Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration

Abstract

The $(9_{3})$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $K_{3,3}$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $m \ge 3$ whose reduced Levi graphs are all labelled versions of $K_{3,3}$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections.