Desargues Configuration - Symmetries

Symmetries

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.