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MATHEMATICS TEACHING RESEARCH JOURNAL 17 SPRING 2020 Vol 12, no 1 Vol 12, no 1 Fractals, Self-Similarity, and Beyond Rohitha Goonatilake, Ray A. Casas [email protected] Department of Mathematics and Physics, Texas A&M International University, 5201 University Boulevard, Laredo, Texas 78041-1900 Abstract: Since Mandelbrot coined the term fractals in 1975, fractals and their fractal dimension have become a sensational topic in mathematics, especially with their creative applications to various fields of science (Mandelbrot, 1975). The study of fractal geometry and chaos theory are two examples of new fields in mathematics that have been popularized in the past 30 years, in large part due to the availability of high-speed computers. The Julia sets, for example, are complex 2 fractals that are formed from the sets of repelling periodic points of the mapping 푔푐(푧) = 푧 + 푐, where 푐 is a fixed complex-valued parameter. This article considers several properties of the Julia sets and fractal dimension of the Julia sets at various values of c. A brief overview of theories dealing with vertex figure and dual of a tessellation is provided to unify some other of the concepts associated with fractals. Attempts are being made to study some basic properties of fractals and self-similarity shapes and to some extent appreciate the attractiveness of fractals, chiefly in analytical points of view using some elementary computations. In conclusions, numerical construction of Julia sets is presented with the examples of finding their factional dimensions. This coverage is adequately considered for possible framework of the course design that could be developed and delivered in a typical college course. Introduction Chaotic dynamics and fractal geometry have gained prominence in mathematics and applications for the past several decades. Due to their unique applications in various fields of science, fractals and their associated fractal dimension have been studied heavily. Because of the varied nature, both of the mathematical insights and the applications, the goal of this article is to make such results available to a larger mathematical audience providing to have an expository flavor. One notable example of a fractal is the Julia set. The Julia sets were discovered and studied by the French mathematicians Gaston Julia and Pierre Fatou during World War I (Kennedy et al., 2015). Due to his contributions in the study of iterations of complex rational functions, the Julia sets were named after Julia (Gulick, 2012). For years now, Julia sets have been studied for their complex Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/ MATHEMATICS TEACHING RESEARCH JOURNAL 18 SPRING 2020 Vol 12, no 1 Vol 12, no 1 dynamical systems. However, there is still much work that can be done concerning their properties as fractals, specifically their fractal dimension. Over the past few decades, there has been significant advances in this area. For example, Shishikura (1998) found that Julia sets with complex constants located on the boundary of the Mandelbrot set had fractal dimension 2 (Gleick, 1987). The primary focus of this article is to determine the fractal dimension of the Julia sets. The calculation of the fractal dimension of a Julia set is actually not as straightforward and of course, is dependent on the dimension and the parameter 푐. For example, a line is one-dimensional, and a plane is two-dimensional. However, what if someone asks for a geometric figure whose dimension is between one and two? In the beginning of the late nineteenth century, mathematicians have created numerous objects of uncertain dimension. The Koch curve is an example, named after the Swedish mathematician, Helge Von Koch, who first described it in 1904 (Dekking & Dekking, 2016). Furthermore, we study various types of functions in particular, their graphs. These functions and others are used to model physical phenomena in problem solving. However, in many instances, such as the shapes of waves and prices in the stock market, behavior does not correspond to reasonable functions. First, some known concepts and related discussion from the references cited are presented to make this article readable for those who are new to the topic. Fractals and Self-Similarity A property that can be clearly observed from the repeated magnification of images of the sets is that of self-similarity. Many fractals express this property. However, when Mandelbrot first defined the term ‘fractal’, he did not express it in terms of only self-similarity. In fact, he considered it in terms of dimensions. Given below are Mandelbrot’s definitions of fractals and self-similarity (Falconer, 2013). Definition 1. Fractals: Fractals are geometrical figures that are generated by starting with a very simple pattern that grows through the application of rules. In many cases, the rules to make the figure grow from one stage to the next involve taking the original figure and modifying it or adding to it. This process can be repeated recursively an infinite number of times. Some other known fractals are Hilbert curve, Stairs, Koch anti-snowflake curve, Peano curve, Peano-Gasper curve, Bush One, Bush Two, Bush Three, Dragon curve, and Jurassic Park fractals (Bourke, 2006). Definition 2. Self-Similarity: Self-similarity means that each small portion, when magnified, can reproduce exactly a larger portion. In 1975, Benoit B. Mandelbrot, a Fellow at IBM Thomas J. Watson Research Center, published the first comprehensive study of the geometry of self-similar shapes such as the Koch curve, the Sierpinski gasket and the Menger sponge (Pickover, 1998). Fractals are not simply abstract Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/ MATHEMATICS TEACHING RESEARCH JOURNAL 19 SPRING 2020 Vol 12, no 1 Vol 12, no 1 creations of mathematicians. Indeed, intense ongoing scientific research suggest that many, and possibly most of the physical and life processes are what can be described as chaotic, and the geometric shapes of chaos are fractals. The fractal geometry of nature popularized a form of geometry that attempts to study chaotic behavior. The following discussion provides a somewhat introduction to the notion of self-similarity, which is part of the modern field of study. Examples of Reptiles Fractals are a relatively new mathematical concept associated with the geometry of irregular shaped objects. Generally, more attractive fractal gaskets result from reptiles with smaller numbers of component tiles. The reptiles are comprised of tiles of more than one size. As we have seen tiles, when arranged in the proper way, produced new tiles of the same shape. Equilateral triangles are simple examples of reptiles. It is also observed that as new shapes are constructed, they are similar to each of the previous shapes. Some trees may also exhibit the idea that begin with two branches and then grow two similar ones at the end of each new branch. A more intricate pattern can be constructed in this manner. To construct the Koch curve, begin with a line segment. Next, replace the middle third by an equilateral triangular bump, resulting in a 4-segment polynomial curve. In the third step, a triangular bump is added to each side of the curve, yielding a curve with 16 sides. In the first iteration of the process, the original segment is replaced by a curve called the generator. At each iteration, every segment of the developing fractal curve is replaced with a suitably scaled version of the generator. Figure 1. Construction of the Koch Curve As displayed in Figure 1, this construction could begin with an equilateral triangle. Next, equilateral triangles are constructed on the middle thirds of the original triangle. The curve formed by repeating this process indefinitely is known as the Koch curve. Notice how each small triangle formed is similar to the proceeding triangle (L-System: Lindenmayer, 1968). The Koch curve is said to be self-similar, since the curve looks the same when magnified. In addition to being self- similar, the Koch curve has a finite area, but an infinite perimeter. These two facts will be proved later in the article. These figures are often referred to as stars, the first one is a three-pointed star and the second one is a six-pointed star. If this pattern of constructing a new equilateral triangle is continued indefinitely, the resulting figure is called Koch curve or Koch snowflakes as displayed Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/ MATHEMATICS TEACHING RESEARCH JOURNAL 20 SPRING 2020 Vol 12, no 1 Vol 12, no 1 in Figure 2. The Koch curve has many interesting properties, but of special interest is self- similarity. That is, if fragments of the curve are viewed with highly powered microscopes, the enlargements all appear identical.
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