You are here

Points and Lines: Characterizing the Classical Geometries

Ernest E. Shult
Publication Date: 
Number of Pages: 
[Reviewed by
Ursula Whitcher
, on

The subject of Points and Lines is incidence geometry; the classical geometries referred to in the subtitle are Lie incidence geometries. Shult has designed the book as a self-contained resource for a graduate student who plans to pursue research in this area.

Shult’s background is in finite group theory, and his treatment is highly algebraic and axiomatic. This approach allows a student to approach open research questions with a minimum of prerequisites. On the other hand, because Shult does not assume his reader has any background in topology or differential and algebraic geometry, historical and topological motivations for constructions can be obscured. For example, Shult does not assume his readers have encountered universal covering spaces, and the explanation he gives for “why universal covering spaces are important” is confined to graph theory and incidence geometries.

In keeping with its role as a self-contained resource, the book gives detailed proofs and offers exercises at the end of each chapter, organized by topic. Diagrams are rather sparse, so geometrically inclined readers will wish to illustrate the text, in addition to working through the official exercises. Though the book’s tone is friendly and even chatty, the intensity of detail and sparsity of illustrations make it more suitable as a handbook for experts and would-be experts than as a first introduction to incidence geometry. 

Ursula Whitcher is an assistant professor at the University of Wisconsin-Eau Claire.


1 Basics about Graphs
2 .Geometries: Basic Concepts
3 .Point-line Geometries
4.Hyperplanes, Embeddings and Teirlinck's Eheory

II.The Classical Geometries

5 .Projective Planes
6.Projective Spaces
7.Polar Spaces
8.Near Polygons


9.Chamber Systems and Buildings
10.2-Covers of Chamber Systems
11.Locally Truncated Diagram Geometries
12.Separated Systems of Singular Spaces
13 Cooperstein's Theory of Symplecta and Parapolar Spaces

IV.Applications to Other Lie Incidence Geometries

15.Characterizing the Classical Strong Parapolar Spaces: The Cohen-Cooperstein Theory Revisited
16.Characterizing Strong Parapolar Spaces by the Relation between Points and Certain Maximal Singular Subspaces
17.Point-line Characterizations of the “Long Root Geometries”
18.The Peculiar Pentagon Property