GEODESICS IN THE SIERPINSKI CARPET AND MENGER SPONGE

Abstract

In this paper, we study geodesics in the Sierpinski carpet and Menger sponge, as well as in a family of fractals that naturally generalize the carpet and sponge to higher dimensions. In all dimensions, between any two points we construct a geodesic taxicab path, namely a path comprised of segments parallel to the coordinate axes and possibly limiting to its endpoints by necessity. These paths are related to the skeletal graph approximations of the Sierpinski carpet that have been studied by many authors. We then provide a sharp bound on the ratio of the taxicab metric to the Euclidean metric, extending Cristea’s result for the Sierpinski carpet. As an application, we determine the diameter of the Sierpinski carpet taken over all rectifiable curves. For other members of the family, we provide a lower bound on the diameter taken over all piecewise smooth curves.