The hat tile (solution of the Einstein problem)

The Einstein problem is a mathematical problem from discrete geometry. A geometric shape is sought that can be joined together in rotated, shifted and/or mirrored copies without overlapping, so that the entire plane is covered (parquetized) without gaps. In addition, this form should have the property that no parquetry with a periodically recurring pattern - as on wallpaper - can be created. The problem was considered unsolved for decades; two solutions were proposed for the first time in 2023. The name of the problem is a humorous reference to Albert Einstein and states that only one shape ("one stone") should parquet the plane.