Some self-similar constructions in two and three dimensions and their neighbor geometry

Self-similar sets are a class of fractals which can be rigorously defined and treated by mathematical methods. Their theory has been developed in n-dimensional space, but we have just a few good examples of self-similar sets in three-dimensional space. This thesis has two different aims. First, to extend fractal constructions from two-dimensional space to three-dimensional space. Second, to study some of the properties of these fractals such as finite type, disk-likeness, ball-likeness, and the Hausdorff dimension of boundaries. We will use the neighbor graph tool for creating new fractals, and studying their properties.In dieser Arbeit werden verschiedene selbstähnliche Konstruktionen von zwei auf drei Dimensionen übertragen, und ihre Eigenschaften werden mittels Nachbar-Abbildungen untersucht