Hat tile
The first aperiodic monotile, an asymmetric 13-gon announced in March 2023.
Discovery
The Hat is an asymmetric polygon that admits tilings of the plane, but none that are periodic. It was the first shape proven to solve the einstein problem — tiling with a single prototile subject to standard monohedral definitions that allow reflected copies.[1]
Why reflections matter
Every known Hat tiling mixes unreflected and reflected tiles. Whether that counts as a true monotile sparked public debate; the authors and standard references (Grünbaum & Shephard) treat reflected congruent copies as the same tile shape.[1][3]
The subsequent Spectre tile answered the stricter question: a shape that tiles aperiodically without any reflected copies at all.
See also
Aperiodic monotile, Spectre tile
Categories: Mathematics · Concepts