On a New $(21_4)$ Polycyclic Configuration

Abstract

When searching for small 4-configurations of points and lines, polycyclic configurations, in which every symmetry class of points and lines contains the same number of elements, have proved to be quite useful. In this paper we construct and prove the existence of a previously unknown $(21_4)$ configuration, which provides a counterexample to a conjecture of Branko Grünbaum. In addition, we study some of its most important properties; in particular, we make a comparison with the well-known Grünbaum-Rigby configuration. We show that there are exactly two $(21_{4})$ geometric polycyclic configurations and seventeen $(21_{4})$ combinatorial polycyclic configurations. We also discuss some possible generalizations.