Configuration

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 A001403; Sterneck 1894, 1895; Wells 1991, p. 72; Colbourn and Dinitz 1996; Gropp 1997; Hilbert and Cohn-Vossen 1999).

A combinatorial configuration has a geometric realization when its abstract points and lines can be represented by points and straight lines with the prescribed incidences. Some authors take the ambient space to be the real projective plane, while others require an affine or Euclidean plane realization (Bokowski and Pilaud 2014, Berman et al. 2024).

The smallest geometric symmetric 4-configurations are the nonisomorphic Bokowski-Schewe configuration and Bokowski-Pilaud configuration; none exists for (Bokowski and Schewe 2005, Bokowski and Schewe 2013, Bokowski and Pilaud 2014). No geometric configuration exists (Bokowski and Pilaud 2015), while Grünbaum (2008) constructed the Grünbaum 20₄ configuration. At order 21, the Grünbaum-Rigby configuration and the Berman-Gévay-Pisanski configuration are the two geometric polycyclic configurations (Berman et al. 2024).

The Fano plane, in which the central point corresponds to the point at infinity, is the unique configuration. It is realizable over the finite field of order 2 , but not over the real or rational numbers (Gropp 1997). 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 (Kantor 1891). This is known as the Möbius-Kantor configuration (Pisanski and Randić 2000).

Kantor (1891) showed that there are three configurations, of which the Pappus configuration (left figure) is one (Coxeter 1950; Wells 1986, p. 75). The other two consist of embedded equilateral triangles (Wells 1991, pp. 159-160).

Kantor (1881) proved that there are exactly 10 configurations , of which the Desargues configuration, illustrated above, is one. However, while not explicitly commented upon in paper, Kantor's contained a line which consists of two line segments oriented in slightly different directions. Schroeter (1889) subsequently proved that exactly one of these configurations cannot be drawn in the real or rational planes (Gropp 1997).

There are 31 configurations (Gropp 1997), as constructed by Martinetti (1887) using recursive construction method, which were subsequently realized in the plane by Daublebsky von Sterneck (1895), though the proof that all are realizable came only with the work of Sturmfels and White (1990). Page and Dorwart (1984) discuss the 31 configurations (Wells 1991, p. 63).

There are 229 configurations , 228 of which had been found by Daublebsky von Sterneck (1895), with the missing one not found until the work of Gropp (1991). One of the configurations is sometimes known as the Coxeter configuration, although it is perhaps better to refer it the Nauru configuration based on its Levi graph (which is the Nauru graph). Coordinates for realizations of all and configurations appeared in an appendix to Crapo et al. (1988).

The Coxeter configuration is a configuration whose Levi graph is the Foster graph .

The Cremona-Richmond configuration, illustrated above, is one of the 245342 configurations.

There is a configuration known as the Cox configuration.

The following table summarizes the Levi graphs of some named configurations.

name	configuration	Levi graph
Berman-Gévay-Pisanski configuration		Berman-Gévay-Pisanski graph
Bokowski-Pilaud configuration		Bokowski-Pilaud graph
Bokowski-Schewe configuration		Bokowski-Schewe graph
Cox configuration		hypercube graph
Coxeter configuration		Foster graph
Cremona-Richmond configuration		Tutte 8-cage
Danzer configuration		Danzer graph
Desargues configuration		Desargues graph
Fano plane		Heawood graph
Georges configuration		Georges graph
Goedgebeur configuration		Goedgebeur graph
Gray configuration		Gray graph
Grünbaum 204 configuration		Grünbaum 204 graph
Grünbaum-Rigby configuration		Grünbaum-Rigby graph
Hesse configuration		Hesse graph
Klein configuration		120-Klein graph
Kummer configuration		Kummer graph
Miquel configuration		rhombic dodecahedral graph
Möbius-Kantor configuration		Möbius-Kantor graph
Nauru configuration		Nauru graph
Pappus configuration		Pappus graph
Pasch configuration		Pasch graph
Reye configuration		Reye graph
Schläfli double six		Schläfli double six graph