Fractals: The Organization of Chaos

Fractals: The Organization of Chaos

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Fractals: The Organization of Chaos


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Fractals are a relatively new concept in geometry. Most concepts for Euclidean geomtery, the division of geometry which deals with lines, circles, triangles, and other standard shapes, stem from the Late Greek and Early Rioman times. Considering the age of mathematics, the study of fractals is new becasue it dates to the beginning of this century. However, the age of computers brought about an explosion into this yet untamed universe of math. As Heinz-Otto Peitgen and Dietmar Saupe remark in the preface for The Science of Fractal Images, "Computer graphics has played an essential role both in its development and rapidly growing popularity" (V). Before this, mathematicians could only visualize what they were discussing (Mandelbrot, Fractals: Form, Chance, and Dimension 2). But now, fractals are the mathematician's answer to chaos and therefore can be used to help scientists better understand nuature and the universe. Scientists can define any structure from a snowflake to a mountain or even an entire planet with this new division in Mathematics. Thus, fractals define our universe.

Benoit B. Mandelbrot is a key figure behind the rise of this new science. A Professor of mathematical Sciences at Yale and an IBM Fellow, Mandelbrot is the man who coined the term "fractal" in 1975. Mathematicians, such as Gaston Julia, only defined them as sets before this and could only give properties of these sets. Also, there was no way for these early fractal researchers to see what they were hypothesizing about. As Mandelbrot states in The Fractal Geometry of Nature, "I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means 'to break...'" (4). Mandelbrot used this particular root because of how he defines fractals. Unlike Euclidean geometry, which has its figures in a particular dimension (e.g. a square is two-dimensional), fractals have fractional dimensions. They do not exist in just one dimension but can encompass part of another. For example, as Mort La Brecque states in his article on fractals in the Academic American Encyclopedia "a natural fractal of fractal dimension 2.8 ... would be a spongelike shape that is nearly three dimensional in its appearance. A natural fractal of fractal dimension 2.2 would be a much smoother object that just misses being flat" (105-106, Mandelbrot "Fractals").

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Therefore, fractals can represent everyday objects such as the sponge mentioned above and many other natural objects which Euclidean geometry can only imitate by glossing over the irregularities. This glossing over occurs because the whole number dimensions of Euclidean geometry, such as the first or third dimensions, are used constantly in our modern world. Defined geometrically, a zero-dimensional figure would just be a point. It has no height, width, or lenght. A one-dimensional figure is characterized by a line, which has only length. A two-dimensional figure would only have length and width, and would be a plane. A three-dimensional figure, like a box, has all three components of length, width, and height. Fractals, meanwhile explore the spaces in between these dimensions, figures which do not fit so nicely into these categories a planes, lines, or boxes do.

Thus, not only is the previously described step into the fractional dimensions of fractals important but also the fact that most fractals are self-similar and infinitely complex. The latter is defined as a picture that has no end to the magnification of itself. It is possible to pick a point and zoom in infinitely many times and still end up with some sort of shape. One will never come to a single point, line, plane, or surface that defines the fractal. The former means that the fractal is made of smaller versions of the larger pictures. Because the fractal is infinitely complex, self-similarity also goes to infinity.

The Sierpinski triangle, named after the Polish mathematician Waclaw Sierpinski, is an excellent example of the infinite complexity and self-similarity of fractals. It is the composition of a series of triangles shrunk and moved repeatedly. This is done by reducing an image by one-half, copying the image three times, and placing the images in an equilateral triangle form. The steps are then repeated with the new figure. As the process is repeated infinitely many times, a Sierpinski triangle starts to form. This can be done with any image as is seen in Figure 1 (Jürgens, et al. 62). Magnifying any portion of the triangle will produce another view of the same triangle. The Sierpinski Triangle is both self-similar and infinitely complex.

The mathematics behind fractals is quite simple considering the fact that these numbers represent chaos, something that we previously thought undefinable and quite complex. The process deals basically with sets and imaginary numbers. Sets are an easy concept to understand and will be left undefined here. However, a brief explanation of imaginary numbers is in order.

Imaginary numbers are just that, imaginary. They do not have any physical representation. Mathematically, they are represented by i, where is defined as . Imaginary numbers are undefined because there is no number that can be squared and equal -1. These numbers are usually written in the form of a+bi, where a and b are real numbers. They are represented graphically in the same coordinate plane as regular numbers; however, the y-axis is now the imaginary axis. The point is represented as the ordered pair (a,bi).

The way a computer knows how to draw fractals is by taking an equation entered in this form and seeing at which points the answer tends to infinity and how many iterations, or repetitions, it takes for this to occur. The iterations consist of putting a number into the equation and either getting an output of a number or an output that tends to infinity. If an answer is received that is a number, then that number is put into the equation until the output tends to infinity. Then the computer assigns certain colors to the number of times it takes for the output to go towards infinity. For example, say one iteration causes the computer to make the point blue, two iterations to make the computer color the point red, and so on until the computer reaches a preset limit. Figure 2 is an example of a computer generated drawing of The Julia set of (1+0.2i)sin(z). As Robert L. Devaney, a former professor at Northwestern University and the University of Maryland as well as chairman of the Boston University Department of Mathematics, explains, "[a] Julia set is the place where all the chaotic behavior of a complex function occurs" (221). The point where the output values that go to infinity meet the output values that do not is the interesting part of the fractal.

