Some partitions in Figueroa planes

Abstract

Using Grundhöfer's construction of the Figueroa planes from Pappian planes  which  have an order $3$ planar collineation ${\widehat \alpha }$, we show that any  Figueroa plane (finite or infinite) has a partition of the complement of any proper (${\widehat \alpha }$)-invariant triangle mostly into subplanes together with a few  collinear  point sets (from the point set view) and a few concurrent line sets (from the  line set  view).  The partition shows that each Figueroa line (regarded as a set of  points) is  either the same as a Pappian line or consists mostly of a disjoint union of  subplanes of the Pappian plane (most lines are of this latter type) anddually. This last sentence is true with "Figueroa" and "Pappian" interchanged. There are many collinear subsets of Figueroa points which are a subset of the set of points of a Pappian conic and dually.