On a smallest topological triangle free (n_4) point-line configuration

Keywords: Finite generalised quadrangles, computational synthetic geometry, point-line configurations, oriented matroids, pseudoline arrangements


We study an abstract object, a finite generalised quadrangle W(3), due to Jacques Tits, that can be seen as the Levi graph of a triangle free (404) point-line configuration. We provide for W(3) representations as a topological (404) configuration, as a (404) circle representation, and a representation in the complex plane. These come close to a still questionable (real) geometric (404) point-line configuration realising this finite generalised quadrangle. This abstract (404) configuration has interesting triangle free realisable geometric subconfigurations, which we also describe. A topological (n4) configuration for n < 40 must contain a triangle, so our triangle free example is minimal.

Bled 2019 – Polytopes, Configurations, and Symmetries (Grünbaum Issue)