n_4 Configuration

An configuration is a symmetric configuration consisting of abstract points and 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. The word "line" here denotes an abstract element of the incidence structure, not necessarily a straight line.

A geometric configuration realizes the prescribed incidences by points and straight lines in the real projective plane, with no additional incidences between the selected points and lines. A topological realization instead uses pseudolines. These realization conditions are stronger than abstract existence.

Abstract configurations exist exactly for , and topological configurations exist exactly for (Bokowski and Pilaud 2014). For real geometric realizations, the complete existence criterion is and (Lu 2026). Lu (2026) settled the last unresolved case by constructing geometric 23₄ configurations.

The smallest real geometric examples are the two combinatorial types of 18₄ configurations, named the Bokowski-Schewe configuration and Bokowski-Pilaud configuration (Bokowski and Pilaud 2014). The 19₄ configuration case shows that topological realizability need not imply the existence of a geometric realization. The following table lists named configurations with real geometric realizations (Grünbaum 2008, Bokowski and Pilaud 2014, Boben et al. 2015, Berman et al. 2024).

name	type
Bokowski-Schewe configuration	184
Bokowski-Pilaud configuration	184
Grünbaum 204 configuration	204
Grünbaum-Rigby configuration	214
Berman-Gévay-Pisanski configuration	214
Danzer configuration

The Grünbaum-Rigby configuration and the Berman-Gévay-Pisanski configuration exhaust the geometric polycyclic configurations of order , not all 21₄ configurations (Berman et al. 2024).