Dense near polygons.- Regular near polygons.- Glued near polygons.- Valuations.- The known slim dense near polygons.- Slim dense near hexagons.- Slim dense near polygons with a nice chain of convex subpolygons.- Slim dense near octagons.- Nondense slim near hexagons.
"Preface (p. ix-x)
In this book, we intend to give an extensive treatment of the basic theory of general near polygons. The subject of near polygons has been around for about 25 years now. Excellent handbooks have appeared on certain important subclasses of near polygons like generalized quadrangles ([82]) and generalized polygons ([100]), but no book has ever occurred dealing with the topic of general near polygons. Although generalized polygons and especially generalized quadrangles are indispensable to the study of near polygons, we do not aim at giving a profound study of these incidence structures.
In fact, this book can be seen as complementary to the two above-mentioned books. Although generalized quadrangles and generalized polygons were intensively studied since they were introduced by Tits in his celebrated paper on triality ([96]), the terminology near polygon ?rst occurred in a paper in 1980. In [91], Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. In [91], also some very fundamental results regarding the geometric structure of near polygons were obtained, like the existence of quads, a result which was later generalized by Brouwer and Wilbrink [16] who showed that any dense near polygon has convex subpolygons of any feasible diameter.
The paper [16] gives for the ?rst time a profound study of dense near polygons. Other important papers on near polygons from the 1980s and the beginning of the 1990s deal with dual polar spaces, the classi?cation of regular near polygons in terms of their parameters and the classi?cation of the slim dense near hexagons. The subject of near polygons has regained interest in the last years. Important new contributions to the theory were the theory of glued near polygons, the theory of valuations and important breakthrough results regarding the classi?cation of dense near polygons with three and four points on every line.
These new contributions will be discussed extensively in this book. This book essentially consists of two main parts. In the ?rst part of the book, which consists of the ?rst ?ve chapters, we develop the basic theory of near polygons. In Chapters 2, 3 and 4, we study three classes of near polygons: the dense, the regular and the glued near polygons.
Our treatment of the dense and glued near polygons is rather complete. The treatment of the regular near polygons is concise and results are not always accompanied with proofs. More detailed information on regular near polygons can be found in the book Distanceregular graphs [13] by Brouwer, Cohen and Neumaier. In that book regular near polygons are considered as one of the main classes of distance-regular graphs. In Chapter 5, we discuss the notion of valuation of a near polygon which is a very important tool for classifying near polygons."
