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. 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 biregularbipartite 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 simplest possible configuration, consisting of a point incident to a line. Often excluded as being trivial.
, the triangle. Each of its three sides meets two of its three vertices, and vice versa. More generally any polygon of sides forms a configuration of type
, the Fano plane. This configuration exists as an abstract incidence geometry, but cannot be constructed in the Euclidean plane.
, the Möbius–Kantor configuration. This configuration describes two quadrilaterals that are simultaneously inscribed and circumscribed in each other. It cannot be constructed in Euclidean plane geometry but the equations defining it have nontrivial solutions in complex numbers.
, the Pappus configuration.
, the Hesse configuration of nine inflection points of a cubic curve in the complex projective plane and the twelve lines determined by pairs of these points. This configuration shares with the Fano plane the property that it contains every line through its points; configurations with this property are known as Sylvester–Gallai configurations due to the Sylvester–Gallai theorem that shows that they cannot be given real-number coordinates.
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 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. Any finite projective plane of order is an and remove a line not passing through and all the points that are on line. The result is a configuration of type 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.
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.
