Aperiodic tiling

In the mathematics of tessellations, a non-periodic tiling is a tiling that does not have any translational symmetry. An aperiodic set of prototiles is a set of tile-types that can tile, but only non-periodically. The tilings produced by one of these sets of prototiles may be called aperiodic tilings.

The Penrose tilings are a well-known example of aperiodic tilings.

In March 2023, four researchers, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, announced the proof that the tile discovered by David Smith is an aperiodic monotile, i.e., a solution to the einstein problem, a problem that seeks the existence of any single shape aperiodic tile. In May 2023 the same authors published a chiral aperiodic monotile with similar but stronger constraints.

Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood.

Several methods for constructing aperiodic tilings are known.