Aperiodic Tiling - Physics of Aperiodic Tilings

Physics of Aperiodic Tilings

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 – so it had to be a crystalline substance with icosahedral symmetry. In 1975 Robert Ammann had already extended the Penrose construction to a three dimensional icosahedral equivalent. In such cases the term 'tiling' is taken to mean 'filling the space'. Photonic devices are currently built as aperiodical sequences of different layers, being thus aperiodic in one direction and periodic in the other two. Quasicrystal structures of Cd-Te appear to consist of atomic layers in which the atoms are arranged in a planar aperiodic pattern. Sometimes an energetical minimum or a maximum of entropy occur for such aperiodic structures. 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. Faraday waves have been observed to form large patches of aperiodic patterns. The physics of this discovery has revived the interest in incommensurate structures and frequencies suggesting to link aperiodic tilings with interference phenomena.