A finite geometry is 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. A finite geometry can have any (finite) number of dimensions.

Finite geometries may be defined via linear algebra, as vector spaces and related structures over a finite field, which is called Galois geometry, or can be defined purely combinatorially. Many, but not all, finite geometries are Galois geometries – for example, any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (the projectivization of a vector space over a finite field), so in this case there is no distinction, but in dimension two there are combinatorially defined projective planes which are not isomorphic to projective spaces over finite fields, namely the non-Desarguesian planes, so in this case there is a distinction.

[edit] Finite planes

The following remarks apply only to finite planes. There are two kinds of finite plane geometry: affine and projective. In an affine geometry, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, and so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms.

An affine plane geometry is a nonempty set X (whose elements are called "points"), along with a nonempty collection L of subsets of X (whose elements are called "lines"), such that:

1. Given any two distinct points, there is exactly one line that contains both points.
2. The parallel postulate: Given a line and a point p not on , there exists exactly one line containing p such that or
3. There exists a set of four points, no three of which belong to the same line.

The last axiom ensures that the geometry is not empty, while the first two specify the nature of the geometry.

Diagram of the finite affine plane of order 2, containing 4 points and 6 lines. "Lines" of the same color are "parallel".

The simplest affine plane contains only four points; it is called the affine plane of order 2. Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered "parallel", or a square where not only opposite sides, but also diagonals are considered "parallel". More generally, a finite affine plane of order n has n² points and n² + n lines; each line contains n points, and each point is on n + 1 lines.

Diagram of the finite affine plane of order 3, containing 9 points and 12 lines. "Lines" of the same color are "parallel" in the sense that the intersection of the set of points in two lines of the same color is empty.

A projective plane geometry is a nonempty set X (whose elements are called "points"), along with a nonempty collection L of subsets of X (whose elements are called "lines"), such that:

1. Given any two distinct points, there is exactly one line that contains both points.
2. The intersection of any two distinct lines contains exactly one point.
3. There exists a set of four points, no three of which belong to the same line.

Diagram of the Fano plane

An examination of the first two axioms shows that they are nearly identical, except that the roles of points and lines have been interchanged. This suggests the principle of duality for projective plane geometry, meaning that any true statement about the geometry remains true if we exchange points for lines and lines for points. While the third axiom only requires the existence of four points, the plane must contain at least seven points in order to satisfy the first two axioms. In this simplest of the projective planes, there are also seven lines; each point is on three lines, and each line contains three points. This particular projective plane is sometimes called the Fano plane. If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. For this reason, the Fano plane is called the projective plane of order 2. In general, the projective plane of order n has n² + n + 1 points and the same number of lines (respecting duality); each line contains n + 1 points, and each point is on n + 1 lines.

A permutation of the Fano plane's seven points that carries collinear points (points on the same line) to collinear points is called a "symmetry" of the plane. The full symmetry group is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2), and general linear group GL(3,2).

[edit] Order of planes

A finite plane of order n is one such that each line has n points (for an affine plane), or such that each line has n + 1 points (for a projective plane). One major open question in finite geometry is:

Is the order of a finite plane always a prime power?

This is conjectured to be true, but has not been proven.

Affine and projective planes of order n exist whenever n is a prime power (a prime number raised to a positive integer exponent), by using affine and projective planes over the finite field with q = p^k elements. Planes not derived from finite fields also exist, but all known examples have order a prime power.

The best general result to date is the Bruck–Ryser theorem of 1949, 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. The non-existence of a finite plane of order 10 was proven in a computer-assisted proof that finished in 1989 – see (Lam 1991) for details.

The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.

[edit] Finite spaces of 3 or more dimensions

For some important differences between finite plane geometry and the geometry of higher-dimensional finite spaces, see axiomatic projective space. For a discussion of higher-dimensional finite spaces in general, see, for instance, the works of J.W.P. Hirschfeld.

[edit] See also

• Galois geometry
• Strong Law of Small Numbers – other properties of small finite sets

[edit] References

[edit] External links

• Weisstein, Eric W., "finite geometry" from MathWorld.
• Essay on Finite Geometry by Michael Greenberg
• Finite geometry (Script)
• Finite Geometry Resources
• J. W. P. Hirschfeld, researcher on finite geometries
  • Books by Hirschfeld on finite geometry
• AMS Column: Finite Geometries?
• Galois Geometry and Generalized Polygons, intensive course in 1998
• Carnahan, Scott (2007-10-27), "Small finite sets", Secret Blogging Seminar, http://sbseminar.wordpress.com/2007/10/27/small-finite-sets/, notes on a talk by Jean-Pierre Serre on canonical geometric properties of small finite sets.