532 CHAPTER 8 Geometry (a) (b) (c) (d) Figure 8.106 The fractal tree If you take a little piece of any branch and zoom in on it, it will look exactly like the original tree. Fractals are scale independent , which means that you cannot really tell whether you are looking at something very big or something very small because the fractal looks the same whether you are close to it or far from it. In Figs. 8.107 and 8.108, we develop two other fractals through the process of recursion. Fig. 8.107 shows a fractal called the Sierpinski triangle, and Fig. 8.108 shows a fractal called the Sierpinski carpet. Both fractals are named after Polish mathematician Waclaw Sierpinski. Figure 8.107 Sierpinski triangle Figure 8.108 Sierpinski carpet Fractals provide a way to study many natural forms, including coastlines, trees, mountains, rivers, weather patterns, galaxies, molecular structures, brain functions, and lung functions. Fractals help organize structures and processes that appear chaotic, such as the blood flow through veins, capillaries, and arteries. The study of such chaotic processes is called chaos theory . Fractals have the potential to revolutionize many fields of science, including medicine, engineering, meteorology, and astronomy. In short, fractals may alter how we study life itself within the vast universe! MATHEMATICS TODAY Fractal Applications Many irregular shapes found in nature can be represented using fractals. Within our own bodies, brain surface, lung tissue, and blood vessels have patterns that can be modeled using fractals. Fractals can also be used to model the growth of bacteria or viruses in the study of various diseases. In addition, fractal images are used in animation in movies. Fractals are also used to increase the amount of data that can be stored on data storage devices. Smartphones, cordless microphones, wireless headphones, and wireless computer networks often use fractal antennas. These antennas maximize the reception of the desired signal. Why This Is Important Fractal geometry plays an important role in scientific research and is involved in many aspects of modern technology, including many products that you use every day! m Fractal antennas make it possible for us to use many of the apps on our cell phones. Instructor Resources for Section 8.7 in MyLab Math • Objective-Level Videos 8.7 • PowerPoint Lecture Slides 8.7 • MyLab Exercises and Assignments 8.7 • Chapter 8 Projects Maxx-Studio/Shutterstock
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