Lecture Notes 2: Some Combinatorial Results

Another Example: Young's Geometry

This affine plane has 9 points and 12 lines. There are 3 points on each line and 4 lines through each point. There are 4 classes of parallel lines (these are color coded above), each class consists of 3 lines and notice that each point of the plane is found on precisely one line of each parallel class.

The numerical relationships found in this example are not accidental. We shall determine them precisely. First, recall that a bijection between two sets is a map (i.e., a function) which is one-one and onto and that the composition of two bijections is another bijection.

Theorem 4: Given two lines l and m in an affine plane, there is a bijection between the points of l and the points of m.

Pf:

Defn: If any line of a finite affine plane contains exactly n points, the plane is said to have order n.(Notice that this concept is well defined by Theorem 4.)

Young's geometry (the example above) is an affine plane of order 3. The example of the smallest affine plane of the last section was an affine plane of order 2.

Defn: A pencil of parallel lines is a maximal set of mutually parallel lines, or equivalently, a set consisting of a line together with all the lines parallel to it. If we define two lines in a plane to be parallel if either they are equal or they do not meet, then parallelism is an equivalence relation on the set of lines in an affine plane. The equivalence classes of this equivalence relation are the pencils of parallel lines.

Theorem 5: Let (,) be an affine plane of order n, then:

  1. has exactly n2 points.
  2. Each point is on n+1 lines.
  3. Each pencil contains n lines.
  4. The total number of lines is n(n+1).
  5. There are n+1 pencils of parallel lines.
Pf:

Note that by Axiom A3, every finite affine plane has order at least 2 and therefore contains at least three pencils of parallel lines.