Fractal transformations of harmonic functions

Michael F. Barnsley; Uta Freiberg

doi:10.1117/12.696052

3 January 2007 Fractal transformations of harmonic functions

Michael F. Barnsley, Uta Freiberg

Proceedings Volume 6417, Complexity and Nonlinear Dynamics; 64170C (2007) https://doi.org/10.1117/12.696052
Event: SPIE Smart Materials, Nano- and Micro-Smart Systems, 2006, Adelaide, Australia

Abstract

The theory of fractal homeomorphisms is applied to transform a Sierpinski triangle into what we call a Kigami triangle. The latter is such that the corresponding harmonic functions and the corresponding Laplacian Δ take a relatively simple form. This provides an alternative approach to recent results of Teplyaev. Using a second fractal homeomorphism we prove that the outer boundary of the Kigami triangle possesses a continuous first derivative at every point. This paper shows that IFS theory and the chaos game algorithm provide important tools for analysis on fractals.

Citation Download Citation

Michael F. Barnsley and Uta Freiberg "Fractal transformations of harmonic functions", Proc. SPIE 6417, Complexity and Nonlinear Dynamics, 64170C (3 January 2007); https://doi.org/10.1117/12.696052

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