Finite Geometries of Fano and Young

Fano's geometry

Undefined terms: point, line, on

Axioms:

1. There exists at least one line.
2. Every line has exactly three points on it.
3. Not all points are on the same line.
4. For each two distinct points, there exists excatly one line on both of them.
5. Each two lines have at least one point on both of them.

Theorem 1.7. Each two distinct lines have exactly one point in common.
Proof. (See the one in the book)

Theorem 1.8. Fano's geometry consists of exactly seven points and seven lines.
Proof. (See the one in the book)

Exercise 14. For Fono's geometry, prove that each point lies on exactly three lines.

Proof. (cuter than the one presented in class!) Let A be a point of Fono's geometry. We need to show that A is on at least three lines and, at the same time, at most three lines. Let us show first that A is on at least three lines. For each of the six remaining points, there is exactly one line on the point and point A. Among these six lines, there must be at least three distinct ones, otherwise a line has to be on more than three points (7/2 > 3) violating Axiom 2.

Next, we verify that there are at most three line on A. Assume there four lines on A. Then, by Axiom 4, point A is the only point on all these four lines, and by Axiom 2, each of these four lines must be on two other points. This will yield 8 points in addition to point A, contradicting Theorem 1.8 (Fano's geometry consists of seven points.). Done.

Young's geometry

Replace Axiom 5 of Fano's geometry by:

Axiom 5: If a point does not lie on a given line, then there exists exactly one line on that point that does not intersect the given line.

Theorem. Young's geometry has nine points and 12 lines.
Proof. (Your home work asks you to show that there are at least nine points.)