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:

{{n+1} \choose {k+1}}_q = \prod_{i=0}^k \frac{q^{n+1-i}-1}{q^{i+1}-1},

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

Read more about this topic:  Finite Geometry

Famous quotes containing the words finite, spaces and/or dimensions:

    “Sisters define their rivalry in terms of competition for the gold cup of parental love. It is never perceived as a cup which runneth over, rather a finite vessel from which the more one sister drinks, the less is left for the others.”
    —Elizabeth Fishel (20th century)

    “In any case, raw aggression is thought to be the peculiar province of men, as nurturing is the peculiar province of women.... The psychologist Erik Erikson discovered that, while little girls playing with blocks generally create pleasant interior spaces and attractive entrances, little boys are inclined to pile up the blocks as high as they can and then watch them fall down: “the contemplation of ruins,” Erikson observes, “is a masculine specialty.””
    —Joyce Carol Oates (b. 1938)

    “Is it true or false that Belfast is north of London? That the galaxy is the shape of a fried egg? That Beethoven was a drunkard? That Wellington won the battle of Waterloo? There are various degrees and dimensions of success in making statements: the statements fit the facts always more or less loosely, in different ways on different occasions for different intents and purposes.”
    —J.L. (John Langshaw)