| Abstract: |
| The Sierpinski carpet and Menger sponge are well-studied generalizations of versions of the Cantor set. They are also members of a two-parameter family of connected higher-dimension fractals that can be constructed iteratively from the $n$-cube. In this talk we focus on determining taxicab paths--piece-wise linear paths that always travel parallel to an axis, possibly with non-trivia 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 a fractal, we explicitly construct taxicab geodesics between them. We then use these data to compare the taxicab metric to the standard Euclidean metric in the fractals we consider. This is joint work with Elene Karangozishvili and Derek Smith. |
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