In 2023, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss discovered a single tile that can tessellate the plane in a non-periodic manner. This shape was named “The Hat” and is considered the first known einstein—a single shape that, by itself (without additional shapes), forces a non-periodic tiling.

A non-periodic tiling is a gapless, overlap-free covering of the plane in which the pattern never repeats through translation—unlike periodic tilings, which are based on motifs that recur regularly. Although “The Hat” fills the plane completely, it does not form a regular grid.

Its tiling uses both the front and back sides of the tile. Geometrically, the asymmetric piece can be constructed from 8 kites or, alternatively, from 4 monogems.

The puzzle shape of the Hat Puzzle Box is made from six Hat tiles and can be assembled with relative ease.

For creating more complex and varied shapes, however, additional tiles are required. Therefore, the download section also includes a storage box for a total of 30 Hat tiles (5 × 6). With this larger set, you can build bigger and significantly more interesting shapes.

The puzzle shape of the slider box exhibits threefold rotational symmetry and includes three reflection axes. Another illustrated shape even achieves sixfold rotational symmetry.

Beyond that, among the symmetric shapes, only shapes with threefold rotational symmetry are known.

The 3D-print templates consist of the Hat puzzle piece and two boxes.