| Abstract: |
| In this talk we address several questions about geodesic paths between two points in a 3-parameter family of fractals that naturally generalize the Sierpinski carpet. This family includes the Menger sponge as well as higher-dimensional path connected fractals. We focus on taxicab paths, which are piecewise linear paths whose components are parallel to coordinate axes, with nontrivial limiting behavior typically necessary to join arbitrary pairs of endpoints. We provide a framework which allows for the construction of a taxicab geodesic between any two endpoints for all members of the family. We also provide a tight upper bound on the ratio of the geodesic path length and the Euclidean distance between the endpoints, answering and extending a question of L. Cristea. |
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