Aperiodic tiling
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An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern. The best known example of aperiodic tiling are the Penrose tilings. Aperiodic tiling is also relevant in the formation of quasicrystals.
Mathematical treatment
Main articles: Wang tiles, Penrose tilings
The question of aperiodic tiling first arose in 1961, when logician Hao Wang tried to decide whether the tiling problem was computable: i.e. whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane. Wang was able to show that such an algorithm exists if it could be shown that every finite set of prototiles that admits a tiling of the plane also tiles it periodically.
Hence, when in 1966 Robert Berger demonstrated that the tiling problem is in fact not computable, it followed that there must exist an aperiodic set of prototiles. The first such set, presented by Berger, consisted of 20,426 Wang tiles. The first compact set was discovered by Raphael Robinson in 1971. Roger Penrose discovered three more sets in 1973 and 1974, and Robert Ammann discovered several new sets in 1977. It is still unknown, however, whether there exists an aperiodic set containing just one prototile.
Physical treatment
Main article: Quasicrystals
One notable property of the tiles discovered by Penrose is that they display a seemingly-impossible quasi-periodic crystallographic structure with fivefold symmetry. Only twofold, threefold, fourfold and sixfold rotational symmetries are possible for periodic crystalline patterns. However, for the Penrose tilings it is possible to find translational and fivefold symmetrical motions that come arbitrarily close to being symmetries.
In 1975 Robert Ammann extended this fivefold symmetry to a three dimensional icosahedral equivalent. Then in 1984, physicist Dan Shechtman announced the discovery of a phase of an aluminiummanganese alloy which seemed to be a crystalline substance with icosahedral symmetry. Such quasicrystalline substances still pose many questions, particularly regarding their method of assembly, which is necessarily non-local.