An aperiodic tiling is a nonperiodic tiling in which arbitrarily large periodic patches do not occur. Thus, every aperiodic tiling
is a nonperiodic tiling, but the converse need
not hold. Nonperiodicity alone excludes a nonzero translation
that leaves a particular tiling invariant. It does not rule out arbitrarily large
periodic patches or prevent the same tiles from forming a periodic tiling. A set
of tiles is said to be aperiodic if they can form only nonperiodic
tilings. The most widely known examples of aperiodic tilings are those formed
by Penrose tiles.
The Federation Square buildings in Melbourne, Australia feature an aperiodic pinwheel tiling attributed to Charles Radin. The tiling is illustrated above in a pair of photographs by P. Bourke.
The longstanding open problem of finding an aperiodic monotile was solved by Smith et al. (2023) with the discovery of the hat polykite.
