Finite geometry

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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.

[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:

Given any two distinct points, there is exactly one line that contains both points.
The parallel postulate: Given a line $l$ and a point $p$ , there exists exactly one line $l'$ containing $p$ such that $l = l'$ or $l \cap l' = \empty$ .
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. 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 $n 2$ points and $n 2 + n$ lines; each line contains $n$ points, and each point is on $n + 1$ lines.

(Figures of affine planes of orders 2 and 3 to be added.)

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:

Given any two distinct points, there is exactly one line that contains both points.
The intersection of any two distinct lines contains exactly one point.
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). For a different representation of the Fano plane that allows study of this full group of 168 symmetries, see The Eightfold Cube.

It is well-established that both affine and projective planes of order n exist when n is a prime power, a prime number raised to a positive integer exponent. It is 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.

[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. For a discussion of the smallest affine 3-space-- 8 points that can be visualized as the subcubes of a 2×2×2 cube-- see Block Designs in Art and Mathematics.