Leah Wrenn Berman: Symmetrically generalizing the Pappus configuration

Povzetek. The (9_3) Pappus configuration can be realized with three-fold rotational symmetry with the property that under the action of Z_3, there are three symmetry classes of points, three symmetry classes of lines, each line passes through one point from each of the three geometric symmetry classes of points and each point has one line from each of three symmetry classes of lines passing through it. It has been an open question as to how to generalize this configuration robustly to produce other 3-configurations with this property with m-fold rotational symmetry for any arbitrary m.

In this talk, we describe a very new geometric construction for producing other 3-configurations with this property; the construction is primarily a ruler-and-compass construction, but also depends on the construction of a particular conic! We show that these configurations are moveable, and we also describe how to generalize the construction to construct 3-configurations with more than three symmetry classes with the property that no line passes through more than one point from any symmetry class of points, and no point lies on more than one line from any symmetry class of lines. We conclude with some open questions.