Fractal Tiling Investigations

Because of their self-similar nature, fractals can easily be used to generate tilings in the plane. In this paper, we use techniques from the article “Fractal Tilings in the Plane” by Darst, Palagallo and Price to investigate various fractal tilings. We consider both examples from the article and our own variations in order to analyze the properties of fractal tilings. We first look at basic tiling configurations before investigating tilings that have radial symmetry and tilings that use a change of basis matrix to create new configurations. All tilings are generated using an iterated function system and the lattice points determined by the column vectors of a 2 by 2 matrix M. In addition, we give a proof of Pick’s Theorem to show the relationship between the number of lattice points (which form the complete residue set of M) contained within a fundamental parallelogram determined by the two column vectors of M