# 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.

## Keywords

Polygon Cubic curve Conic Triple system Flex## References

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