Finite Geometry - Finite Spaces of 3 or More Dimensions

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. The study of these higher-dimensional spaces ( n ≥ 3) has many important applications in advanced mathematical theories.

A projective space S can be defined axiomatically as a set P (the set of points), together with a set L of subsets of P (the set of lines), satisfying these axioms :

• Each two distinct points p and q are in exactly one line.
• Veblen's axiom: If a, b, c, d are distinct points and the lines through ab and cd meet, then so do the lines through ac and bd.
• Any line has at least 3 points on it.

The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an incidence structure (P,L,I) consisting of a set P of points, a set L of lines, and an incidence relation I stating which points lie on which lines.

In order to obtain a finite projective space, one more axiom is needed:

• The set of points P is a finite set.

In any finite projective space, each line contains the same number of points and the order of the space is defined as one less than this common number.

A subspace of the projective space is a subset X, such that any line containing two points of X is a subset of X (that is, completely contained in X). The full space and the empty space are always subspaces.

The geometric dimension of the space is said to be n if that is the largest number for which there is a strictly ascending chain of subspaces of this form:

There is a standard algebraic construction of systems which satisfy these axioms. For a division ring D construct an n + 1 dimensional vector space over D (vector space dimension is the number of elements in a basis). Let P be the 1-dimensional subspaces and L the 2-dimensional subspaces (vector space dimension) of this vector space. Incidence is containment. If D is finite then this constructs a finite projective space. For finite projective spaces of geometric dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, GF(q), whose order (that is, number of elements) is q (a prime power). A finite projective space defined over such a finite field will have q + 1 points on a line, so the two concepts of order will coincide. Such a finite projective space is denoted by PG(n, q), where PG stands for projective geometry, n is the geometric dimension of the geometry and q is the size (order) of the finite field used to construct the geometry.

In general, the number k-dimensional subspaces of PG(n, q) is given by the product:

which is a Gaussian binomial coefficient, a q analogue of a binomial coefficient.