Imagine an infinitely large sheet of paper, a straight ruler of infinite length and a pencil with an infinitely sharp point (a tall order). With these tools you may draw infinitely small points and infinitely long straight lines which are infinitely thin. These are the points and lines of `classical' geometry. They have the following properties:

- Two points are joined by exactly one straight line.
- Two lines intersect in at most one point.

- Given a line and a point not on that line, there is exactly one
line through that point which is parallel to the original line. In
other words, all lines through that point
intersect the first line in a single point,
*except one line*which is parallel to the first line.

Semi-Linear spaces which satisfy these axioms are called *affine
planes*. The `classical' plane is an example of an infinite affine
plane. We shall be more interested in *finite* affine planes.

An affine plane of order q has q points on every line, q+1 lines
through every point, q^2 (q squared) points in total and q^2+q
lines. Parallel lines come in groups of q that are all parallel to
each other. Such a group is called a *parallel class*. The lines of an affine plane can be partitioned into q+1 disjoint parallel classes.

**The affine plane of order 2.**

There are four points and six lines. Each line contains two points and each point lies on three lines. There are three parallel classes (black, red and blue) with two lines to each class.

**The affine plane of order 3.**

There are nine points and twelve lines. Each line contains three points and each point lies on four lines. There are four parallel classes (black, red, blue and green) with three lines to each class.

The picture on the left, for example, was obtained by removing the top middle point from the affine plane of order 3. It should look familiar, for it is a picture of the semi-linear space that corresponds to the CDS {0,1,3} modulo 8. Now there are eight points and eight lines, every line contains three points and every point lies on three lines. There are still four parallel classes, but now each class contains only two lines.

In general, the plane obtained in this way has the following properties:

- Two points are connected by at most one line.
- Two lines intersect in at most one point.
- Given a line and a point not on that line, every line through that point intersects the first line, except one.
- Given a point and a line not through that point, every point on that line is connected to the first point, except one. (Consider for example the point 0 and the line {3,4,6} in the picture above, then 0 is connected to 3 and 6, but not to 4.)

A semi-linear space satisfying these four axioms is called a
*semi-affine plane*. In general, removing a point from an
affine plane together with all the lines through it, results in a
semi-affine plane with q^2-1 points and q^2-1 lines, q points on every
line and q lines through every point, q+1 parallel classes of q-1
lines each and likewise q+1 parallel classes of q-1 points each.

In our last example, the parallel classes of points are {0,4}, {1,5}, {2,6} and {3,7}, while the parallel classes of lines are listed below. Lines in the same column belong to the same parallel class:

0 6 5 | 0 1 3 | 0 2 7 | 1 6 7 1 2 4 | 4 7 5 | 3 6 4 | 3 2 5In the following pages we will show how to construct (large) cyclic semi-affine planes of this kind. But first we need to make a small detour...

- Remove all green lines from the affine plane of order 3. Prove that you obtain a semi-affine plane with 9 points and 9 lines.
- Is this semi-linear space cyclic? (Can you number its points in such a way that the lines form shifts of a CDS?)
- Prove that removing an entire parallel class of lines from an affine plane always results in a semi-affine plane.

And now for something completely different

*97/01/01 -
Kris Coolsaet
*