Coincidence indices of sublattices and coincidences of colorings

Abstract

Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice ${\Lambda }_1$ and to a sublattice ${\Lambda }_2$ may differ. Here, we examine the coloring of ${\Lambda }_1$ induced by ${\Lambda }_2$ to identify how the coincidence indices with respect to ${\Lambda }_1$ and to ${\Lambda }_2$ are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.

15 pages, 1 Figure

References in corpus (2)

• Multiplicativity in the theory of coincidence site lattices
• The coincidence problem for shifted lattices and crystallographic point packings