Intersecting the sides of a polygon

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