Mathematical modeling of geometric fractals using R -functions
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MATHEMATICAL MODELING OF GEOMETRIC FRACTALS USING R -FUNCTIONS K. V. Maksymenko-Sheyko a† and T. I. Sheykoa‡
UDC 532.5+536.24
Abstract. Based on new constructive means of the theory of R-functions, new approaches are proposed to the construction of equations for objects of fractal geometry. Equations for some of the most well-known fractals, namely, the Koch curve, snowflake, and cross, Sierpinski carpet, Levy fractal, and Pythagoras tree are presented. Keywords: R-function, geometrical fractal. INTRODUCTION At the present time, fractals are widely used in radio engineering in designing antenna devices (the Koch curve and Sierpinski carpet) and waveguides (the Koch snowflake), in computer graphics, and in image compression. In physics, fractals naturally arise in modelling nonlinear processes such as a turbulent liquid flow, complex diffusion and adsorption processes, etc. Fractals are used in modelling porous materials, for example, in petrochemistry. They are used in biology for modelling populations and for describing systems of internal organs (a blood vessel system). Of increased interest are problems of mathematical modeling of physicomechanical fields in objects of a fractal nature. The definition of a fractal proposed by B. Mandelbrot is as follows: “A fractal is understood to be a structure consisting of parts that are similar to the whole in some sense” [1]. Geometric fractals are most demonstrative [2, 3]. It is precisely these fractals with which the history of fractals began. First ideas of fractal geometry have arisen in the XIXth century when G. Cantor transformed a line into a collection of unconnected points with the help of a simple recursive procedure (the so-called Cantor dust or Cantor set). He took an interval, eliminated its central third, and then repeated the same procedure with the remained intervals. G. Peano has drawn a curve of a peculiar form. At the first step, he took an interval and replaced it by nine intervals whose length was by a factor of three smaller than the length of the initial interval. Then he repeated the same operation with each interval of the obtained curve. The Peano curve and Cantor dust went beyond the scope of conventional geometric objects. They had no precise dimension. The Cantor dust was constructed from a one-dimensional straight line but consisted of points (dimension 0), and the Peano curve was constructed from a one-dimensional line and, as a result, a plane was obtained. For self-similar sets, the Hausdorff dimension is computed as follows: if a set is partitioned into n parts similar to the initial set with coefficients r1 , r2 , r3 , K, rn , then its dimension S is the solution of the equation r1S + r2S + r3S + K + rnS = 1. For example, the Hausdorff dimension of the Cantor set is equal to ln 2 / ln 3 (it is partitioned into two parts, and the similarity factor equals 1/3), and the Hausdorff dimension of the Sierpinski triangle equals ln 3 / ln 2 (it is partitioned into three parts, and the similarity factor equals 1/2). In all deterministic fractals
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