only if \(ax + by + cz = 0\). It is then easy to verify that this is indeed a projective plane, which I’ll denote \(P_2(D)\). One can also verify that the extended Euclidean plane is isomorphic to \(P_2(\mathbb{R})\), where \(\mathbb{R}\) denotes the field of real numbers.

The question then arises: given an axiomatically defined projective plane, when can we recognize this plane as being isomorphic to some \(P_2(D)\) for a division ring \(D\)? The lovely answer links algebra to two classical theorems of geometry, namely the theorems of Desargues and Pappus, both of which are true in the extended Euclidean plane, but neither of which is provable from the axioms of a projective plane. It turns out (after a good deal of work) that a particular projective plane is isomorphic to \(P_2(D)\) if and only if Desargues’ Theorem is true in that plane; moreover, we can take \(D\) to be a field (i.e., commutative division ring) if and only if the plane satisfies Pappus’s Theorem. So, in a projective plane, algebraic considerations imply that the truth of Pappus’s Theorem implies that of Desargues’ Theorem (a result that can also be given a fairly hard direct geometric proof) and, thanks to Weddeburn’s Theorem (any finite division ring is a field), any finite plane satisfying Desargues also satisfies Pappus. (There is, to my knowledge, no known purely geometric proof of this latter fact.)

There are finite planes that do not satisfy Desargues’ theorem, but one thing that can be said is that for *any* given finite projective plane, there is a positive integer \(n\) with the property that every line contains exactly \(n+1\) points, every point lies on exactly \(n+1\) different lines, and the total number of points (and also lines) in the plane is \(n^2+n+1\). In this case, we say that our plane has order \(n\). So, for example, \(P_2(D)\), where \(D\) is the field with two elements, produces a plane of order 2 containing exactly 7 points and 7 lines. This plane is often referred to as the Fano plane.

The theory of finite planes leads to interesting and very difficult combinatorial questions, among them: for which integers \(n\) does there exist a projective plane of order \(n\)? When \(n\) is the power of a prime, say \(n=p^m\), there does exist such a plane; just consider \(P_2(D)\) where \(D\) is the field with \(p^m\) elements. There are no known projective planes of order \(n\) where \(n\) is not the power of a prime, but on the other hand nobody has yet proven that this obviously sufficient condition is also necessary. One celebrated result in this direction is the famous Bruck-Ryser theorem: if \(n\) is a positive integer that is congruent to either 1 or 2 mod 4, and \(n\) is not the sum of two squares, then no projective plane of order \(n\) exists. (So, for example, there is no projective plane of order 6.)

With this as background (additional facts will be stated later, as needed), we can now discuss the book in more detail. It begins in chapter 1 with the axioms defining a projective plane. The basic properties of projective planes (not necessarily finite at this point) are discussed, as are related topics like the principle of duality; Desargues’ Theorem makes its first appearance in this chapter as well. In chapter 2, the basic combinatorial facts about finite planes are established, after which collineations (i.e., bijective incidence-preserving mappings of a plane onto itself) are defined and shown to form a group. The concept of a loop (a more general notion than a group; associativity is not assumed) is also introduced but this more general notion does not get used until later.

Chapter 3 discusses planes defined by fields, as described above; as only finite fields are considered, there is no need at this point to discuss division rings. Matrix-induced collineations of these fields are defined and discussed.

In field planes, the points are coordinatized by triples of field elements; the next chapter discusses the possibility of coordinatizing an arbitrary projective plane by an algebraic structure more general than a division ring. In particular, it turns out that given *any *plane, it is possible to create an algebraic object called a ternary ring that can be used to coordinatize it. Unlike the “real” rings that undergraduates learn about in their first semester of abstract algebra and which are defined by binary operations of addition and multiplication, a ternary ring \(R\) is defined by a ternary operation \(F\), which associates to every ordered triple \((x, y, z)\) of elements of \(R\) an element \(F(x, y, z)\in R\); of course \(F\) must also satisfy a bunch of identities, which we need not write down here. Suffice it to say that any division ring \(D\) can be viewed as a ternary ring by defining \(F(x, y, z)\) to be \(xy + z\). Ternary rings are somewhat cumbersome to work with, but they do offer a way of coordinatizing any projective plane.

In the next two chapters, the authors work towards establishing the connection, mentioned earlier, between Desargues’ theorem and the coordinatization of a projective plane by a field. This requires initial analysis of central collineations in a projective plane. Since the authors work in the context of finite planes here, they do not (thanks to Weddeburn’s theorem) have to deal with division rings and can instead prove the result that a finite projective plane is a field plane if and only if Desargues’ theorem holds. However, at the end of chapter 7, the connection between Pappus’s theorem and commutativity of the field is explained.

The last chapter in the book constructs classes of examples of non-Desarguesian finite projective planes. These constructions also involve a fair amount of abstract algebra, and the details in the various constructions are often just sketched or left to the reader.

A natural question to ask about any book that is almost half a century old is: how has it held up over the years? In some respects, this text does show its age. There have been advances in mathematics since this book was written, and at least one statement made in the text (that the existence of projective planes of order 10 remains open) is no longer true: in 1989, Clement Lam, and others, established, with significant computer assistance, that no such planes exist.

In addition, the notation used by the authors struck me as somewhat clunky: upper-case German letters like \(\mathfrak{M}\) are used to denote projective planes (try writing *that *on a blackboard), and the statement that a point \(P\) lies on line \(l\) is expressed not by some simple and intuitive expression like \(P\in l\) but instead by a more cumbersome expression with a fancy letter “I”: P* I* *l*.

The original volume contained a few typos, and, since this is an unaltered reprint, they still exist. The most serious one that I noted occurs on page 47, where the definition of a ternary ring misstates one of the conditions that must be satisfied: the statement should read \(F(a, 0, c) = F(0, b, c) = c\), not \(0\).

The authors have made a real effort to keep prerequisites to a minimum and do not assume much in the way of technical background. They also intersperse helpful exercises throughout the text. Nevertheless, this book was written for students of a different era, from whom more was expected. More recently, other books have appeared that address the subject matter of this text; good accounts can be found, for example, in Bennett’s *Affine and Projective Geometry* and Casse’s *Projective Geometry: An Introduction*, neither of which is limited to finite planes. I suspect that the current generation of undergraduate students would find these texts somewhat easier to read than the book under review. However, it is always interesting to read a trailblazing book, especially one by a renowned mathematician, and the short, succinct nature of this text allows a fairly quick entrée into this area of mathematics. For these reasons alone, the book deserves a recommendation. Anybody with an interest in projective geometry will want to look at this text.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.