Rotary Polygons in Configurations

Abstract

A polygon $A$ in a configuration $\mathcal{C}$ is called rotary if $\mathcal{C}$ admits an automorphism which acts upon $A$ as a one-step rotation. We study rotary polygons and their orbits under the group of automorphisms (and antimorphisms) of $\mathcal{C}$. We determine the number of such orbits for several symmetry types of rotary polygons in the case when $\mathcal{C}$ is flag-transitive. As an example, we provide tables of flag-transitive $(v_3)$ and $(v_4)$ configurations of small order containing information on the number and symmetry types of corresponding rotary polygons.