Discrete extremal lengths of graph approximations of Sierpinski carpets

Abstract

The study of mathematical objects that are not smooth or regular has grown in importance since Benoit Mandelbrot's foundational work in the in the late 1960s. The geometry of fractals has many of its roots in that work. An important measurement of the size and structure of fractals is their dimension. We discuss various ways to describe a fractal in its canonical form. We are most interested in a concept of dimension introduced by Pierre Pansu in 1989, that of the conformal dimension. We focus on an open question: what is the conformal dimension of the Sierpinski carpet? In this work we adapt an algorithm by Oded Schramm to calculate the discrete extremal length in graph approximations of the Sierpinski carpet. We apply a result by Matias Piaggio to relate the extremal length to the Ahlfors-regular conformal dimension. We find strong numeric evidence suggesting both a lower and upper bound for this dimension.