Aperiodic Tiling with a Single Monotile: Hats, Turtles, and SAT Solvers

In 2023, David Smith, a retired printer technician, discovered a single shape, dubbed the "Hat," capable of aperiodically tiling the infinite plane, creating a sensation in mathematics. This blog post explores this discovery and delves into using SAT solvers—a relatively unknown family of algorithms—to solve the tiling problem for finite regions. It uses Sudoku as a warm-up example, then explains how SAT solvers find tilings for the Hat and a related shape, the "Turtle." The article culminates in the introduction of the "Spectre," a new monotile that alone aperiodically tiles the plane, solving a long-standing mathematical problem. An accompanying app lets readers experiment with these fascinating tiles and their tiling patterns.