The Butterfly Effect of Fractals

This presentation applies concepts in fractal geometry to the relatively new field of mathematics known as chaos theory, with emphasis on the underlying foundation of the field: the butterfly effect. We begin by reviewing concepts useful for an introduction to chaos theory by defining terms such as fractals, transformations, affine transformations, and contraction mappings, as well as proving and demonstrating the contraction mapping theorem. We also show that each fractal produced by the contraction mapping theorem is unique in its fractal dimension, another term we define. We then show and demonstrate iterated function systems and take a closer look at chaotic and dynamical systems, defining terms including terms transitivity, sensitivity to initial conditions, and density of spaces, as well as presenting examples that clearly show these concepts. We go on to define the butterfly effect, specifying that the initial condition is of the upmost importance, as well as introduce attractors, including strange attractors and the Lorenz system (as well as its fractal nature), as well as show applications of these concepts to disciplines such as art, science, and philosophy. We close by reaffirming the fractal nature of the universe, reassuring the reader that chaos is not simply the absence of order, but rather possesses unpredictable qualities which we can observe and ponder