Fractals


There is no doubt that just about everyone on this planet has seen a fractal in their lifetime, even though many do not even know it. Every natural thing on our planet can be described in mathematical terms. This is where the subject of fractals comes into play. Fractals occur everywhere in nature and can sometimes be depicted as forming the core of our lives. Being able to comprehend the structure of a mathematical fractal opens up the ability to understand how everything in this world was formed. A fractal is defined to be a rough or fragmented shape that can be broken up into smaller parts, which can be seen as a smaller copy of the original shape. It is almost impossible to describe the natural things in our world, such as the clouds, trees, plants and so on, as geometric images. This is why fractals are such an important part of how nature is structured. This history of fractals is quite interesting, considering that the “father” of fractals has just passed away a few weeks ago. Also looking at different kinds of sets that describe fractals is of great importance in the overall application of them. Finally, I will write about the computer programming of fractals and how to create them on your own.

Benoit Mandelbrot is considered to be the father of fractal geometry. He has said that the first thing that made him start to even think about the idea of fractals was when he was trying to figure out how long the coast of Britain was. What he discovered was that if you look at a map and keep on zooming in on it, repeated patterns will appear. [Hoffman 2010]. The idea that he used to get the most accurate measure of the coastline of Britain was determined by what length of ruler he would use. He showed that smaller rulers are more accurate because they can fit better into the irregular patterns of the coast, rather than using one large ruler. He concluded that as the scale of measurement he used decreased in size, the actual length of the coastline increased [Laubender 1999]. This shows that we can zoom into the coastline an infinite number of times, using a smaller unit of measurement and keep getting a more accurate estimate. Mandelbrot always said to not think about what you see, but what it took to make what you see. “The key to fractal geometry…is that if you look on the surface, you see complexity and it looks very non-mathematical” [Jersey and Shwarz 2008]. His studies about the Britain coastline lead into one of the main ideas of fractals, known as self-similarity. “A set S is called self-similar if S can be subdivided into k congruent subsets, each of which can be magnified by a constant factor M to yield the whole set S” [Shapiro 2010]. By looking at the coastline of Britain from a far distance and then zooming up extremely close, the images would look similar. Self-similarity is one huge principal idea when classifying what a fractal is. Although Mandelbrot was the one to coin the term “fractal geometry” in 1975, there were many mathematicians before his time that noticed this property of self-similarity.

A simple start to understanding the formation of fractals is to look at the Sierpinski Triangle. WacLaw Sierpinski was a polish mathematician whose most important work was in the fields of set theory, number theory, and point set topology [Riddle 2010]. What Sierpinski came up with was to first look at a large equilateral triangle. He then started to divide that large triangle into four smaller equilateral triangles. This action is repeated over and over again leaving the center triangle open each time [Lauwerier 1991]. Looking at this triangle and dividing it up an infinite number of times displays the idea of self- similarity, upon which fractals are based upon. By looking at the design of the Sierpinski triangle, it can be concluded that if the number of triangles is increased, the length and area of the triangles will decrease. If we let $N_k$ denote how many triangles we have within the main triangle, $L_k$ denote the length of the sides of each triangle, and $A_k$ be the area of the triangle, we have the following equations: : \beginequation\begin{align} N_k=3^k\nonumber\\ L_k=(1/2)^k=2^(-k)\nonumber\\ A_k=L_k^2\timesN_k=(3/4)^k\nonumber\\\ \end{align}\endequation We have to understand that we can take one portion of this divided up triangle and it will look exactly like the whole triangle itself, and as the number of iterations tends to infinity, the area of each triangle tends to zero, however will never equal zero.

