The Fun Filled Fractal Phenomenon A fractal is a type of geometric figure. It is generated by starting with a very simple pattern such as a triangle and, through the application of many repeated rules, adding to the figure to make it more complicated.
Often, an input will be entered into a recursive function and it will yield an output. This output is then inserted back into the function as an input and the process is repeated infinitely. Fractals often exhibit self-similarity. This means that each small section of the fractal can be viewed as a reduced-scale replica of the whole. Some famous fractals include Sierpinski’s triangle, Koch’s snowflake and the length of a coastline. Fractals were brought to the public’s attention by the work of French mathematician Benoit B.
Mandelbrot in the 1970’s. Mandelbrot discovered how to calculate fractal dimensions. The formula for fractal dimension is N=2D where N equals the number of copies of the original figure, which is calculated by doubling its size and D is the dimension. Mandelbrot named his creations fractals because each part is a fraction of the whole figure. The Chaos Theory describes the complex and unpredictable motion of systems that are sensitive to their initial conditions.
Chaotic systems follow precise laws but their irregular behavior can appear to be random to the casual observer. For example, weather is a chaotic system. If the rays of the sun bounce off the hood of a car in a certain way, causing a breeze, the breeze could blow a leave off a tree, which starts a series of additional events that could alter the weather in some other part of the world. Chaos can be related to fractals. In a fractal if one tiny change occurs in a repeated pattern, the entire fractal will change. The above picture is an example of a strange attractor that charts the trajectory of a system in chaotic motion.
It is a fractal. The fractal exhibiting chaos is predictably unpredictable. This is because, in a chaotic system, it is predictable that there will be minute changes that will alter the entire shape. Koch’s snowflake, (above ) exhibits the concept of an infinite perimeter with a finite area.
Koch’s snowflake is created by dividing each of the sides of an equilateral triangle into three equal parts. Next, the center part of each side is taken out and replaced with two sides of equal length to that of the original centerpiece. This pattern is repeated infinitely. Each time the process is completed the perimeter gradually increases to infinity by increments of 4/3. However, the area of this snowflake is finite.
If you draw a circle enclosing the original triangle that contains the vertices of the triangle, the area of the snowflake will never exceed the area of that circle no matter how many times its perimeter increases. Therefore, it has a finite area. Fractals exhibit self-similarity. This is the concept that each small portion of the fractal can be viewed as a reduced-scale replica of the whole. For example, in Sierpinski’s Triangle, each small triangle inside is similar to the large one on the outside.
A real life example of self-similarity is a tree. The tree has a trunk on which limbs grow. Branches grow from the limbs, and twigs grow from the branches, which is followed by sticks on the twigs and so on. The sticks growing on the twigs are just a smaller version of the twigs growing on the branches, which are a smaller version of the branches growing on the limbs, which are a smaller version of the limbs growing on the trees. Another example is a universe, which is composed of a collection of spinning galaxies, which are composed of a collection of spinning solar systems which is a collection of spinning plants and so on.
Each step is self-similar to the universe. Finally, a cloud exhibits self-similarity. A cumulus cloud is a collection of smaller puffs, which, in turn, are a compilation of smaller puffs and so on. Each puff is a smaller replica of the large puff. Fractals are often formed by an iterative process. That means that an operation is preformed on one figure to create a new figure.
Then this operation is performed on the new figure to make another figure and so on. Each step of this process is called an iteration. An illustration of this is the diagram of Koch’s snowflake on page two. It begins with a triangle Then an operation is performed on it and it becomes the Star of David. As the operation is repeated infinitely on the figure, it becomes an increasingly complex snowflake.
Once a fractal, such as Sierpinski’s Triangle, is created it is crucial to find out its dimension. The dimension of this fractal is greater than a line and less than a plane, so it is between 1 and 2. To find the exact dimension, one needs to follow a simple formula: The dimension (d) of a shape is the log of the number of copies (n) that are produced when the figures sides are doubled, divided by the log of 2 (logn / log2 =d or n=2d). The dimension of Sierpinski’s Triangle would be the Log of 3, because you get three copies of the triangle when you double its sides, divided by the Log of 2. The final dimension is 1.58496250072115618145373894394782.
The human body is composed of many fractals. From the moment of fertilization, the cells of the egg and the sperm break up into two more cells, which, in turn, break up into two additional cells and so on. Each cell is self-similar to the entire collection of cells. This compilation exhibits the chaos theory.
If one link in this collection is incorrect or missing, the entire organism can be ruined. It will collapse on itself creating a sickle cell. Some African’s have a disease called sickle cell anemia in which their blood cells have one bad amino acid chain in a protein of many hundred amino acids. These sickle cells clot and create a lot of pain for the person afflicted with this disease.
A body as a whole is a fractal. It is a group of dissimilar systems working together, which are composed of groups of dissimilar organs working together, which, in turn, are composed of groups of dissimilar tissues working together, which is a group of dissimilar cells working together, which is a group of dissimilar organelles working together. The body begins with the creation of cell organelles that are formed together to make a cell. These cells, as stated above, duplicate to form tissues, which duplicate to form organs and so on until a human body is conceived. Fractal research can be used to predict how complicated organ systems in the body will respond to changes. This is important for understanding how to treat diseases.
Bibliography “Chaos,” Encarta Encyclopedia, 2000. “Choas Theory,” Encarta Encyclopedia, 2000. “Fractals,” Encarta Encyclopedia, 2000. “Fractals: An introduction” Available. (online) http://www.planetclick.com/ratebar.mpl?siteID=1000000000024998. Lampton, Christopher, Science of Chaos (New York: Franklin Watts, 1992) 9-16.
Lanius, Cynthia, “Fractals” Available. (online) http://math.rice.edu/lanius/frac. Laplante, Phil, Fractal Mania (New York: Windcrest/McGraw-Hill, 1994) 1-22.