An Introduction to Incidence Geometry

It is always a pleasure to see familiar mathematics presented from a different perspective. In my upper-division geometry class, we spend one chapter out of eight studying incidence geometry as an example of a simple axiomatic system and the very beginnings of an axiomatic development of Euclidean geometry. What An Introduction to Incidence Geometry represents to me is a thorough expansion of what “incidence geometry” encompasses; it stands as a comprehensive exploration of the numerous ideas that flow from the simple notion of a point-line geometry.

This includes ideas from graph theory, which runs throughout the book from an introduction at the start to a treatment of some elements of design theory in the final chapter. These are topics that I have studied at one time or another — but not in this setting. The geometric approach to designs is a fascinating counterpart to the combinatorial approach with which I am more familiar.

At the same time, there are new-to-me topics in abundance: near-polygons, generalized quadrangles, and polar spaces, to name three. While reading this excellent book is not likely to lead to major changes in how I teach incidence geometry (aside, perhaps, from a lot of new answers for students who ask “What is this good for?” — which is not an insignificant benefit), it has certainly expanded my view of the subject and of areas of mathematics adjacent to it, and that cannot be a bad thing.

Mark Bollman (mbollman@albion.edu) is professor of mathematics and chair of the department of mathematics and computer science at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. Mark’s claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.