In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Gérard Desargues, and closely related to Desargues' theorem, which proves the existence of the configuration.
Construction from perspective triangles
Two triangles ABC and abc are said to be in perspective centrally if the lines Aa, Bb, and Cc meet in a common point (the so-called center of perspectivity). They are in perspective axially if the crossing points of the lines through pairs of corresponding triangle sides X = AB·ab, Y = AC·ac, and Z = BC·bc all lie on a common line, the axis of perspectivity. Desargues' theorem in geometry states that these two conditions are equivalent: if two triangles are in perspective centrally then they must also be in perspective axially, and vice versa. When this happens, the ten points and ten lines of the two perspectivities (the six triangle vertices, three crossings points, and center of perspectivity, and the six triangle sides, three lines through corresponding pairs of vertices, and axis of perspectivity) together form an instance of the Desargues configuration.
Although Desargues' theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more symmetric: any of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity. The Desargues configuration has a symmetry group of order 120; that is, there are 120 different ways of permuting the points and lines of the configuration in a way that preserves its point-line incidences. One way of constructing the same configuration in a way that makes these symmetries more readily apparent is to choose five planes in three-dimensional space, and to form the Desargues configuration as the set of ten points where three planes meet and the set of ten lines where two planes meet. Then, the 120 different permutations of these five planes each correspond to symmetries of the configuration.
The Desargues configuration is self-dual, meaning that it is possible to find a correspondence from points of one Desargues configuration to lines of a second configuration, and from lines of the first configuration to points of a second configuration, in such a way that all of the configuration's incidences are preserved Coxeter (1964).
The Levi graph of the Desargues configuration, a graph having one vertex for each point or line in the configuration, is known as the Desargues graph. Because of the symmetries and self-duality of the Desargues configuration, the Desargues graph is a symmetric graph.
As a projective configuration, the Desargues configuration has the notation (103103), meaning that each of its ten points is incident to three lines and each of its ten lines is incident to three points. Its ten points can be viewed in a unique way as a pair of mutually inscribed pentagons, or as a self-inscribed decagon (Hilbert & Cohn-Vossen 1952). The Desargues graph, a 20-vertex bipartite symmetric cubic graph, is so called because it can be interpreted as the Levi graph of the Desargues configuration, with a vertex for each point and line of the configuration and an edge for every incident point-line pair.
There also exist eight other (103103) configurations (that is, sets of points and lines in the Euclidean plane with three lines per point and three points per line) that are not incidence-isomorphic to the Desargues configuration, one of which is shown at right. In all of these configurations, any chosen point has three other points that are not collinear with it. But in the Desargues configuration, these three points are always collinear with each other (if the chosen point is the center of perspectivity, then the three points form the axis of perspectivity) while in the other configuration shown in the illustration these three points form a triangle of three lines. As with the Desargues configuration, the other depicted configuration can be viewed as a pair of mutually inscribed pentagons.