Group Members: Mark Chambliss, Jason Stewart, and Jason Basri
Fractals are relatively new observances to the math world. It was not until 1985 that Michael Barnsley developed the first practical way for generating fractals. Fractals however have been around since before man. They are seen in biology and nature ranging from microorganisms to mountain ranges.
Fractals in its basic form is an object made up of smaller copies of itself. In a more mathematical sense, a fractal is a subset of a Euclidean space whose Hausdorff dimension and topological dimension are not equal (Klang). We as linear algebra students were shown linear transformations and a few applications of them. We would like to take what we have learned and apply it to our own project of creating fractals. With the help of Matlab, this will be possible.
For our application project we will look into the problem “How can computers be used to create fractal images?”. As stated earlier, fractals are images created by performing repetitive linear transformations on points and images. Computers are very adept at performing repetitive tasks and thus will be well suited for performing the necessary repetitive operations for creating fractal images. To achieve our desired goal, we will study algorithms that have been developed by others in the past to create fractal images and determine the differences and benefits of each approach. Lastly, we will reflect back upon how the fractal image algorithms can be used in modern applications. An example of a simple fractal image that was created with matlab is displayed below (Seiler):
Asking a more basic question is “what processes are used by various math based software to create complex fractal images?” To help understand how fractals are used to create an image like this, we will use the basic fractal of the sierpinksi carpet. This fractal takes a 2x2 matrix and turns it into a carpet like image comprised of a pattern made up of many small squares. This is done through linear transformations called similitudes, which take that original 2x2 matrix, scale it down, and transform it to another area of the graph. A process of n similar similitudes, typically 8 in the case of the sierpinski carpet, continue with this resizing and translation of the matrix until many “squares” appear on the graph. These similitudes, many of them quite complex, form the basis of many fractals, which can be comprised of thousands of similitudes. Because of this, to form many of the more advanced fractals, such as the one seen above, the Monte Carlo approach is used, which involves performing many similitudes upon many randomly selected coordinates in the graph. This is where the fractal mapping software becomes ideal, as it can perform these various complex functions (McKeeman).
Klang, Jesse. An introduction to Fractals. http://home2.fvcc.edu/~dhicketh/LinearAlgebra/studentprojects/fall2004/jesseklang/Fractalsproject.htm.
McKeeman, Bill. Wild Fern. http://www.mathworks.com/matlabcentral/fileexchange/19141-wind-fern
Seiler, Ben. Fractal Generation. http://wso.williams.edu/~skaplan/fractal/chapter3.htm