Harmonic Mappings of the Sierpinski Gasket

Greg Padowski

As part of the Cornell Summer 2000 REU program, I worked on harmonic mappings of the Sierpinksi Gasket (SG) under the supervision of Professor R. Strichartz. I investigated harmonic mappings of SG to both the sphere (x^2 + y^2 + z^2 = 1) and the hyperboloid (- x^2 - y^2 + z^2 = 1). A sample of a harmonic mapping to the sphere, produced using Maple 6, appears below.

Here I give an intuitive definition of a harmonic mapping of SG. A harmonic mapping of the mth level approximation to SG into a range D is a mapping that would arise when one replaced the edges of the mth level approximation with springs constrained to lie on geodesics of D and allowed the structure to assume a minimal energy configuration. These springs obey Hooke's Law and have zero length at equilibrium. A mapping of SG is a harmonic mapping if it is the limit of some sequence of harmonic mappings on the mth level approximation to the gasket as m goes to infinity.

The harmonic map below is a mapping of SG to the sphere. The images of the boundary points lie equally spaced on a line of latitude slightly north of the equator. The graphs below are actually 5th level approximations to the harmonic map. The algorithm used to compute this map used a fractal formulation of gradient flow.

The C++ source code for the functions that generated these harmonic mappings may be viewed by clicking on this link: 2aug00.cpp The code was written and compiled in a Linux environment.

A harmonic mapping from SG to the unit sphere

The projection of this same map to the xy plane

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Last Update: 14 August 2000