TITLE: Limit of a sequence of polygons.
QUESTION [12 upvotes]: Begin with a polygon $P_0$.
Place two points on every edge of the polygon such that they divide each side equally into three parts. Create a new polygon $P_1$ by connecting all new points with lines.
If we begin with a square and iterate this process, what is the limit as the number of iterations approaches infinity? It is clearly not a circle, but what the correct answer is, I do not know.

REPLY [3 votes]: As noted, goes back to de Rham.  Found also in:

Georges de Rham, "Sur quelques courbes définies par des equations functionnelles". Univ. e Politec. Torino. Rend. Sem. Mat. 16 1956/1957 101–113.   

English tranlation in my book CLASSICS ON FRACTALS.