Intersecting the sides of a polygon

TL;DR

This paper studies the dynamics of a polygon transformation map related to the pentagram map, proving that convex polygons almost never remain convex under iteration, with the convex-preserving set being measure zero and algebraically defined.

Contribution

It proves that the set of polygons remaining convex under the map has measure zero and characterizes this set as an algebraic subvariety, advancing understanding of polygon dynamics.

Findings

Convex polygons almost never stay convex under the map S.

The set of polygons that remain convex forms a measure-zero algebraic subvariety.

The paper provides equations and geometric interpretation for this subvariety.

Abstract

Consider the map $S$ which sends a planar polygon $P$ to a new polygon $S(P)$ whose vertices are the intersection points of second nearest sides of $P$. This map is the inverse of the famous pentagram map. In this paper we investigate the dynamics of the map $S$. Namely, we address the question of whether a convex polygon stays convex under iterations of $S$. Computer experiments suggest that this almost never happens. We prove that indeed the set of polygons which remain convex under iterations of $S$ has measure zero, and moreover it is an algebraic subvariety of codimension two. We also discuss the equations cutting out this subvariety, as well as their geometric meaning in the case of pentagons.