Configuration (geometry)


In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.
Although certain specific configurations had been studied earlier, the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book Anschauliche Geometrie, reprinted in English as.
Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes, or as a type of abstract incidence geometry. In the latter case they are closely related to regular hypergraphs and biregular bipartite graphs, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point. That is, the girth of the corresponding bipartite graph must be at least six.

Notation

A configuration in the plane is denoted by, where is the number of points, the number of lines, the number of lines per point, and the number of points per line. These numbers necessarily satisfy the equation
as this product is the number of point-line incidences.
Configurations having the same symbol, say, need not be isomorphic as incidence structures. For instance, there exist three different configurations: the Pappus configuration and two less notable configurations.
In some configurations, and consequently,. These are called symmetric or balanced configurations and the notation is often condensed to avoid repetition. For example, abbreviates to .

Examples

Notable projective configurations include the following:
The projective dual of a configuration is a configuration in which the roles of "point" and "line" are exchanged. Types of configurations therefore come in dual pairs, except when taking the dual results in an isomorphic configuration. These exceptions are called self-dual configurations and in such cases.

The number of () configurations

The number of nonisomorphic configurations of type, starting at, is given by the sequence
These numbers count configurations as abstract incidence structures, regardless of realizability.
As discusses, nine of the ten configurations, and all of the and configurations, are realizable in the Euclidean plane, but for each there is at least one nonrealizable configuration. Gropp also points out a long-lasting error in this sequence: an 1895 paper attempted to list all configurations, and found 228 of them, but the 229th configuration, the Gropp configuration, was not discovered until 1988.

Constructions of symmetric configurations

There are several techniques for constructing configurations, generally starting from known configurations. Some of the simplest of these techniques construct symmetric configurations.

Cyclic configurations

Some self dual configurations are cyclic configurations and can be constructed by one "generator line", like, with vertices indexed from zero, and where indices in following lines are cycled forward modulo p. This is guaranteed to produce a symmetric configuration when valid. An invalid generator line produces disconnected configurations, or it may break the axiom requiring at most one line between any two points.
Every polygon as configuration is trivially a cyclic configuration with generator line. A triangle has lines.
The Fano plane,, the smallest self-dual order 3 symmetric configuration, can be defined by generator line as lines. They also can be represented in configuration table:
0123456
1234560
3456012

The smallest self-dual order 5 symmetric configuration is is a cyclic configuration and can be generated by the line.
0123456789101112
3456789101112131415
45678910111213141516
910111213141516171819200
11121314151617181920012

Finite projective planes

Any finite projective plane of order n, PG, is an configuration. Since projective planes are known to exist for all orders n which are powers of primes, these constructions provide infinite families of symmetric configurations.
Automorphisms for PG, with n=''qm'' are /.
Not all symmetric configurations are realizable. Specifically n must be a power prime. For instance, PG or configuration does not exist. However, has provided a construction which shows that for, a configuration exists for all, where is the length of an optimal Golomb ruler of order.

Unconventional configurations

Higher dimensions

The concept of a configuration may be generalized to higher dimensions, for instance to points and lines or planes in space. In such cases, the restrictions that no two points belong to more than one line may be relaxed, because it is possible for two points to belong to more than one plane.
Notable three-dimensional configurations are the Möbius configuration, consisting of two mutually inscribed tetrahedra, Reye's configuration, consisting of twelve points and twelve planes, with six points per plane and six planes per point, the Gray configuration consisting of a 3×3×3 grid of 27 points and the 27 orthogonal lines through them, and the Schläfli double six, a configuration with 30 points, 12 lines, two lines per point, and five points per line.

Topological configurations

Configuration in the projective plane that is realized by points and pseudolines is called
topological configuration. For instance, it is known that there exists no point-line configurations, however, there exists a topological configuration with these parameters.

Configurations of points and circles

Another generalization of the concept of a configuration concerns configurations of points and circles, a notable example being the Miquel configuration.