Configuration (geometry)

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 (for instance by Thomas Kirkman in 1849), 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 Geometry and the Imagination).

Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be realizable in that geometry), or as abstract incidence structures. 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 (the Levi graph of the configuration) must be at least six.

Read more about Configuration (geometry):  Notation, Examples, Duality of Configurations, configurations">The Number of (n3) Configurations, Constructions of Symmetric Configurations, Higher Dimensions

Other articles related to "configuration, configurations":

Configuration (geometry) - Higher Dimensions
... The concept of a configurationmay be generalized to higher dimensions, for instance to points and lines or planes in space ... Notable three-dimensional configurationsare the Möbius configuration consisting of two mutually inscribed tetrahedra, Reye's configuration consisting of twelve points and twelve planes ... a given configurationmay be represented by the matrix The principle extends generally to n dimensions, where 0 ≤ j < n ...