Geodesics in generalizations of the Sierpinski carpet

Abstract

We analyze geodesics in a 3-parameter family of fractals that contains the Sierpinski carpet and Menger sponge, generalizing results in the literature. Between any two points in one of these fractals, we construct a geodesic path, where path lengths are induced by the 1-norm. For any of these fractals we then determine the maximum possible ratio of the geodesic metric to the Euclidean metric, and we provide examples to show that this upper bound is sharp.