Projective Plane

A Projective Plane is an incidence structure denoted by \( \pi = (\mathscr{P}, \mathscr{L}, \mathcal{I}) \) satisfying the following axioms:

1. P1: Two distinct points determine exactly one line.
2. P2: Two distinct lines intersect in exactly one point.
3. P3: Every line contains at least three points.
4. P4: There exists at least one point not on a given line.
5. P5: There exists at least one line.

Example: Fano Plane (Smallest Projective Plane)
The Fano Plane has 7 points and 7 lines. Each line contains exactly 3 points, and each point lies on exactly 3 lines.

Example: Extension of Young’s Configuration
Adding points at infinity to an affine grid creates a projective plane with 13 points and 13 lines. Each line has 4 points.

Theorem: In a projective plane, two distinct lines determine exactly one point.

Let \( \pi = (\mathscr{P},\mathscr{L},\mathcal{I}) \) be a projective plane.
Also let \( L \) and \( M \) be two distinct lines in \( \pi \).
By P2, there is at least one point \( p \) incident with both \( L \) and \( M \).
Suppose there is another point \( q \) on both \( L \) and \( M \).

Then by P1, there is exactly one line through \( p \) and \( q \), contradicting that both \( L \) and \( M \) contain them.
Hence, the intersection point is unique.

Theorem: Every projective plane contains a four-point.

Let \( \pi = (\mathscr{P},\mathscr{L},\mathcal{I}) \) be a projective plane.
By P5, there is a line \( L \).
By P3, there are three points, say \( p, a, b \in L \).
By P4, there is a point, say \( c \notin L \).
By P1, line \( pc \) exists.

By P3, line \( pc \) has a third point, say \( d \neq p, c \).
Then \( \pi \) contains \( a, b, c, d \), which is a four-point

Theorem: If \( p \) is a point in a projective plane, there are at least three lines through \( p \).

Let \( \pi = (\mathscr{P},\mathscr{L},\mathcal{I}) \) be a projective plane, and \( p \) is a point in \(\pi\).
There exists a line \( L \) not through \( p \).
By P3, \( L \) has at least three points: \( a, b, c \).
By P1, lines \( ap, bp, cp \) exist and are distinct.
Hence, at least three lines pass through \( p \).

Theorem: An incidence structure \( \pi = (\mathscr{P},\mathscr{L},\mathcal{I}) \) is a projective plane if it satisfy:

1. Two points determine a unique line,
2. Two lines meet in a unique point,
3. There exists a four-point.

Let \( \pi = (\mathscr{P},\mathscr{L},\mathcal{I}) \) be aan incidence structure satisfying (1)-(3). We verify the five axioms using (1)-(3).

• P1: Given in (1).
• P2: Given in (2).
• P5: A four-point implies at least one line (via P1).
• P4: The four-point ensures not all points are on one line.
• P3: Assume that, \(p,q,r,s\) is a four point.
  For any line \( L \), pick a point \( p \notin L \) (from four-point). Join \( p \) to three points of the four-point; their joins with \( p \) meet \( L \) in three distinct points.

Hence, \( \pi \) is a projective plane.

Note:
These three conditions are often easier to verify than all five axioms.

Note:
Every affine plane can be extended to a projective plane by adding "points at infinity".