Matching rules for quasicrystals

A tiling is a covering of the plane by interior disjoint compact called tiles. Matching rules are local constraints on the way tiles can locally fit together. Motivated by computability issues, the first tiles with matching rules which allow only non-periodic tilings of the plane have been found in 1964. Several other examples have been found in the following years (e.g., Penrose tiles), until the discovering of quasicrystals in 1982 boosted the interest in a complete characterization of the non-periodic tilings that are enforced by matching rules (in this context, matching rules model finite range energetic interaction). In this talk we shall review the results obtained in this direction in the 90's (notably in Moscow by L. Levitov, S. Burkov and T. Q. T. Le) and discuss recent improvements as well as some related issues.