Fractal Geometry and Non-Integer Dimensions

Matt McClinton
PhD Graduate Student
University of Wisconsin-Milwaukee

Popularized in the 1980s, fractals have become something of a household name. These fractal sets often demonstrate peculiar topological properties. One such property is the notion of a fractal dimension. Sets such as the Cantor set, Sierpinski Gasket (SG), and the von Koch curve are traditionally visualized in 2D images. However, these sets actually exist in-between dimensions 1 and 2!

Certain fractals can be built using what is known as an Iterated Function System (IFS), and there is a powerful theorem stating that having an IFS representation of a fractal provides a simple means of determining the fractal dimension. I will begin by stating the IFS that generates the Sierpinski Gasket. There are two transformations on the Gasket to which creates the Level-n Stretched Sierpinski Gasket (SSG^n). I will demonstrate how one constructs the IFS for SSG^n, as well as provide the highlights to a theorem in which I prove the fractal dimension of SSG^n.