Harmonize your Fractals

Matt McClinton
Graduate Student
University of Wisconsin-Milwaukee

The Sierpinski Gasket (SG) is a known fractal object. A simple observation shows that SG is path connected. Unfortunately, the infinitely jagged structure of the Gasket prevents these paths from being differentiable. If only there existed a means of smoothing out SG into an object where continuous and differentiable paths existed between pairs of points. As a matter of fact there is!

I will demonstrate the technique known as “minimizing the graph energy” as described in the late Robert Strichartz’s book “Differential Equations on Fractals”. This technique involves finding the solution to a system of equations where the solution produces a graph that has differentiable paths, and even better satisfies the Laplacian. Using a homeomorphic mapping defined by Jun Kigami in 1989, by finding the graph energy minimizing values on level sets of SG, we produce a fractal object known as the Harmonic Sierpinski Gasket (HSG).

This talk is intended for those that are interested in analysis, algebra, combinatorics, topology, fractal geometry, and/or graph theory. Any necessary background information will be provided during the talk, and I will end with some open problems.