An Investigation of Iterated Function Systems and Fractals

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Michael Zielinski


This paper explores some of the fascinating ideas involved in the generation of fractals. A linear algebra and strong calculus background is sufficient to understand the main ideas, however many of the proofs require analysis. I examined metric spaces, and in particular, the metric space H(X), which consists of the compact subsets of the space X. The completeness of this space is proven, enabling the construction of fractals as fixed points of contractive transformations on H(X). I use the classical Cantor set and the Sierpinski triangle to illustrate the construction of fractals using iterated function systems (IFS's) which consist of a space together with a finite number of contractive transformations. I also prove that for at least one case, a transformation consisting of an infinite number of contraction mappings is not a contraction mapping.

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