Geodesic taxicab paths in the Sierpinski carpet, Menger sponge, and their generalizations

The Sierpinski carpet and Menger sponge are well-studied connected generalizations of the Cantor set.  They are also members of a two-parameter family of connected higher-dimensional fractals that can be constructed from the $n$-cube as iterated function systems.  

In this talk we focus on taxicab paths—piece-wise linear paths that always travel parallel to a coordinate axis with possible limiting behavior at the endpoints—between any two points $x$ and $y$ in members of this fractal family.  In particular, given points $x$ and $y$ in such a fractal, we show how to explicitly construct taxicab geodesics between them.  As an application, we compare the taxicab metric to the standard Euclidean metric in the fractals we consider, providing a sharp bound on their ratio.  We will also briefly address non-taxicab geodesics.  

This is joint work with Elene Karangozishvili and Derek Smith.