Affine and semi-affine planes

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.

Lines that do not intersect are called parallel. There is a third property which is slightly more involved. It is often called `the axiom of parallelism':

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.

Finite affine planes

Affine planes with only a finite number of points are called finite. Every line in such a plane has only a finite number of points, and it can be proved that this number is the same for every line. This number (traditionally written as q) is called the order of the plane.

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.

Some examples

The smallest affine planes have order two and three. We present them in the pictures below. Note that we have used different colours for the lines. Each colour corresponds to a different parallel class. Remember that lines need not be straight and that an intersection of two lines only counts when there is a point (a circle) on it.

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.

Semi-affine planes

Affine planes cannot be cyclic, for they do not have the same number of lines and points. That is why we did not put numbers inside the circles in the pictures above. However, if we remove a single point from an affine plane, together with all the lines through that point, we obtain a semi-linear space which has a good chance of being cyclic.

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.)

We use the same term `parallel' for points that are not connected by a line (0 and 4 are parallel points). Parallel points also come in groups called `parallel classes'. In fact, every concept that exists for lines also exists for points.

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 5

In 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...

Problems

Not all semi-affine planes are the result of removing a single point from an affine plane. There are various other constructions:

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