Finite geometry

Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes. Similar results hold for other kinds of finite geometries.

Finite planes

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

Finite affine planes

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. For every two distinct points, there is exactly one line that contains both points.
2. Playfair's axiom: Given a line ${\displaystyle \ell }$ and a point ${\displaystyle p}$ not on ${\displaystyle \ell }$, there exists exactly one line ${\displaystyle {\ell }^{\prime }}$ containing ${\displaystyle p}$ such that ${\displaystyle \ell \cap {\ell }^{\prime }=\varnothing .}$
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 trivial (either empty or too simple to be of interest, such as a single line with an arbitrary number of points on it), 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. (The order of an affine plane is the number of points on any line, see below.) 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".

The affine plane of order 3 is known as the Hesse configuration.

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.

Finite projective planes

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. For every 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.

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 geometries, meaning that any true statement valid in all these geometries remains true if we exchange points for lines and lines for points. The smallest geometry satisfying all three axioms contains seven points. 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. The Fano plane is called the projective plane of order 2 because it is unique (up to isomorphism). In general, the projective plane of order n has n² + n + 1 points and the same number of lines; 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 collineation of the plane. The full collineation group is of order 168 and is isomorphic to the group PSL(2,7) ≈ PSL(3,2), which in this special case is also isomorphic to the general linear group GL(3,2) ≈ PGL(3,2).

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.

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 n = p^k elements. Planes not derived from finite fields also exist (e.g. for ${\displaystyle n=9}$), 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.

History

Individual examples can be found in the work of Thomas Penyngton Kirkman (1847) and the systematic development of finite projective geometry given by von Staudt (1856).

The first axiomatic treatment of finite projective geometry was developed by the Italian mathematician Gino Fano. In his work on proving the independence of the set of axioms for projective n-space that he developed, he considered a finite three dimensional space with 15 points, 35 lines and 15 planes (see diagram), in which each line had only three points on it.

In 1906 Oswald Veblen and W. H. Bussey described projective geometry using homogeneous coordinates with entries from the Galois field GF(q). When n + 1 coordinates are used, the n-dimensional finite geometry is denoted PG(n, q). It arises in synthetic geometry and has an associated transformation group.