Aperiodic Tiling

An aperiodic monotile, humorously dubbed an ‘einstein’ (from the German term “einstein,” meaning “one stone” or “one tile”), is a single tile that covers a surface without repeating patterns. This posed a challenging question for some time: could such a tile exist, or was it impossible?

In 2023, David Smith and his team provided an answer. They discovered a simple tile called a “hat” that can achieve this aperiodic tiling. The geometric shape of the “hat” tile is based on the symmetry and edges of a hexagon, as shown in the picture. According to Smith, this tile, along with its reflection (shown in blue), enables an array of unique, non-repeating tile arrangements. The “hat” falls within the broader category of Smith–Myers–Kaplan–Goodman-Strauss tiles.