Is it possible to draw a set of 7 points and 7 lines so that each point lies on exactly three of the lines and each line contains exactly three of the points? What about 10 points and 10 lines? What if you want each line to contain exactly four points? If you are intrigued by these types of questions, then I have a book for you. Branko Grünbaum’s new book, Configurations of Points and Lines studies questions about points and lines which intersect in prescribed ways and their generalizations. While many of the questions could be posed to an elementary school student, the answers get very sophisticated very quickly, and there is ample material to fill a book and leave many open questions for the reader to work on.
The first chapter of Grünbaum’s book serves as an introduction, leading the reader through the history of these questions from the work of Pappus in the fourth century through the formal definitions of the questions given by Reye in 1876 and on to the “renaissance” of work that continues in the area today. This section summarizes some of these results, and gives a number of the concepts and definitions used in the study of configurations. Most importantly, it introduces the distinction between combinatorial configurations (which are set theoretic objects), topological configurations (in which we use a more general definition of “line”), and the geometric configurations which can actually be realized by points and lines as above. The study of these different notions of configurations each require different tools, which Grünbaum proceeds to work through.
The second chapter is the longest, and discusses a classification of all (n3) configurations — that is, configurations of n points and n lines, with three points on each line and three lines through each point. In this chapter, Grünbaum classifies the existence of combinatorial, geometric, and topological configurations for various n and describes various results to count the number of such configurations and to classify them. Special attention is given to configurations that admit various symmetries, and the chapter ends with a number of open questions.
(n4) configurations are considered in the next chapter, and one quickly realizes how much less is known in this case. In particular, while these first appeared in an 1879 paper by Felix Klein, in which he proved the existence of a (214) configuration, it is still unknown whether (224) geometric configurations exist. This chapter summarizes the results that are known, and also points out a number of things that we do not know. Several sections discuss in detail the situation of astral configurations, which are a class of geometric configurations with nice symmetry properties.
With a lengthy chapter dedicated to 3-configurations and a shorter chapter dedicated to 4-configurations, it is not surprising that only a single section is written about 5-configurations and there is another section for k-configurations for all k > 5. This section includes the first images of 6-configurations in print anywhere, involving 110 and 120 points! There are also a handful of sections dealing with unbalanced configurations, such as the situation where you have 9 points and 12 lines, with each line containing exactly three points and each point lying on exactly four lines. A final chapter looks at configurations which admit certain properties, such as those that are connected or those that admit Hamiltonian circuits.
This book is written in a style which is very different from most books you will find in the “AMS Graduate Studies in Mathematics” series. In particular, it is much chattier and friendlier to its readers, with loads of color illustrations and digressions into historical and philosophical asides. But this is not to say that the mathematics it contains is not interesting and sophisticated: at various times, Grünbaum’s techniques involve combinatorics, algebraic geometry, group theory, and topology. But the book is almost entirely self-contained, and the author has a tone that is extremely readable; it is probably the only book in this series that I have considered bedtime reading! There are plenty of topics in mathematics that it is hard to imagine writing about in this style — motivic cohomology or lattice delay equations, for example — but I think many authors could learn a thing or two about readability from Grünbaum, and reading his book was a real pleasure.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College, whose mathematical interests include Cryptography, Galois Theory, and Arithmetic Geometry. He can be reached at email@example.com.