Incidence axioms for affine geometry

Abstract

In terms of incidence alone it is possible to define an affine plane, as Artin does [I], by calling lines parallel if they do not intersect, and basing the definition on the Euclidean axiom that there is a unique parallel to a line through a point not on the line. In higher dimensions we can define afine geometry by deleting the points and lines of a hyperplane from a projective geometry, using the axioms of Veblen and Young [4]. It is an easy exercise to show that the Artin approach and that of Veblen and Young agree in the definition of an afIine plane. Sut in higher dimensions it is not clear how an afline geometry can be defined directly so that it can be shown to arise from a projective geometry by deleting the points and lines of a hyperplane. This paper gives a set of axioms which have this property. We must define parallelism in such a way that nonintersecting lines in a plane are parallel. But the real surprise is that this is not enough. In Section 5 an example is given of a geometry in which every plane is an afhne plane, but the geometry as a whole is not an afline geometry. If we think of the embedding of an affine geometry into a projective geometry, lines are parallel if and only if they pass through the same projective point at infinity. In particular, the abstract property of parallelism between lines must be transitive, and three parallel lines need not lie in a plane. Thus the axiom of transitivity of parallelism (Axiom A4 in Section 2) is a three-dimensional axiom. The axioms are given in Section 2. Affine planes are defined in Section 3, and their main properties are developed there. The issue as to whether or not the plane is Desarguesian does not arise. In Section 4, the ideal (infinite) points and lines are defined and adjoined to the affine geometry. It is shown that the resulting geometry is a projective geometry. From the treatment here, it follows that if we have an incidence geometry, consisting of points and certain distinguished subsets of points which we call