Another type of fractal image is called the Koch curve. Helge von Koch was a mathematician who studied curves without any tangents. He came up with the idea that any line segment can be described as being infinitely long. The idea of the Koch curve, or fractal, is to take one line segment with some length, $$l$$. Then take that line segment and divide it into three different segments of equal length. By taking the middle segment and splitting it into two and configuring an equilateral triangle out of it, we expand the length of the entire line segment when added together. For example, if we took a line that had length 1 and took out the middle third of the segment and added in an equilateral triangle, we would have four smaller line segments of length, 1/3. So the original line of length 1 has turned into length 4/3. Doing this repeatedly will keep expanding the length of that original line segment [Lauwerier 1991]. If we were to apply this process to a particular shape, the equilateral triangle for instance, the result would be a figure that looks like a snowflake. This Koch snowflake is shown to have infinite perimeter but finite area. In order to see this we let $$N_k$$ be the number of sides the snowflake has after completing the kth step of the process. We start out with three sides because it is an equilateral triangle: \beginequation\begin{align} N_0=3\nonumber\\ N_1 =4×3=12\nonumber\\ N_2 =4×12=4^2 ×3\nonumber\\N_3 = 4^3 × 3\nonumber\\ N_k = 4N_(k−1) = 4^k × 3\nonumber\\ \end{align}\endequation For each step in the process, each line segment is divided three times, so if we nlet L_k define the length of each segment after the kth step, then we have $$ L_k=1/3^k$$. Then, for the perimeter we have to multiply the number of sides by the length of each side so we let $$P_k$$ be the perimeter after the kth step. We have $$P_k=N_kL_k=3(4/3)^k$$ By looking at this equation we can see that as k tends to infinity, so does the perimeter. We can then conclude that the Koch snowflake as finite area but infinite perimeter [Shapiro 2010].

The German mathematician, Greg Cantor, who was one of the founders of set theory, discovered the Cantor fractal. This fractal is similar to the Koch Curve. Similar to the Koch curve, we start out with one straight-line segment and divide it into three parts. Then remove the middle third, but keep the end points. So we started out with one line and two endpoints, and turned it into two lines and four endpoints. This process is to be repeated over and over, and the outcome after many times, would be points. \begin{center} \begin{tabular}{|c|c|} \hline Steps & Number of Line Segments& Length of Line Segments\\ \hline 1& 2^1=1 & 1/3^(1)=1/3\\ \hline 2& 2^2=4 & 1/3^(2)=1/9\\ \hline 3& 2^3=8 & 1/3^(3)=1/27\\ \hline n & 2^n& 1/3^(-n)\\ \hline \end{tabular} \end{center} As this table shows, the length of the lines tends to approach zero, as we increase the amount of steps we apply. Therefore this is why the Cantor Fractal is also known as the Cantor point-set. Another way of drawing the Cantor set to make it more clear, is to use horizontal bars rather than just straight lines. By repeating the steps of removing the middle third, we get a picture that looks more like a comb [Lauwerier 2010]. In 1977, when Benoit Mandelbrot was exploring the concept of what a fractal was, he also discovered that fractals have dimension. He explained that fractals have capacity and it can be defined by a formula. “The concept of fractal dimension provides a way to measure how rough fractal curves are. The more jagged and irregular a curve is, the higher ts fractal dimension, a value between one and two. Fractional dimension is related to self-similarity in that the easiest way to create a figure that has fractional dimension is through self-similarity” [Laubender 1999]. Before explaining what the formula depends on, we need to be sure that when we are zoomed into a fractal image, the boundary to what we are zoomed in at must match that of the entire fractal. After that is confirmed, the number of pieces or line segments that fit into the larger one must be defined. We will call this number n. We also need to know the scale or how many times we have magnified into the entire fractal itself, which we will call M. We can then define the fractal dimension to be: \beginequation\begin{align}D= (log(n)) /logM/nonumber//\end{align}\endequation [Shapiro 2010] Using this equation, we can apply it to the fractal that was just discussed, the Koch curve. First we can look at the dimension of the Koch curve after doing the process just one time. So, there is one line fragment that is divided up into four with an equilateral triangle in the middle. Now we have four line segments and we can say $$n=4$$. Because these four pieces are 1/3 the length of the original line segment, then the scale or magnification is 3. So using the definition for fractal dimension, we have $$D=log4/log3$$, which gives us D= 1.26185… Because this number has a dimension greater than 1, and is an integer, then we can conclude that the Koch curve is indeed a fractal.

