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.