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 arrangement for the orchard-planting problem with 22 points and exactly four points per line is obtained by adding the center of the configuration. The original 21 four-point lines remain, and the seven additional three-point lines through the center now contain four points each, giving 28 in all. Five of these lines pass through each of the 21 original points, and seven pass through the added point, so the arrangement fails the uniform incidence requirement for a configuration.

Du (2019a) posted coordinates for a projective realization of the arrangement on the eMath forum. The forum user "wayne" used a projective transformation to redraw it and recognized the heptagonal pattern (wayne 2019). Du (2019b) subsequently posted the symmetric drawing of the 22-point arrangement shown above.