When one considers “tiling,” one may find it more appropriate to consult a contractor rather than a mathematics textbook. This concept, though, is important, both in geometry and in one’s restroom. In mathematics, a series of plane figures which contain no space between them and which do not overlap each other is called a tiling, and the plane figures are called tiles. There is certainly an analogue here to tiling a floor, wall, or other space. The goal is to cover the entirety of the space, ideally without leaving gaps. In real life, of course, this might not be an absolutely perfect process; a “tiling” in one’s house may have caulk or other binding substances between the tiles, as shown below.
A tiling is periodic if it has a pattern which repeats itself regularly; otherwise it is called aperiodic. For a natural example of a true periodic tiling, consider honeycomb. The individual cells within the saccharine structure fit together perfectly, and the repeating shape of a hexagon continues within.
A Penrose tiling, named after Roger Penrose, is a tiling that has the remarkable property of always being aperiodic regardless of the configuration of the tiles, a far cry from what we often think of when we think of tilings. In this project, we will explore the history of these tilings, how they can be generated, some of the specific shapes which create them and how they relate to the golden rule, and the connection of Penrose tilings to an extraordinary discovery in physics, quasicrystals.
