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.
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 1 Assuming 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:
- 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.