Finite Plane

Definition:

An finite plane is just an affine plane that has a finite number of points and lines. They obey the same axioms as affine planes:

Over a finite field F a finite set of points (x, y) such that x, y ∈ F together with the set of lines y = mx + b and x = b such that m and b ∈ F. The size of the field F is the order of the finite plane.

1. Any two points are on a line.

   y = mx + b is an equation in two unknowns: m and b. Therefore, two equations are necessary to uniquely determine m and b for given values of x and y.

2. Given a point and a line such that the point is not on the line, there is another line the point is on, and the two lines do not share any points.

   It is easy to construct parallel (here, parallel means that the two lines do not have any common points: L₁ ∩ L₂ = ∅) lines in finite planes. Given a point (x₀, y₀) and a line y - mx = b where y₀ - mx₀ ≠ b, clearly the line with the same slope as L₁ that passes through (x₀, y₀) is parallel to it. This assumes that enough points exist so that the parallel line exists.

3. There exist three points not on the same line.

Result 1: Enumerating the points of a finite plane

Claim: If the size of the field F over which the plane π is constructed is n, there are n² points in π

Remark: Sometimes π is written as F × F -- × being the cartesian product. One of the properties of the cartesian product is that if A = B × C, |A| = |B| × |C|. That is, the number of elements in the product is equal to the product of the number of elements in the factors. |F| = n, so |{points in π}| = n².

Result 2: Enumerating the lines of a finite plane

Claim: A finite plane π of order n has n² + n lines.

Remark: There are two kinds of lines on π

Type 1: All lines of the form y = mx + b. m and b are both free variables, and each has n possible values. Therefore, there are n² lines of type 1 on π

Type 2 - infinite slope parallel class: All lines of the form x = b. There are then n lines of type 2 on π

So, in total, there are n² + n lines on π.

Result 3: Planes of order n = p^k, p prime, exist.

Remark: Finite affine planes of a certain order n exist whenever projective planes of order n exist. (This is because projective planes are supersets of affine planes].) It is known that finite planes of order 2, 3, 4, 5, 7, 8, 9, and 11 exist, along with every prime power. No finite plane of order 6 exists, and it was recently discovered (by Lam, Thiel and Swiercz in 1989)¹ that no finite plane of order 10 exists. (It took a hell of a lot of calculation, too.)

Result 4: All finite planes of order n are (n², n, 1) designs.

Remark: This follows from the definition of a (v, k, λ) design - v is the order, k is the number of elements in a block (i.e, number of points on a line) and λ is the number of blocks to which every pair of distinct points belongs.

Result 5: Pictures!

A finite plane of order two: (the little plus signs are points)

      /
 +---+
 |\  |
 | \ |
 |  \|
 +---+
/

The line going from the upper-left to the lower-right wraps around.

A finite plane of order three:

      ------
 --- /      \
/   / \      \
|  +---+---+ |
|  |\ /|\ /|\|
|  | \ | \ | |
|  |/ \|/ \|/ \
|  +---+---+   |
\ /|\ /|\ /|   |
 | | \ | \ |   |
 |\|/ \|/ \|   |
 | +---+---+   |
 |     \  /    |
 |      \-----/
  \-----/

¹ CWH Lam, L Thiel, S Swiercz. "The non-existence of finite projective planes of order 10", Canadian Journal of Mathematics, 1989.