10_3 Configuration

A configuration is an abstract configuration of 10 points and 10 lines, with exactly three lines incident with each point and exactly three points incident with each line. Two distinct points are incident with at most one common line. A geometric realization represents these incidences by points and straight lines in the real projective plane, without adding incidences between the selected points and lines.

There are ten abstract configurations up to isomorphism, of which nine have real geometric realizations. The remaining type is topologically realizable using pseudolines, but not geometrically realizable using straight lines (Gropp 1997, Bokowski and Pilaud 2014). The Desargues configuration, arising from Desargues' theorem, is one of the nine realizable types.