Introduction via the Fano Plane

Abstract

In this book a geometry will usually consist of a nonempty point set and a nonempty line set Here a line is a subset of the point set containing at least two points and a point is contained in at least two lines. For example, the Euclidean plane is a geometry whose point set is the xy-plane and whose lines are the straight lines. Sometimes, for historical reasons, lines will also be called blocks or circles. For example, in the geometry of circles on a sphere the lines are called circles. Usually our geometries also satisfy a number of axioms. For example, both the Euclidean plane and the geometry of circles satisfy an “axiom of joining”; that is, two points in the Euclidean plane are always contained in a unique line, and three points on a sphere are always contained in exactly one circle. In fact, various versions of the axiom of joining are the most important kind of axioms that we will come across, and virtually all the geometries considered in this book satisfy an axiom of joining.