We (most of us) are familiar with the Euclidean Geometry and the (really ) axioms it is based on. Let’s consider a simpler system here called **Affine plane**.

** Affine Plane **

Def : *An Affine plane is a triple where is a set of points, a set of lines and an incidence relation (just a fancy name) such that : *

- Each line is incident with at least two points and each point is incident with at least three lines.
- Any two distinct points determine a unique line.
- (Playfair axiom) Given any point and a line , there exists a unique line incident to which is parallel to .

Note that two lines , are called parallel if either or there is no point incident to both and . The first axiom simply rules out some trivial cases like , , and all points incident to , etc.

An alternate view here would be to consider as a class of subsets of , then instead of the incidence relation we say belongs to or contains where is a point and is a line. See this for more details.

Lemma 1Assuming the first two axioms to be true, playfair axiom is equivalent to the statement “The parallel relation (defined above) is an equivalence relation and each parallel class is a partition of in the sense that each point is incident with a unique line in a parallel class”.

The most common example of an Affine plane is the Euclidean plane . Here is another example:

Let be a division ring. Consider the system called whose point set is and lines are all sets of the form and for .

It is an easy exercise to see that this defines an Affine plane. Things start getting interesting when we consider the possibility where set is finite. Here’s a non trivial theorem for the case of finite Affine plane:

Theorem 2Let be a finite Affine plane, then there is a number (called the order of the plane) such that :

- Each line is incident with exactly points.
- Each point is incident with exactly lines.
- Each parallel class has lines.
- There are parallel classes.
- There are points.
- There are lines.

*Proof:* We’ll first show that any two points are incident to the same number of lines and call that number . Let , be two points. They determine a unique line which is incident to both, call it . Now, because of the Playfair axiom, there is an obvious bijection between the lines incident to and lines incident to other than . This proves 2.

Now, Let be a line and a point not incident to then there is a unique line parallel to and passing through . Each of the other lines incident to intersect at a unique point. And corresponding to each point on we have a unique line determined by that point and . Hence, the number of points on must be .

Let be a line in a parallel class and a point incident to it. Let be a line different from and incident to . There are other points on . Through each of these points there is a line parallel to and distinct from , and hence number of lines in the parallel class is at least . Also, has a common point with all the members of this parallel class and hence the number of lines is at most .

Let be a point. It has lines incident to it and all of them must lie in different parallel class. Also, is incident to a unique line in each parallel class. Hence, number of parallel classes must be .

5. Follows from 1. and 3. while 6. follows from 3. and 4. Therefore, we have shown the existence of such an .

The simplest example of finite Affine plane is the structure with and = all -subsets of . Another one is the family of Affine planes given by where is the Galois field of order . Therefore, for any prime power we have a finite Affine plane of order .

Given a finite affine plane, must its order be a prime power? Given a prime , is there a unique affine plane of order ? These are two important open problems in the area of finite geometry.

Left maths a long time ago….maybe this will rejuvenate some of it.

And, I have my first follower. Thank you 🙂

I’ll be posting the second part soon. You can ask if something is unclear.

Pingback: My first publication | Anurag's Math Blog