On a new (21_4) polycyclic configuration

TL;DR

This paper introduces a new (21_4) polycyclic configuration, providing a counterexample to a conjecture, and analyzes its properties, including geometric and combinatorial classifications and potential generalizations.

Contribution

It constructs and proves the existence of a novel (21_4) configuration, challenging previous conjectures and expanding understanding of polycyclic configurations.

Findings

Existence of exactly two geometric (21_4) configurations.

Existence of seventeen combinatorial (21_4) configurations.

Comparison with the Gr"unbaum-Rigby 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) conguration, which provides a counterexample to a conjecture of Branko Gr\"unbaum. In addition, we study some of its most important properties; in particular, we make a comparison with the well-known Gr\"unbaum-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.