18_4 Configuration

A configuration is an abstract configuration of 18 points and 18 lines, with exactly four lines incident with each point and exactly four 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.

The two real geometrically realizable incidence types are the Bokowski-Schewe configuration and the Bokowski-Pilaud configuration (Bokowski and Pilaud 2014). This is a classification up to combinatorial isomorphism, not a count of coordinate realizations up to projective equivalence. They have the smallest possible order among geometric n₄ configurations. There are also abstract configurations that do not have real geometric realizations.