Related Configurations
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 left. 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.
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