G 8
FINITE GEOMETRIES*
Finite or "miniature" geometries have only a few axioms and theorems and a finite number of elements; that is, a finite number of points or lines or "things to work with." The study of finite geometries provides an opportunity to study geometries with a simple structure. Historically, the first finite geometry was a three-dimensional geometry, each plane containing seven points and seven lines. This finite geometry was explored by Gino Fano in 1892. Finite projective geometries were studied by Oswald Veblen beginning in 1906. Since that time, a great many finite geometries have been or are being studied.
In general, all finite geometries have point and line as undefined terms. A line in finite geometry is assumed to have more than one, but only a finite number of points.
Three-Point Geometry
Undefined terms: point, line, on.
Axioms:
1. There exists exactly three distinct points.
2. Two distinct points are on exactly one line.
3. Not all the points are on the same line.
4. Two distinct lines are on at least one point.
Definition 1. Two lines with a point in common are called intersecting lines.
Theorem 1. Two distinct lines are on exactly one point.
*Source: Smart, J. Modern Geometries. 3rd Ed. Belmont, CA: Brooks/Cole, 1988.