Aperiodic Monotiles: New Shapes Just Dropped

A longstanding unsolved problem in geometry asks whether it is possible for a single shape to be aperiodic: to tile the plane without ever permitting translational symmetries. We recently proved that a shape called the “hat” solves this problem, making it the first known aperiodic monotile. In this talk I will introduce some background concepts from tiling theory and summarize the history of the search for aperiodic shapes. I will then relate the story of our discovery of the hat and the proof of its aperiodicity. Along the way I will also talk about two other closely related shapes, the “turtle” and the “spectre”, which allow us to derive additional results about aperiodic tilings.