In the 1970's, Roger Penrose found a set of tiles that cover the plane in beautiful and remarkable ways. Perhaps even more surprising was the realization a few years later that these tilings could explain new phenomena in crystallography. This talk will describe Penrose tilings and explain the fundamental idea responsible for their behavior as well as a method for constructing them. Don't be left out on the tiles!

In the latter nineteenth century, geometry underwent a radical shift from a study of shapes to a study of the transformations that change one shape into another, or that leave a shape unchanged. For particular shapes, these transformations can have a delicate relationship with one another, and just the abstract enumeration of all the possible combinations of the transformations can be difficult to accomplish in both theoretical and practical senses. We will discuss some of the modern theory of groups of transformations and their practical enumeration by finite state automata, and the beautiful computer graphic images that come from these groups.