At work, Wendy and I have been talking about the Cantor set, and a related 3D doohicky called the Menger sponge. It’s easiest to describe a Menger sponge by showing the iterative process which creates one:
In the limit, a Menger sponge has infinite surface area, but zero volume. Neat!
I just got linked here from a search “surface area menger sponge.” Zero and infinite, pretty cool. Try checking out the Mandelbrot set if you haven’t already, it’s another neat mathematical oddity.