Start with an equilateral triangle with unit area. Trisect each of the sides and then cut-off the corners. In this case, we get a regular hexagon. Next, trisect each of the sides of the hexagon and cut-off the corners. This will give a dodecagon, but not a regular one.
Continue this process ad infinitum.
What is the shape of the limiting "polygon"?
I know the polygon must be continuous but isn't differenciable, and that it has symmetry $S_3$. I also know the area ($4/7$). It's not a circle, constant diameter shape, or a finite collection of beizier curves. It is convex.