Colourings of lattices and coincidence site lattices

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.

9 pages, 6 figures

References in corpus (4)

• Planar coincidences for N-fold symmetry
• Coincidence rotations of the root lattice $A_4$
• Colourings of planar quasicrystals
• A Note on Coincidence Isometries of Modules in Euclidean Space

Cited by in corpus (1)

• Coincidence indices of sublattices and coincidences of colorings