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The word configuration is sometimes used to describe a finite collection of points , , where is a Euclidean space.

The term "configuration" also is used to describe a finite incidence structure with the following properties (Gropp 1992).

1. There are points and lines.

2. There are points on each line and lines through each point.

3. Two different lines intersect each other at most once and two different points are connected by a line at most once.

The conditions

are necessary for the existence of a configuration. For , these conditions are also sufficient, and for this is probably also the case (Gropp 1992). The necessary conditions hold, but there is no . For and 7, the above conditions are not sufficient, as illustrated by the affine projective plane of order 6 (, ) and the projective plane .

Configurations are among the oldest combinatorial structures, having been defined by T. Reye in 1876. An -regular graph can be regarded as a configuration by associating nodes with the points, and edges with the lines.

A symmetric configuration consists of lines and points arranged such that lines pass through each point and there are points on each line. All symmetric configurations are known for (Betten et al. 2000). The number of , , , ... configurations are 1, 1, 3, 10, 31, 229, 2036, 21399, 245342, ..., correcting an error of von Sterneck for (OEIS Belgazou Rousseau Grey Womens Jerome C SwHqRTnUt; Sterneck 1894, 1895; Wells 1991, p. 72; Colbourn and Dinitz 1996; Gropp 1997; Hilbert and Cohn-Vossen 1999).

The Fano plane, in which the central point corresponds to the point at infinity, is the unique configuration. There are no configurations using points all at finite distances (Wells 1986, p. 75).

There are no configurations using points all at finite distances (Wells 1986, p. 75), but a single configuration exists with a point at infinity. This is known as the Möbius-Kantor configurationPatent Ladies Fashion Blue Leather Polka Dots BalaMasa Slippers ORvWzUHO (Pisanski and Randić 2000).

There are three configurations, of which the Waterproof Platform Women's Brown Snow Shoes Winter Boots Flat Duberess Lace up Sneaker XUq8Uw (left figure) is one (Coxeter 1950; Wells 1986, p. 75). The other two consist of embedded equilateral triangles (Wells 1991, pp. 159-160).

The Desargues configuration, illustrated above, is one of the ten configurations. Page and Dorwart (1984) discuss the 31 configurations (Wells 1991, p. 63).

There is a configuration known as the Coxeter configuration.

The B Shoes Sneakers Canvas HUAN Shoes Men's Fall Tulle Outdoor Shoes Spring for Walking Comfort xnpfTnRW, illustrated above, is one of the 245342 configurations.

There is a configuration known as the Cox configuration.

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