Short Paths and Long Titles: Travels through the Sierpinski carpet, Menger sponge, and beyond.

The Sierpinski carpet and Menger sponge are fractals which can be thought of as two and three dimensional versions of the Cantor set.  Like the Cantor set, each is formed by starting with a shape (a square for the carpet, a cube for the sponge) and then recursively removing certain subsets of it.  Unlike the Cantor set, what remains is connected in the following sense: given any two points s and f in the carpet or sponge, there is a path from s to f that stays in the carpet or sponge.  In this talk, we’ll discuss what we know about the shortest path from s to f in the carpet, sponge, and even higher dimensional versions of these fractals.