Colourings of lattices and coincidence site lattices

Abstract

The relationship between the coincidence indices of a lattice Γ₁ and a sublattice Γ₂ of Γ₁ is examined via the colouring of Γ₁ that is obtained by assigning a unique colour to each coset of Γ₂. In addition, the idea of colour symmetry, originally defined for symmetries of lattices, is extended to coincidence isometries of lattices. An example involving the Ammann–Beenker tiling is provided to illustrate the results in the quasicrystal setting.

Keywords:

• coincidence site lattice
• coincidence index
• colour coincidence
• colour symmetry

Acknowledgements

M. Loquias would like to thank the Deutscher Akademischer Austausch Dienst (DAAD) for financial support during his stay in Germany. This work was supported by the German Research Council (DFG), within the CRC 701.