Cremona–Richmond configuration
In mathematics, the Cremona–Richmond configuration is a configuration of 15 lines and 15 points, having 3 points on each line and 3 lines through each point, and containing no triangles. It was studied by Cremona (1877) and Richmond (1900). It is a generalized quadrangle with parameters (2,2). Its Levi graph is the Tutte–Coxeter graph.
The points of the Cremona–Richmond configuration may be identified with the unordered pairs of elements of a six-element set; these pairs are called duads. Similarly, the lines of the configuration may be identified with the 15 ways of partitioning the same six elements into three pairs; these partitions are called synthemes. Identified in this way, a point of the configuration is incident to a line of the configuration if and only if the duad corresponding to the point is one of the three pairs in the syntheme corresponding to the line.
The symmetric group of all permutations of the six elements underlying this system of duads and synthemes acts as a symmetry group of the Cremona–Richmond configuration, and gives the automorphism group of the configuration. Every flag of the configuration (an incident point-line pair) can be taken to every other flag by a symmetry in this group.
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