Super-coloured tilings: A novel class of two-dimensional limit-periodic structures

A two-dimensional (2D) periodic coloured tiling has a colour symmetry if it is invariant against a symmetry operation involving the change of colours. The idea of colour symmetry may also be adopted for another type of symmetry, namely the inflation symmetry for self-similar structures. A general method for constructing substitution rules that generate a class of 2D aperiodic coloured tilings is presented, where we assume that there are only two colours and a single shape for the tiles. These structures are limit-periodic, and we will call them super-coloured tilings (SCTs). SCTs form an important class of aperiodic ordered structures with a perfect long-range order. Since their aperiodicity relies on the colours of the tiles, an SCT reduces to a periodic tiling if the colours of the tiles are disregarded.