  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.
