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Concept# Configuration (geometry)

Summary

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 .
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 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 (the Levi graph of the configuration) must be at least six.
A configuration in the plane is denoted by (pγ lpi), where p is the number of points, l the number of lines, γ the number of lines per point, and pi the number of points per line. These numbers necessarily satisfy the equation
as this product is the number of point-line incidences (flags).
Configurations having the same symbol, say (pγ lpi), need not be isomorphic as incidence structures. For instance, there exist three different (93 93) configurations: the Pappus configuration and two less notable configurations.
In some configurations, p = l and consequently, γ = pi. These are called symmetric or balanced configurations and the notation is often condensed to avoid repetition.

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Levi graph

In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942. The Levi graph of a system of points and lines usually has girth at least six: Any 4-cycles would correspond to two lines through the same two points.

Incidence geometry

In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure.

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