Desargues configuration

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 Girard Desargues, and closely related to Desargues' theorem, which proves the existence of the configuration.

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 intersection points of the 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 crossing 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 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 (Barnes 2012). This construction is closely related to the property that every projective plane that can be embedded into a projective space obeys Desargues' theorem. This three-dimensional realization of the Desargues configuration is also called the complete pentahedron (Barnes 2012).

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. Each line intersects the hyperplane in a point, and each plane intersects the hyperplane in a line; these ten points and lines form an instance of the Desargues configuration (Barnes 2012).

...
Wikipedia