Lecture Notes 2

Pappus' Theorem: If points A,B and C are on one line and A', B' and C' are on another line then the points of intersection of the lines AC' and CA', AB' and BA', and BC' and CB' lie on a common line called the Pappus line of the configuration.

Axioms for the Finite Geometry of Pappus

1. There exists at least one line.
2. Every line has exactly three points.
3. Not all lines are on the same point. [N.B. Change from the text]
4. If a point is not on a given line, then there exists exactly one line on the point that is parallel to the given line.
5. If P is a point not on a line, there exists exactly one point P' on the line such that no line joins P and P'.
6. With the exception in Axiom 5, if P and Q are distinct points, then exactly one line contains both of them.

Pf. Let X be any point. By corrected axiom 3, there is a line not containing X. This line contains points A,B,C [Axiom 2]. X lies on lines meeting two of these points, say B and C [Axiom 5]. There is exactly one line through X parallel to BC [Axiom 4]. There can be no other line through X since by Axiom 4 it would have to meet BC at a point other than A, B or C [Axioms 6 and 5], and this would contradict Axiom 2.

Pappus geometry has 9 points and 9 lines.

Desargues' Theorem: In a projective plane, two triangles are said to be perspective from a point if the three lines joining corresponding vertices of the triangles meet at a common point called the center. Two triangles are said to be perspective from a line if the three points of intersection of corresponding lines all lie on a common line, called the axis. Desargues' theorem states that two triangles are perspective from a point if and only if they are perspective from a line.

Desargues' Configuration has 10 points and 10 lines.

Local Definitions for this geometry only!

The line l is a polar of the point P if there is no line connecting P and a point on l.

The point P is a pole of the line l if there is no point common to l and any line on P.

Axioms for the Finite Geometry of Desargues

1. There exists at least one point.
2. Each point has at least one polar.
3. Every line has at most one pole.
4. Two distinct points are on at most one line.
5. Every line has exactly three distinct points on it.
6. If a line does not contain a point P, then there is a point on both the line and any polar of P.

Theorem 1.11 Every line in the geometry of Desargues has exactly one pole.

Theorem 1.12 Every point in the geometry of Desargues has exactly one polar.

Def: PG(n,q)

Correct definition of order in the text.

Talk about ovals.

Axioms for Euclidean Geometry

Euclid's unstated assumptions.

• Lack of continuity.
• Pasch's Axiom.
• order of points and betweeness
• problems with superposition.

Modern systems:

• Pasch
• Hilbert - in the style of Euclid
• Birkhoff - a different approach

Desirable properties of Axiom Systems

1. consistent
2. complete
3. independent