Aperiodic tiling
Form of plane tiling without repeats at scale / From Wikipedia, the free encyclopedia
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An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings.
The Penrose tilings are a well-known example of aperiodic tilings.[1][2]
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.[3] In May 2023 the same authors published a chiral aperiodic monotile with similar but stronger constraints.[4]
Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman[5] who subsequently won the Nobel prize in 2011.[6] However, the specific local structure of these materials is still poorly understood.
Several methods for constructing aperiodic tilings are known.