Perfect Polygons and Geometric Triple Systems

Abstract

A perfect n-gon is an abstraction of a regular n-gon when regarded in the real projective plane. The vertices of a regular n-gon P lie on n parallel classes of lines. The lines in any parallel class meet at a point at infinity. We call these points the perspective points of P. The vertices of P lie on a circle and the perspective points of P lie on the line at infinity in the projective plane, so we can say that the combined set of vertices and perspective points lie on a (reducible) cubic curve consisting of a line and a circle. In our Main Theorem we show that the combined set of vertices and perspective points of any perfect polygon lie on a cubic curve which may be irreducible. In case the cubic is irreducible, a well-known algebra which we call a geometric triple system can be defined on its points. We show that perfect polygons can be obtained as translates of these algebras.