Dimension of an affine plane

Question

In geometry, an affine plane is defined as a system of points which fullfill:

1) Any two distinct points lie on a unique line. 2) Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. 3) There exist three non-collinear points.

How does this definition tell me that an affine plane is an affine space which has dimension 2. Every book says that an affine space has dimension 2, but I cannot proof it.

My definition of an affine space A is that: Given a set A, whose elements are taken as points. Furthermore, let a K-vector space V_A be given and a map C: A x A, which uniquely assigns a connection vector pq to each ordered pair (p, q) with p, q in A; such that the following axioms are satisfied: a) For every vector v in V_a and every point p in A there is exactly one point q in A with v = {pq}. b) {pq} + {qr} = {pr}.

Answer 1

If the points in the affine plane in your first paragraph are points in ordinary three dimensional Euclidean space then the two dimensionality follows from the fact that if you had four noncoplanar points you could find skew lines that violate the uniqueness in

there is a unique line which contains the point and does not meet the given line.

If your affine plane is an abstract object (not a part of Euclidean space) there's a fair amount of work to do to get from that definition to your definition of an affine space.

Consider the four vertices of a tetrahedron. Define the lines to be the six edges, thought of as pairs of points. The conditions for an affine plane are satisfied. This structure is in fact an affine space: it is the entire two dimensional plane over the two element field $K = \mathbb{Z}_2$.