# Configuration

## Etymology

Late Latin configuration-, configuratio similar formation, from Latin configurare to form from or after, from com- + figurare to form, from figura figure

## Definitions

• 1 a : relative arrangement of parts or elements: as (1) : shape (2) : contour of land <configuration of the mountains> (3) : functional arrangement <a small business computer system in its simplest configuration>
b : something (as a figure, contour, pattern, or apparatus) that results from a particular arrangement of parts or components
c : the stable structural makeup of a chemical compound especially with reference to the space relations of the constituent atoms

## Description (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.

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 Anschaulische 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, or as abstract incidence structures. In the latter case they are closely related to regular hypergraphs and regular bipartite graphs.[1]

## References

• Berman, Leah W., "Movable (n4) configurations", The Electronic Journal of Combinatorics 13 (1): R104, https://www.combinatorics.org/Volume_13/Abstracts/v13i1r104.html . See also Berman's animations of movable configurations.
• Betten, A; Brinkmann, G.; Pisanski, T. (2000), "Counting symmetric configurations", Discrete Applied Mathematics 99 (1–3): 331–338, doi:10.1016/S0166-218X(99)00143-2 .
• Coxeter, H.S.M. (1948), Regular Polytopes, Methuen and Co .
• Gropp, Harald (1997), "Configurations and their realization", Discrete Mathematics 174 (1–3): 137–151, doi:10.1016/S0012-365X(96)00327-5 .
• Grünbaum, Branko (2006), "Configurations of points and lines", in Davis, Chandler; Ellers, Erich W., The Coxeter Legacy: Reflections and Projections, American Mathematical Society, pp. 179–225 .
• Grünbaum, Branko (2009), Configurations of Points and Lines, Graduate Studies in Mathematics, 103, American Mathematical Society, ISBN 978-0-8218-4308-6 .
• Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), Chelsea, pp. 94–170, ISBN 0-8284-1087-9 .
• Kelly, L. M. (1986), "A resolution of the Sylvester–Gallai problem of J. P. Serre", Discrete and Computational Geometry 1 (1): 101–104, doi:10.1007/BF02187687 .