Finite geometry describes any geometric system that has only a finite number of points. Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact precisely the same number of points as there are real numbers. There are two main kinds of finite geometry: affine and projective. In an affine geometry, the parallel postulate holds, meaning that the normal sense of parallel lines applies. In a projective geometry, by contrast, any two lines intersect at a unique point, and so parallel lines do not exist. Both finite affine geometry and finite projective geometry may be described by fairly simple axioms.
For affine geometry, the axioms are as follows:
- Given any two distinct points, there is exactly one line that includes both points.
- The parallel postulate: Given a line L and a point P not on L, there exists exactly one line through P that is parallel to L.
- There exists a set of four points, no three colinear.
(Figures of affine planes of orders 2 and 3 to be added.)
The axioms of projective geometry are:
- Two distinct points lie on exactly one line.
- Two distinct lines intersect at exactly one point.
- There exists a set of four points, no three colinear.
Diagram of the Fano plane |
It is well-established that both affine and projective planes of order n exist when n is a prime number raised to a positive integer exponent. It is believed (more precisely, conjectured) that no finite planes exist with orders that are not prime powers, although this statement has not been proved. The best result to date is the Bruck-Ryser Theorem, which states: If n is a positive integer of the form 4k+1 or 4k+2 and n is not equal to the sum of two integer squares, then n does not occur as the order of a finite plane. The smallest integer that is not a prime power and not covered by the Bruck-Ryser Theorem is 10; 10 is of the form 4k+2, but it is equal to the sum of squares 12+32. Using sophisticated techniques and computer analysis, it has been shown that 10 is also not the order of a finite plane. The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.