The aperiodic Ein-Stein Tiling

The term Ein-Stein tiling has no connection to Albert Einstein. These tilings, which mathematicians searched for nearly 60 years without success, are named after the German phrase "Ein Stein" (meaning "one stone"), because they are intended to be formed from a single shape that completely covers the plane without overlaps or gaps, while never forming repeating patterns. Einstein tilings exhibit no symmetry whatsoever, not even translational symmetry.

In 2022, the printing technician David Smith from the United Kingdom discovered the so-called Hat tile with the required properties. However, its tiling also requires mirrored versions of the hat shape. After verification by the mathematical community, its aperiodicity was confirmed in 2023. Later that year, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss developed the Vampire tile (Spectre) and other shapes that do not require mirroring, making the term "Ein Stein" tiling even more fitting. The research team has since proven that there are infinitely many Ein-Stein tiles. The video provides a visual introduction to the topic.