| We investigate the topological, similarity and Hausdorff dimensions of self-similar fractals that are the invariant sets of iterated function systems. We start with the Contraction Mapping Theorem, which will give us a constructive method in which to find fractals using iterated function systems. We then define the Hausdorff metric in order to use the Contraction Mapping Theorem to prove that each iterated function system has a unique invariant set.;Next, we discuss three different types of dimensions: topological, similarity, and Hausdorff dimension. The main theorem of this thesis tells us that the similarity dimension equals the Hausdorff dimension if the iterated function system satisfies the Open Set Condition. This is important since the similarity dimension is much easier to compute than the Hausdorff dimension.;Finally, we apply the theory that we have developed to some famous examples. For each example we give its construction, discuss the three dimensions, and explain the strange properties each fractal possesses. We begin with the Cantor set. Then move on to the Sierpi'{n}ski gasket and Koch snowflake. Next, we discuss the Menger sponge. Last, we present an example, Barnsley's wreath, whose associated iterated function system does not satisfy the Open Set Condition. |
