Honors Capstone Project
Date of Submission
Arts and Science
Capstone Prize Winner
Won Capstone Funding
Sciences and Engineering
Algebra | Algebraic Geometry | Applied Mathematics | Other Applied Mathematics
For my capstone project, I analyzed fractals. A fractal is a picture that is composed of smaller images of the larger picture. Each smaller picture is self- similar, meaning that each of these smaller pictures is actually the larger image just contracted in size through the use of the Contraction Mapping Theorem and shifted using linear and affine transformations.
Fractals live in something called a metric space. A metric space, denoted (X, d), is a space along with a distance formula used to measure the distance between elements in the space. When producing fractals we are only concerned with metric spaces in R2, which is the collection of all ordered pairs. Also, we only use infinite sequences, in specific Cauchy sequences, which are sequences in which the distance between elements decreases as the sequence progresses.
Fractals live in a special metric space called (H (X), h). Every element in (H (X), h) is actually a compact set. A compact set is closed, bounded, and every infinite sequence has a subsequence that has a limit in the space. A space is closed if it contains all of its points and limit points. A space is bounded if you can draw a circle around the space enclosing all of its elements. When producing fractals, the Hausdorff distance is used to measure the distances between compact sets in (H (X), h), which is the maximum distance between two compact sets in the space.
Fractals are produced using contraction mappings. This means that fractal pictures are contracted using a contractivity factor, between 0 and 1 that contracts the size of the image. The Contraction Mapping Theorem states that all contraction mappings or iterated function systems, which are sets of contraction mappings, have one fixed point to which the sequence of points in the fractal converge.
My final fractal is a picture of my initials, KMM. To make this picture, I used linear and affine transformations, affine transformations being linear transformations plus a shift. I drew my initials in a square. Using the Contraction Mapping Theorem, I then contracted the square to the origin, and then used shifts to move the picture to the twelve different sections to which each part belonged. I calculated the twelve different contraction mappings then used the program written by Professor Banerjee for Maple to apply The Random Iteration Algorithm. This algorithm starts with any point then randomly chooses one of the twelve contraction mappings, applies it to the point and finds a new point in the picture. It repeats this process infinite times, and the sequence of all points found converge to the fixed point, producing my final picture.
Mesznik, Kara, "Analyzing Fractals" (2010). Syracuse University Honors Program Capstone Projects. 353.
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