Iterative function systems and constructing fractals

Neli Blagus (2010) Iterative function systems and constructing fractals. EngD thesis.

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Abstract

A fractal is a mathematical object, that can be split into several parts, each of which is a minuscule image of the fractal itself. This feature is called self-similarity. The second typical feature of fractals is their infinite complexity of details, because of which they are appropriate for modelling different natural structures. Fractals are too irregular in form to be described with classical geometrical shapes, even though they are often symmetric. The branch of mathematics concerned with fractals is called fractal geometry. It introduces effective methods of approximating fractals as well as natural structures described by them on the computer screen. Fractals can also be applied in computer graphics, geography and even biomedicine. The main goal of this diploma thesis is to describe fractals in the plane and the mathematical background necessary to define them and to and explain their structure. One way of constructing fractals is through hyperbolic iterated function systems. A hyperbolic iterated function system is a system that consists of a finite set of contraction mappings on some metric space. This type of mapping brings points closer to each other and makes shapes smaller. This enables the construction of self similar fractals with the use of a random iteration or a deterministic algorithm. In this diploma thesis, both of these algorithms are described. We also decribe the colour stealing algorithm, which is a variation of the random algorithm. The thesis consists of three parts. The first part is an introduction and motivation of the topics introduced later. The second part is of theoretical nature, describing the mathematical concepts, important for defining fractals and representing algorithms for fractal construction. The last part of the thesis deals with algorithms, as well as possibilities for further work. The results obtained with the algorithms are represented mostly with pictures generated with Mathematica.

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