2nd Stage, Lecture 3: Fano s and Young s Geometries

Fano’s Geometry Ax1. There exists at least one line. Ax2. Every line of the geometry has exactly 3 points on it. Ax3. Not all points of the geometry are on the same line. Ax4. For two distinct points, there exists exactly one line on both of them. Ax5. Each two lines have at least one point on both of them.

Theorem 3.4.1. Each two lines have exactly one point in common. Proof. Assume that two distinct lines l ? m have two distinct points in common P and Q. From Ax4, (C!), since these two points would then be on two distinct lines. Theorem 3.4.2. Fano s geometry consists of exactly seven points and seven lines.

Proof. First, we have to show that there are at least 7 points and seven lines. Assume that there is an 8th point. By Ax4, it must be on a line with point 1. By Ax5, this line must meet the line containing points 3,4 and 7. But, the line cannot meet at one of these points (C! Ax4). So, the point of intersection would have to be a fourth point on the line 347(C! Ax2). Thus, there are exactly seven points and seven lines.




Young’s Geometry Ax1. There exists at least one line. Ax2. Every line of the geometry has exactly 3 points on it. Ax3. Not all points of the geometry are on the same line. Ax4. For two distinct points, there exists exactly one line on both of them. Ax5. 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.