Abstract. We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by identifying the square lattice with the ring of Gaussian integers. To illustrate them, we calculate the set of coincidence isometries, as well as generating functions for the number of coincidence site lattices and coincidence isometries, for specific examples.

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.Comment: 15 pages, 1 Figur

The relationship between the coincidence indices of a lattice Γ 1 and a sublattice Γ 2 of Γ 1 is examined via the colouring of Γ 1 that is obtained by assigning a unique colour to each coset of Γ 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.

If G is the symmetry group of an uncolored pattern then a coloring of the pattern is semiperfect if the associated color group H is a subgroup of G of index 2. We give results on how to identify and enumerate all inequivalent semiperfect colorings of certain patterns. This is achieved by treating a coloring as a partition fhJ i Y i : i 2 I; h 2 Hg of G, where H is a subgroup of index 2 in G, J i H for i 2 I, and Y ¼ [ i2I Y i is a complete set of right coset representatives of H in G. We also give a one-to-one correspondence between inequivalent semiperfect colorings whose associated color groups are conjugate subgroups with respect to the normalizer of G in the group of isometries of R n .