Aperiodic tiling

[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.

[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.

In order to rule out such boring examples, one defines an aperiodic tiling to be one that does not contain arbitrarily large periodic parts.

[7] The first specific occurrence of aperiodic tilings arose in 1961, when logician Hao Wang tried to determine whether the domino problem is decidable – that is, whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane.

In 1964, Robert Berger found an aperiodic set of prototiles from which he demonstrated that the tiling problem is in fact not decidable.

[12] The number of tiles required was reduced to one in 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss.

After the discovery of quasicrystals aperiodic tilings become studied intensively by physicists and mathematicians.

de Bruijn for Penrose tilings eventually turned out to be an instance of the theory of Meyer sets.

[7] An einstein (German: ein Stein, one stone) is an aperiodic tiling that uses only a single shape.

[17][18] The tilings which have been found so far are mostly constructed in a few ways, primarily by forcing some sort of non-periodic hierarchical structure.

[17] For a tiling congruent copies of the prototiles need to pave all of the Euclidean plane without overlaps (except at boundaries) and without leaving uncovered pieces.

However, the tiling produced in this way is not unique, not even up to isometries of the Euclidean group, e.g. translations and rotations.

A complete tiling of the plane constructed from Robinsion's tiles may or may not have faults (also called corridors) going off to infinity in up to four arms and there are additional choices that allow for the encoding of infinite words from Σω for an alphabet Σ of up to four letters.

However, the tiles shown below force the chair substitution structure to emerge, and so are themselves aperiodic.

Joshua Socolar,[22][23] Roger Penrose,[24] Ludwig Danzer,[25] and Chaim Goodman-Strauss[20] have found several subsequent sets.

Shahar Mozes gave the first general construction, showing that every product of one-dimensional substitution systems can be enforced by matching rules.

[18] Charles Radin found rules enforcing the Conway-pinwheel substitution tiling system.

[26] In 1998, Goodman-Strauss showed that local matching rules can be found to force any substitution tiling structure, subject to some mild conditions.

The Penrose tiles are the first and most famous example of this, as first noted in the pioneering work of de Bruijn.

[27] There is yet no complete (algebraic) characterization of cut and project tilings that can be enforced by matching rules, although numerous necessary or sufficient conditions are known.

Notably, Jarkko Kari gave an aperiodic set of Wang tiles based on multiplications by 2 or 2/3 of real numbers encoded by lines of tiles (the encoding is related to Sturmian sequences made as the differences of consecutive elements of Beatty sequences), with the aperiodicity mainly relying on the fact that 2n/3m is never equal to 1 for any positive integers n and m.[29] This method was later adapted by Goodman-Strauss to give a strongly aperiodic set of tiles in the hyperbolic plane.

[31] Block and Weinberger used homological methods to construct aperiodic sets of tiles for all non-amenable manifolds.

[32] Joshua Socolar also gave another way to enforce aperiodicity, in terms of alternating condition.

Aperiodic tilings were considered as mathematical artefacts until 1984, when physicist Dan Shechtman announced the discovery of a phase of an aluminium-manganese alloy which produced a sharp diffractogram with an unambiguous fivefold symmetry[5] – so it had to be a crystalline substance with icosahedral symmetry.

Steinhardt has shown that Gummelt's overlapping decagons allow the application of an extremal principle and thus provide the link between the mathematics of aperiodic tiling and the structure of quasicrystals.

[35] The physics of this discovery has revived the interest in incommensurate structures and frequencies suggesting to link aperiodic tilings with interference phenomena.

Sometimes the term described – implicitly or explicitly – a tiling generated by an aperiodic set of prototiles.