North Georgia College and State University

Dustin Tench

North Georgia College and State University

MATH 6120

Dr. Cooper

Fractals, Why Study Them?

Imagine you are in space looking at the Eastern coast of the United States. From such a great distance, the coast looks perfectly smooth.

As you approach the earth you notice that the coast is not perfectly smooth. It has many curves to it. You could not see these curves from a distance.

As you get closer, you see that there are several bays, capes, peninsulas, etc. that you could not see before.

Before you know it you are standing on the sand on the coast.

But, we can still get closer and there are tiny gaps between the sand particles.

If we get closer still, there are tiny pits on the surface of the sand particles.

The fact that any small part of the coast will look similar to the whole things was first described by Benoit Mandelbrot. He named shapes like this fractals. Fractals are shapes with an infinite amount of detail. The idea of fractals was first developed to study nature. In nature, the more you zoom in on an object, the more detailed it becomes, just like a fractal does. Here is an example of a fern leaf. The fern leaf itself appears to show up as each of the smaller leaves.

Fractals are a relatively new topic in mathematics. Typically, material taught to students is knowledge that we have known for thousands of years. Fractals have been studied for about 500 years, again, relatively new. There are many different types of fractals. There are Base-Motif Fractals, Mandelbrot Sets, paper folding fractals and many more. There are many applications of fractals which include data compression, special effects, diffusion, and many more.

There are several classifications of fractals. One particular type is Base-Motif Fractals. Here’s how a Base-Motif fractal is composed. Take a shape that is composed of line segments. Call this shape the base. Now take another shape and call it the motif. Substitute each line segment in the base with the motif. Do this with the resulting figure and continue the process. Each resulting figure is called an iteration. If you continue the process an infinite number of times, you get a base-motif fractal. An example of a base-motif fractal occurs in the novel Jurassic Park. At the beginning of each chapter, there is an iteration. The picture gets more complicated the further you get into the novel.

As you can see, the process works. Each iteration gets more and more complicated. If this process were continued an infinite number of times, we would get a shape that would be infinitely complex.

Another type of fractal belongs to Mandelbrot sets. This type of fractal is based on a formula. We take a point z on the complex plan such that z = 0 + 0i. We take another point c and make z = z2 + c. We repeat this number many times. If this number does not go to infinity, it belongs to the set and we can mark it. If it does go to infinity, we can color the point depending on how fast it escaped to infinity. This set produces the following on the complex plane.

We can use other formulas as well. Suppose z = cos-1z + log z and we did this process, the following picture is produced on the complex plane.

Another type of fractal is a paper folding fractal. Consider a long strip of paper. I will represent it as a line. Fold this strip in half and then unfold it so that the angle between the two halves is 90ᵒ.

Now, do the same to each half of the paper. Then, to each quarter.

The more iterations we do, the more complicated the picture will get. If we were to do this an infinite number of folds, we would get the Dragon Fractal.

The idea that a fractal is just a repeatable pattern no matter what scale you are in has some applications. One application is data compression. In 1992, Microsoft released its compact disk version of Encarta Encyclopedia which contained thousands of articles, 7000 photographs, 100 animations, and 800 color maps. This all combined was less than 600 megabytes of data. The way the engineers pulled this off was with fractal data compression. If we consider the Mandelbrot set, a full-screen color .gif image of it occupies about 35 kilobytes. If you store the formula the produces the graph, z = z2+ c, it takes up no more than 7 bytes. That’s a 99.98% compression. If this works for the Mandelbrot Set, perhaps it could work for a flower diagram, map of Europe, or a picture of Bill Clinton. The idea is to find functions, each of which produces some part of the image. For a complex image that is not a fractal, you might need hundreds of such functions. However, it would still take up less space than hundreds of thousands of colored pixels.

Fractals can be used to describe the spreading of substances, such as gases diffusing. Diffusion fractals are created by starting from the center and spread the points outwards. The way this is done is by randomly moving points around the screen. When a point hits the center point, it stays there permanently. When some other points hits either the center point, or the new point, you make it stay there as well. What you see when the shape is being formed is a growing fractal.

Another application of fractals is special effects in movies. Because little storage space is needed due to easy compression, fractal landscapes have been able to be published for a while. An artist named Loren Carpenter was able to generate a computer movie of a flight over a fractal landscape. He was hired by Pixar. Fractals were used in Star Trek II: The Wrath of Khan to generate the landscape on Genesis. Fractals were also used in Return of the Jedi to create the geography of the moons of Endor and the Death Star outline. Today, software is easily available that allows you to create such art.

We’ve looked at mostly two-dimensional fractals with the majority of those occurring on the complex plane. The Megner Sponge is an example of a three dimensional fractal that doesn’t involve complex numbers. Take a cube and cut it into 27 smaller cubes. Remove the center cube and the centers of the six sides so your cube looks like this.

Do this again to each of the 20 remaining cubes. Do this process an infinite number of times to produce the following cube.

The Megner Sponge has a network of infinitely complex lines. The overall length of the lines will be infinite. The surface area of the Megner Sponge would be zero. The length of each small square is

, thus the surface area would be zero. Each time we remove 7 cubes, we take away 7/27 of the volume. This leaves 20/27 of the volume at each iteration. In order to find the volume of the Megner Sponge, we have to raise 20/27 to the infinity power, which becomes 0.

Fractals can show up in many places if you know where to look. We have discussed several types of fractals which include Base-Motif Fractals, Mandelbrot Sets, and paper-folding fractals. We have looked at several applications of fractals which are data compression, special effects, and diffusion. The Megner Sponge was an example of a three dimensional fractal. Because fractals are a new topic in mathematics, studies are still being done on them to see what other applications might exist.