Grünbaum-Rigby Configuration

The Grünbaum-Rigby configuration is a 21₄ configuration consisting of 21 points and 21 lines such that four points lie on each line and four lines pass through each point (left figure above). It was first described by Klein (1878), but a geometric realization in the Euclidean plane was not found until Grünbaum and Rigby (1990) constructed one by overlaying three regular heptagrams. Note that the points also determine seven additional lines through three points (illustrated in red in the right figure above) which are not however part of the configuration.

Grünbaum and Rigby (1990) conjectured that no other geometric 21₄ configuration exists and that no geometric n₄ configuration exists for . Grünbaum (2008) disproved the second conjecture by constructing the Grünbaum 20₄ configuration. Two further exceptions were discovered subsequently: the Bokowski-Schewe configuration and Bokowski-Pilaud configuration are exactly the two geometric configurations up to combinatorial isomorphism (Bokowski and Pilaud 2014).

Berman et al. (2024) disproved the first conjecture by constructing the nonisomorphic Berman-Gévay-Pisanski configuration, denoted in their paper. More specifically, the Grünbaum-Rigby and Berman-Gévay-Pisanski configurations are the two geometric polycyclic configurations, among 17 combinatorial polycyclic configurations of order . This is not a classification of all 21₄ configurations.

The Levi graph of the Grünbaum-Rigby configuration may be termed the Grünbaum-Rigby graph.

The best known solution to the orchard-planting problem for 22 points with 4 points per line is obtained by adding a point at the center of the configuration through which opposite points of the configuration lie, yielding 28 lines (E. Pegg, Jr., pers. comm., Dec. 11, 2023). Note that this arrangement is not a configuration since while five lines pass through 21 of the points, seven lines pass through the central point.