Definition 1.1.1. Intersect.
Two lines intersect if and only if they share a point.
Section 1.1 Finite Geometries
Definition 1.1.2. Parallel.
Two lines are parallel if and only if they do not intersect.
Definition 1.1.3. Four Point Geometry.
The four point geometry is defined by the following axioms and definitions.
- There exist exactly four points.
- Any two distinct points have exactly one line on both of them.
- Each line is on exactly two points.
Checkpoint 1.1.4.
Explore the four point geometry as follows.
- Draw and label four points.
- Use axiom 2 to draw as many lines as possible.
- How many lines exist in this geometry?
- Find a pair of parallel lines.
- Can you find three lines that are pairwise parallel?
- Can you find a point that is on three lines?
The four point geometry is an affine geometry. We are also interested in projective geometries.
Definition 1.1.5. Projective Geometry.
A finite projective geometry satisfies the following conditions.
- Every two distinct points have exactly one line on them.
- There are at least four points with no three on the same line.
- Every two lines have at least one point on them both.
Checkpoint 1.1.6.
Construct a projective geometry with exactly 4 points on every line.
