Desargues configuration

It is also self-dual, meaning that if the points are replaced by lines and vice versa using projective duality, the same configuration results.

They are in perspective axially if the intersection points of the corresponding triangle sides,

[1] Although it may be embedded in two dimensions, the Desargues configuration has a very simple construction in three dimensions: for any configuration of five planes in general position in Euclidean space, the ten points where three planes meet and the ten lines formed by the intersection of two of the planes together form an instance of the configuration.

This three-dimensional realization of the Desargues configuration is also called the complete pentahedron.

[2] The 5-cell or pentatope (a regular simplex in four dimensions) has five vertices, ten edges, ten triangular ridges (2-dimensional faces), and five tetrahedral facets; the edges and ridges touch each other in the same pattern as the Desargues configuration.

Extend each of the edges of the 5-cell to the line that contains it (its affine hull), similarly extend each triangle of the 5-cell to the 2-dimensional plane that contains it, and intersect these lines and planes by a three-dimensional hyperplane that neither contains nor is parallel to any of them.

[1] Kempe (1886) draws a different graph for this configuration, with ten vertices representing its ten lines, and with two vertices connected by an edge whenever the corresponding two lines do not meet at one of the points of the configuration.

Alternatively, the vertices of this graph may be interpreted as representing the points of the Desargues configuration, in which case the edges connect pairs of points for which the line connecting them is not part of the configuration.

This publication marks the first known appearance of the Petersen graph in the mathematical literature, 12 years before Julius Petersen's use of the same graph as a counterexample to an edge coloring problem.

[5] 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.

A tenth configuration exists as an abstract finite geometry but cannot be realized using Euclidean points and lines.