Editing Finite geometry (section)

=== Finite affine planes ===

An affine plane geometry is a nonempty set ''X'' (whose elements are called "points"), along with a nonempty collection ''L'' of subsets of ''X'' (whose elements are called "lines"), such that:
# For every two distinct points, there is exactly one line that contains both points.
# [[Playfair's axiom]]: Given a line <math>\ell</math> and a point <math>p</math> not on <math>\ell</math>, there exists exactly one line <math>\ell'</math> containing <math>p</math> such that <math>\ell \cap \ell' = \varnothing.</math>
# There exists a set of four points, no three of which belong to the same line.
The last axiom ensures that the geometry is not ''trivial'' (either [[empty set|empty]] or too simple to be of interest, such as a single line with an arbitrary number of points on it), while the first two specify the nature of the geometry.

The simplest affine plane contains only four points; it is called the ''affine plane of order'' 2. (The order of an affine plane is the number of points on any line, see below.) Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered "parallel", or a square where not only opposite sides, but also diagonals are considered "parallel".

The affine plane of order 3 is known as the [[Hesse configuration]].

More generally, a finite affine plane of order ''n'' has ''n''<sup>2</sup> points and {{nowrap|''n''<sup>2</sup> + ''n''}} lines; each line contains ''n'' points, and each point is on {{nowrap|''n'' + 1}} lines.