Sean Flaherty, an incoming freshman at Texas Tech University, is currently pursuing a dual bachelor's degree in Mathematics and Computer Science and hopes to pursue graduate education in the future. His research interests are in areas where you might find something as unproven, or simply conjectured, but particularly in the fields of algebraic topology and non-Euclidean geometry. Sean attended the 2014 Mathematica Summer Camp at Bentley University, and is also a Texas Tech University PRISM scholar. Sean also enjoys playing the clarinet and practicing the little-known and esoteric art of pen spinning.
Project: Calculating Dimensions of Fractals through Image Processing
The coastline paradox, stating that a shape as complex as the coast of Great Britain cannot have a well-defined length, appears in both naturally occurring structures as well as mathematically generated ones. This phenomenon is a result of the increasing complexity of the image as one zooms in, causing intuitive definitions of dimension to break down. Using the box-counting method, the Minkowski–Bouligand dimensions of some fractals can be estimated, though finding the true fractal dimension of more complex fractals like the Mandelbrot set is less straightforward. If self-similar patterns can be recognized in an image, the precise Hausdorff dimension can be calculated easily, but computers struggle more than humans with this type of perception. The fractal dimension is approximated for any image as the logarithm of the rate at which the number of boxes to cover the image increases as the side length of the boxes decreases.