Is this geometry?

There is a branch of mathematics called `Combinatorial Geometry' which deals with properties of certain special sets of certain special elements. Often these elements are called `points' and these sets are called `lines' and special requirements (called `axioms') are introduced which must be met by these points and lines. Sometimes these axioms are even reminiscent of properties of real-life points and lines!

For example, there are structures called semi-linear spaces where only the following properties are required:

Every line contains at least two points.
Every point belongs to at least two lines.
For two different points there is at most one line to which they both belong.
Two different lines have at most one point in common.

(The last axiom is a consequence of the other three, and could therefore have been omitted.)

The straight lines and points `as we know them' clearly satisfy these requirements. So, we all live in a semi-linear space! In fact, we live in a so-called 3-dimensional affine geometry, a structure which is required to satisfy even more axioms. We will meet the word `affine' again later.

What has all this to do with CDSs?

Consider any CDS of size s (larger than 1) and modulus n. A combinatorial geometer would call the numbers from 0 up to n-1 points and the shifts lines. Now have a look at the axioms of `semi-linear space' rephrased in the terminology for CDSs:

Every shift has size at least 2.
Every point belongs to at least 2 shifts.
Any pair of different numbers (in the range 0..n-1) occurs in at most one shift.
Two shifts have at most one element in common.

Rule 3 and 4 are properties which were already listed on the previous page. Rule 1 and 2 can even be strengthened: every shift has the same size s and every number lies in the same number (s) of shifts.

Eureka! The shifts of a CDS form a semi-linear space, be it with points that cannot be drawn on a piece of paper and lines to which the adjective straight is totally irrelevant. In fact, in our semi-linear space there are only a finite number (n) of points and only a finite number (the same n) of lines. We are doing `Finite' geometry.

How about an example ?

Have a look at the picture on the left. (Incidentally, this is a solution to the last problem of the previous page.) The numbered circles are `points' and the curves and straight lines are `lines'. You can easily check that no two `lines' ever intersect in more than a single `point' (an intersection point only counts when it has a numbered circle on it). This picture therefore represents a semi-linear space.

If for every `line' in a picture you make a list of all `points' on that line, then you obtain the following list:

We have carefully ordered this list so you will immediately recognize this as the answer to problem 3 of the previous page. These are the shifts of a CDS of size 3 and modulus 8.

Is this any help?

96/12/30 - Kris Coolsaet