Gaston Julia was a French mathematician who was interested in looking the behavior of the orbit of a complex number when it is iterated under a function. This means that for a function, namely f, we apply it to a complex number. Whatever the result is, apply the same function to the new value. Julia repeated this action over and over again to see how the results acted. He came up with the idea of the prisoner set and the escape set by studying how iterating certain functions will give bounded or unbounded sets. A prisoner set refers to all the complex numbers in its orbit is bounded and an escape set refers to the complex numbers that are unbounded in its orbit under a certain function. An example of a prisoner is the value $$z_0 = 2$$ under the function $$ f(z)=z^2-z+1. By iterating this function starting with $$(1+i)$$ we get: \beginequation\begin{align} z_0=1+\imath\nonumber\\ z_1=f(1+\imath)=(1+\imath)^2-(1+\imath)+1=\imath \nonumber \\ z_2=f(\imath)=\imath^2-\imath+1=-\imath\nonumber\\ z_3=f(-\imath)=(-\imath)^2-(-\imath)+1=\imath\nonumber\\ z_4=f(\imath)=-\imath\nonumber\\ \end{align}\endequation Because the outputs switch back and forth, we call the value $$z_0=1+\imath$$ a prisoner. If we were to pick a value to start with and iterate it under a function and the values were to get infinitely large or small, then we would have an escapee [Spitznagel 2000].

Understanding what prisoner and escape sets are gives a better understanding of what Julia sets are. “The Julia set is defined to be the boundary between the prisoner set and the escape set” [Spitznagel 2000]. The functions that Julia looked at were of the form $$f(z)=z^2 +c$$ where c represents a complex constant. For each different c that we use in this function we will get a different Julia set. We choose $$z_0$$ to start with whatever we choose for c. It has been shown that if the orbit of the starting value, $$z_0$$, is ever located outside of the circle which has radius 2, then the rest of the orbit will be unbounded and therefore an escapee [Spitznagel 2000]. After choosing a value of c that we want to use in our function we can generate the Julia set on computer. We use whatever complex number we have chosen as the starting value of $$z_0$$. For each pixel on the computer screen, the color black is assigned if the value is bounded or a prisoner and is colored white if it is not bounded and tends to infinity [Bourke 2001]. This is where all these colorful art images that we see comes from. Many use different color schemes, instead of black and white, to get these vibrant images.

Another famous set that in a way relates to the Julia set was discovered by Benoit Mandelbrot and is called the Mandelbrot set. Mandelbrot also used the same exact function that Julia used, $$f(z)=z^2 +c$$. Mandelbrot wanted to find the values of c that made the set bounded and similarly the values of c that made the orbit unbounded when this function was iterated always starting with $$z=0$$. However, what was drastic difference between what Gaston Julia could do with his data and what Mandelbrot could do with his. The time that Mandelbrot was studying fractals was at a much later date than when Julia was, at a time when computers were in existence. As a result, Mandelbrot could do hundreds of thousands of iterations of a function using a computer, where Julia could only compute a small number of them by hand. After Mandelbrot gathered his Julia sets, he plotted them on a graph, which results in the Mandelbrot set [Jersey and Shwarz 2008]. Following is a set of steps, or an algorithm, for developing a Mandelbrot set by computer: \begin{enumerate} \item Choose a part of the complex plane and divide it up into a grid of c values. \item Define n to be how many points of each orbit it will take you to decide whether it is bounded or unbounded. \item Use the function $$f(z)=z^2+c to iterate the first n points of the orbit starting with the value, 0. \item If the orbit is unbounded, then color the corresponding c value on the grid a certain color.\item If the orbit is bounded, then color the corresponding c value on the grid a different color. \item Move on to the next c value and keep repeating until all c values are accounted for\end{enumerate} There are many algorithms for creating your own fractals on computer and many websites that can show the properties of self-similarity in the Mandelbrot sets, and in creating original fractals by just choosing certain complex numbers.

There are many advantages in knowing and understanding what a fractal is, no matter what subject area you are working in. Fractals have been observed in just about every living thing in nature from trees in the rainforest to our human bodies. But fractals have also helped us propel forward immensely in the field of technology and entertainment. Fractals, or more specifically the Koch snowflake, were used to make the antennas of our cell phones smaller while increasing the amount of frequencies they can receive [Jersey and Shwarz 2008]. Fractals are also used vastly in the medical field. One such example is that the healthy heartbeat when recorded on paper has a fractal pattern. This has helped doctors anticipate people who might have heart problems in the future [Jersey and Shwarz 2008]. One other example of where fractals are used is in computer science to help increase the believability of special effects and graphics in today’s movies [Jersey and Shwarz 2008].

