Here's a fractal Sophie and I worked on together, guided by the "Calculus by and for Young People" program we've been dabbling in. If you click to go to the full-size image, you'll see that we actually began the insane task of drawing in the 6th iteration in the lower left corner.
Sophie hypothesized that the series that defined both perimeter and area of the fractal through successive iterations would diverge. In fact, we were able to work out that only the perimeter diverged, while the area converges.
You can see that I have been having fun with the "scan" function on my new copier.