On the Hausdorff Dimension of the Boundary of a Self-Similar Tile

In our everyday experiences, we have developed a concept of dimension, neatly expressed as integers, i.e. a point, line, square and cube as 0-, 1-, 2-, and 3-dimensional, respectively. Less intuitive are dimensions of sets such as the Koch Curve and Cantor Set. The formal definition of toplogical dimension in a metric space conforms to our intuitive concept of dimension, but it is inadequate to describe the dimension of fractals. The purpose of this thesis is to develop notions of fractal dimension and in particular, to explore Hausdorff dimension with regard to self-similar tiles in detail. Methods of calculation of Hausdorff dimension will be discussed