One specific application of fractals in special effects in the movies involves the movie, Star Wars: Episode III. The part of the movie is when the two heroes run onto the end of a giant mechanical platform and a huge substance of lava comes crashing down in front of them. The initial process that they used to produce this lava was to make it appear that the lava is being shot up from a jet down below the mechanical platform. At first the graphics of the lava looked extremely unrealistic, and emerged as just a straight cylinder of lava flowing up through the air. The creators wanted this to look more realistic, so they took the idea of the fractal and applied it to this cylinder spiral shape of the lava. They took the original shape, shrunk it down and reapplied it. They repeated this over and over again to get a extremely realistic huge ball of fire and lava \cite{ Jersey and Shwarz2008}. Many may not know or understand how movies that are solely computer generated suddenly started. The first movie to have a complete computer generated sequence was Star Trek II: The Wrath of Khan\cite{ Jersey and Shwarz2008}. This movie was created in 1982, less than ten years after Mandelbrot made the idea of fractals public. Fractals is what makes computer generated imagery possible, and without Mandelbrot’s discoveries, we would not have the amazing graphics in our movies and games today.

The grand discovery that Benoit Mandelbrot had made about fractals has changed the way we make movies, video games, computer games, etc. Fractals has made it possible to make real life images through a computer. Loren Carpenter, who is now works at Pixar animation studios, is known for his computer science work taking fractals and using them to create computer generated images. He used to work for Boeing aircraft and his job was to see how the planes they were idealizing and creating looked while in flight. He wanted to make mountains to put in the background to make it look as realistic as possible. So after reading Fractals: Form, Chance, and Dimension by Benoit Mandelbrot, he learned that every surface can be broken down into smaller simpler shapes. He was then able to create images triangles into four smaller ones and kept on iterating until he had the jagged form of mountains [Jersey and Shwarz 2008].

In conclusion, fractals are geometrical figures that have identical repeating patterns on a scale that reduces infinitely. At first, when looking at the colorful picture of a fractal, one might think that it is just a creative piece of artwork. However after studying the mathematical background behind them, they have so much more depth than being just a piece of art. Benoit Mandelbrot is considered to be the father of fractal geometry and coined the term, “fractal.” He showed that every living and non-living thing in this world can be broken down into mathematical terms using fractal geometry. His first experiment of trying to calculate the length of Britain’s coastline led to this theory. The two main principles that define a fractal are self-similarity and fractal dimension. Self-similarity simply means that a pattern looks the same no matter how much it magnified. Fractal dimension is important because it shows that fractals have capacity and that they are not just flat images. Along with Benoit Mandelbrot, WacLaw Sierpinski ,Gaston Julia, Helge von Koch Greg Cantor are all very important mathematicians in the history of fractal geometry. All of these mathematicians have their own forms or types of fractals that they have discovered. One of the most popular is the Mandelbrot set, which describes the iteration of a function using complex numbers. Even though the subject of fractal geometry is purely math, many people have found ways to take it outside math into computer science, health science, and even fashion and the arts. Fractals are used to make our movies look more amazing than life, and to make our computer and video games feel like we are right there in them. The discovery of fractals have allowed us to decrease the size of our cell phones every year and at the same time they have helped doctors anticipate heart problems in our bodies way before they happen. Without the discovery of fractals, our technology, entertainment, our health, etc. would not be where it is today.

References Bourke, P. Julia Set Fractal (2D)(2001). http://local.wasp.uwa.edu.au/~pbourke/fractals/juliaset/

Hoffman,J. Benoit Mandelbrot, Novel Mathematician, Dies at 85. The New York Times (2010). http://www.nytimes.com/2010/10/17/us/17mandelbrot.html?_r=1

Jersey,B.,And Shwarz,M.Hunting the Hidden Dimension.(2008). http://www.pbs.org/wgbh/nova/fractals/

Laubender, P. "What is a Fractal?" Fractaline (1999). http://www.peter-laubender.de/fractaline/what_is_a_fractal.htm

Lauwerier, H.Fractals Endlessly Repeated Geometrical Figures.Princeton, New Jersey. pgs 13,15-16, 32-33,(2010).

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Spitznagel,C.R "Julia Sets" Mathematical Vignettes (2000). http://www.jcu.edu/math/vignettes/Julia.htm