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

Abstract

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.

Acknowledgements

The authors thank Dr. L. Lück and Dr. T. Ogawa for their helpful comments.

Notes

^†Since the tiles take several orientations for a non-Bravais case, two tiles with the same colour but with different orientations can be distinguished. Accordingly the inflation matrix can be expanded; it is 4 × 4 and 6 × 6 for the last two of the above five examples. This has, however, no impact on the present conclusions.

Note

During Aperiodic'06, Dr. L. Lück pointed out a similarity between the first SCT based on the square lattice (figure 2) and a certain pattern presented by Dr. T. Ogawa in his book in Japanese. After the conference, we have identified the book which is a monograph on fractal geometry 12 but has been overlooked by the present authors. The book includes two limit periodic structures, so called the Kelp-Peano curve and the Hexagonal-Peano curve, discovered by Dr. Ogawa on the basis of a heuristic method. Scrutinizing them, we have found that the former of the two traces exactly the connectivity of the black tiles in our tetragonal SCT. On the other hand, the latter has no relationship with any hexagonal SCT in the present paper. However, we have found that the curve is included in our systematic formalism by slightly extending it.