The Mandelbrot set is the most interesting fractal discovered to date (Figure 3). First seen in 1980 by Benoit B. Mandelbrot, it has been found to hold every connected Julia set inside of its picture. Heinz-Otto Peitgen, et al. states of the Mandelbrot set, in The Science of Fractal Images: "[it is] the pictorial manifestation of order in the infinite variety of Julia sets" (177). A connected Julia set is as Hans Lauwerier, former Professor of Mathematics at the University of Amsterdam, states "if [a fractal] is connected, [it] consists of a succession of lines" (148). This means that the points in the Mandelbrot set are all tied together with no single points off on their own. The different sets can be found by zooming into certain regions on the picture. Figure 3a shows one of the Julia sets which occurs in the boundary area of Mandelbrot's discovery. Mandelbrot's fractal is also infinitely complex while being self-similar.

Fractals go beyond the pure mathematics of the concept as the practical uses are just starting to be found. By being able to identify natural structures with mathematical formulas, we can predict and hypothesize about the future of our environment, species, or many other natural events. We could predict volcanic eruptions or the way the stock market behaves. Fractals are all around us, from the weather, to a pile of fallen leaves, to the human body. With the knowledge that we are surrounded by these shapes we can make many predictions about our planet or universe. The use of fractals can help us predict the weather, predict the way a rock will fracture, and represent nearly limitless other possibilities (Briggs 13-25).

John Briggs, in his book entitled Fractals: The Patterns of Chaos, cites many objects in our universe that behave as fractals. The most interesting topic of discussion he goes into is that of the fractals present within our own body. Briggs quotes Homer Smith, a computer engineer, as saying, "if you like fractals, it is because you are made of them. If you can't stand fractals, it's because you can't stand yourself" (123). This statement rings true because "from the cascade of ever smaller blood vessels feeding the heart" (Figure 4) to the folds on the surface of the brain, one can see the pattern of "self-similar scaling" (Briggs 124). Even the very beating of the heart is shown to be chaotic. Briggs says that when a heartbeat from an electrocardiograph is graphed using a special type of plot called phase-space plots, one can see the irregularity of the healthy heart. Only when the heart becomes regular is there a problem; the patient has a heart attack (126). The heart and other bodily functions thrive on irregularity. Thus, chaos reigns throughout the body. With our knowledge of fractals, we can perhaps learn more about ourselves.

Fractals have also come to help the medium that brought them to life. Computer designers are always looking for easier ways to store digitized photographs so as to take up less memory. A mathematician by the name of Dr. Michael F. Barnsley wrote a program which identifies fractal structures in digitized still and video images and then codes them to help save space. This process bore fractal image compression that is used in many forms of multimedia applications (Mandelbrot "Fractal"). The beautiful landscapes that fractals made possible through computer imaging helped gain its entrance into Hollywood. After seeing these landscapes by Mandelbrot, Loren Carpenter decided he wanted to use these in animation. Fractals were used in the Star Wars films as well as in the Star Trek series. Figure 5 shows, as Briggs states "the mountains in the 'Genesis Demo' segment of Star Trek II: The Wrath of Khan" (84).

Fractals permeate our lives. The very planet we stand on is one huge fractal waiting to be discovered. Scientific discoveries make our world smaller but few tend to bring the universe into our grasp while pushing it out of our reach at the same time. Though we are probing into the deepest secrets the universe holds, as with fractals of infinite complexity, we will never reach the end. From the smallest amoebae to the outermost galaxy, we have only begun to define our universe.


Works Cited:


Brecque, Mort La. "Geometry, fractal." Academic American Encyclopedia. 1994 ed.

Briggs, John. Fractals: The Patterns of Chaos. New York: Simon and Schuster, 1992.

Devaney, Robert L. A First Course in Chaotic Dynamical Systems. Reading: Addison-Wesley Publishing Company, Inc., 1992.

Jürgens, Hartmut, et al. "The Language of Fractals." Scientific American August (1990): 60-67.

Lauwerier, Hans. Fractals: Endlessly Repeated Geometrical Figures. Trans. Sophia Gill-Hoffstadt. Princeton: Princeton University Press, 1991.

Mandelbrot, Benoit B. "Fractal." Microsoft Encarta 97 Encyclopedia CD-ROM

Mandelbrot, Benoit B. Fractals: Form, Chance, and Dimension. San Francisco: W.H. Freeman and Co., 1977.

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman and Co., 1983.

Peitgen, Heinz-Otto. The Science of Fractal Images. New York: Springer-Verlag, 1988.
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