Finite geometry

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 as many points as there are real numbers. A finite geometry can have any (finite) number of dimensions.

Finite geometries may be constructed via linear algebra, as vector spaces over a finite field, and called Galois geometries, 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.

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 $\ell$ and a point p not on $\ell$, there exists exactly one line $\ell'$ containing p such that $\ell \cap \ell' = \varnothing.$
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.

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.

Finite affine plane of order 3, containing 9 points and 12 lines.

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.

Duality in the Fano plane: Each point corresponds to a line and vice versa.

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), and general linear group GL(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, 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 n = 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.

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.

Finite three-spaces

Associated with every field K is a (3-dimensional) projective space whose points, lines, and planes can be identified with the 1-, 2-, and 3-dimensional subspaces of the 4-dimensional vector space over the field K. There is a set of axioms for projective spaces. The smallest 3-dimensional projective space over the field GF(2), denoted by PG(3,2), has 15 points, 35 lines, and 15 planes. Each of the 15 planes contains 7 points and 7 lines. As geometries, these planes are isomorphic to the Fano plane. Every point of PG(3,2) is contained in 7 lines and every line contains three points. In addition, two distinct points are contained in exactly one line and two planes intersect in exactly one line. In 1892, Gino Fano was the first to consider such a finite geometry – a three dimensional geometry containing 15 points, 35 lines, and 15 planes, with each plane containing 7 points and 7 lines.

In synthetic projective geometry the undefined elements are taken as points and lines. A plane and a three-space may then be defined using the postulates of incidence and existence:

Postulates of Incidence

P-1: If A and B are distinct points, there is at least one line on both A and B.

P-2: If A and B are distinct points, there is not more than one line on both A and B.

P-3: If A, B, and C are points not all on the same line, and D and E are distinct points such that B, C, and D are on a line and C, A, and E are on a line, there is a point F such that A, B, and F are on a line and also D, E, and F are on a line.

Figure 1

Postulates of Existence

P-4: There exists at least one line.

P-5: There are at least three distinct points on every line.

P-6: Not all points are on the same line.

P-7: Not all points are on the same plane.

P-8: If S3 is a three-space, every point is on S3.

In particular Postulates P-1 through P-8 are satisfied by the points, lines, and planes of the three-space whose points are indicated in Figure 1. This three-space contains exactly 15 points. There are also many other finite projective three-spaces for which these Postulates hold.

Finite n-spaces

In general, for any positive integer n, a geometry on an n-space is called an n-dimensional geometry. A four-dimensional projective geometry may be obtained by replacing P-8 by P-8’: Not all points are on the same three-space and by a postulate of closure P-8’’: If S4is a four-space, every point is on S4.

In general, an n-dimensional projective geometry (n = 4, 5, …) may be obtained by replacing P-8 by postulates stating that: (i) Not all points are on the same S3, S4, …, Sn-1, (ii) If Sn is an n-space, every point is on Sn.

These projective geometries are denoted by PG(n, q) where n is the geometric dimension of the geometry and q is the size (order) of the finite field used to construct the geometry. The study of these higher-dimensional spaces ( n > 3) has many important applications in advanced mathematical theories.

Kirkman's Schoolgirl Problem

PG(3,2) can arise as a solution of Kirkman's schoolgirl problem, which states: “Fifteen schoolgirls walk each day in five groups of three. Arrange the girls’ walk for a week so that in that time, each pair of girls walks together in a group just once.” (See answer in link.) There are 35 different combinations for the girls to walk together. There are also 7 days of the week, and 3 girls in each group. Two of the seven non-isomorphic solutions to this problem can provide a visual representation of the Fano 3-space. Some diagrams for this problem can be found at [1]:

Each color represents the day of the week (seven colors, blue, green, yellow, purple, red, black, and orange). The definition of a Fano space says that each line is on three points. The figure represents this showing that there are 3 points for every line. This is the basis for the answer to the schoolgirl problem. This figure is then rotated 7 times. There are 5 different lines for each day, multiplied by 7 (days) and the result is 35. Then, there are 15 points, and there are also 7 starting lines on each point. This then gives a representation of the Fano 3-space.

See also

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

References

• Lynn Margaret Batten : Combinatorics of Finite Geometries. Cambridge University Press
• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3540617868, MR0233275 
• Hall, Marshall (1943), "Projective planes", Transactions of the American Mathematical Society (American Mathematical Society) 54 (2): 229–277, ISSN 0002-9947, JSTOR 1990331, MR0008892 
• Lam, C. W. H. (1991), "The Search for a Finite Projective Plane of Order 10", American Mathematical Monthly 98 (4): 305–318, http://www.cecm.sfu.ca/organics/papers/lam/ 
• Eves, Howard. A Survey of Geometry: Volume One. Boston: Allyn and Bacon Inc., 1963.
• Meserve, Bruce E. Fundamental Concepts of Geometry. New York: Dover Publications, 1983.
• Polster, Burkard. Yea Why Try Her Raw Wet Hat: A Tour of the Smallest Projective Space. Volume 21, Number 2, 1999. [2]
• “Problem 31: Kirkman's schoolgirl problem” [3]

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.