People have an instinctive understanding of what it means for an object to be three-dimensional: it has height, width, and length. There is similar familiarity with two-dimensional shapes like squares and triangles, as well as one-dimensional shapes such as lines. The dimensions of some objects, on the other hand, cannot be understood in this familiar way. Take, for example, a set of points in the xy-plane that do not form a straight line or any familiar shape from geometry. One way to measure the dimension of these objects is to calculate the Hausdorff dimension of the set of underlying points. Intuitively, the Hausdorff dimension is a measure of how many small boxes it would take to cover an object. Our research aimed to calculate the Hausdorff dimensions of more complicated curves, such as those arising from Brownian motion, which is a probabilistic model for a randomly moving particle. In particular, we set out to corroborate known results about the Hausdorff dimensions of Brownian motion and the Brownian frontier. Further, we wanted to strengthen unsolved conjectures regarding the Hausdorff dimension of the Brownian earthworm model, which is a model of mass redistribution on the d-dimensional integer lattice introduced by Professor Krzysztof Burdzy. We attempted to develop answers to these problems by simulating Brownian motion in two dimensions and performing statistical analysis on our results. Our model correctly calculated the known dimensions and was in line with Professor Burdzy’s previous conjecture regarding the Brownian earthworm. It is our goal that this research will ultimately contribute to a greater understanding of the earthworm model.