Four point Geometry
Undefined Terms
Points
Lines
Belongs to
Axioms
1. There are exactly four distinct points
2. Any two distinct points have exactly one line
3. Each line is exactly on two points
Theorems
1. If two distinct lines intersect, they contains exactly one point
2. There are exactly six lines
3. Each point lies exactly on three lines
4. Each distinct line has exactly one line parallel to it
Note:
The figure given above is an example of this model because it satisfy all axioms.
The geometry in Figure above has four points and six lines and obeys the "Playfair" parallelism axiom. To help you understand the meaning of this diagram, note that if one chooses the point a1 which is not on line L1' (which consists of points a2 and a3), then the line L1 (consisting of points a1 and a0) contains a1 and is parallel to L1'.
It does not look as if L1 and L1' are parallel in Figure above but these lines are parallel in this four point geometry.
The reason is that the point where these lines appear to meet is not a point of the geometry. Only the four points represented by dark dots are points of the geometry!