Colourings of lattices and coincidence site lattices

TL;DR

This paper explores how colouring lattices based on sublattices relates to their coincidence indices and extends colour symmetry concepts to coincidence isometries, with applications to quasicrystals.

Contribution

It introduces a novel connection between lattice colourings and coincidence indices, extending colour symmetry to isometries and illustrating with quasicrystal examples.

Findings

Established relationship between coincidence indices and lattice colourings.

Extended colour symmetry concept to coincidence isometries.

Provided example with Ammann-Beenker tiling in quasicrystals.

Abstract

The relationship between the coincidence indices of a lattice $\Gamma_1$ and a sublattice $\Gamma_2$ of $\Gamma_1$ is examined via the colouring of $\Gamma_1$ that is obtained by assigning a unique colour to each coset of $\Gamma_2$. 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.