Coincidence isometries of a shifted square lattice

CSLs of the root lattice $\mathbf{A_4}$

Recently, the group of coincidence isometries of the root lattice $A_4$ has been determined providing a classification of these isometries with respect to their coincidence indices. A more difficult task is the classification of all CSLs, since different coincidence isometries may generate the same CSL. In contrast to the typical examples in dimensions $d \leq 3$, where coincidence isometries generating the same CSL can only differ by a symmetry operation, the situation is more involved in 4 dimensions. Here, we find coincidence isometries that are not related by a symmetry operation but never…

Similarity Isometries of Point Packings

A linear isometry $R$ of $\mathbb{R}^d$ is called a similarity isometry of a lattice $Γ\subseteq \mathbb{R}^d$ if there exists a positive real number $β$ such that $βRΓ$ is a sublattice of (finite index in) $Γ$. The set $βRΓ$ is referred to as a similar sublattice of $Γ$. A (crystallographic) point packing generated by a lattice $Γ$ is a union of $Γ$ with finitely many shifted copies of $Γ$. In this study, the notion of similarity isometries is extended to point packings. We provide a characterization for the similarity isometries of point packings and identify the corresponding similar subpac…

Coincidence indices of sublattices and coincidences of colorings

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 $Λ_1$ and to a sublattice $Λ_2$ may differ. Here, we examine the coloring of $Λ_1$ induced by $Λ_2$ to identify how the coincidence indices with respect to $Λ_1$ and to $Λ_2$ 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.

Planar coincidences for N-fold symmetry

The coincidence problem for planar patterns with $N$-fold symmetry is considered. For the N-fold symmetric module with $N<46$, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case $N \ge 46$ is briefly discussed and N=46 is described explicitly. The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.

相似子模与重合位置模

Similar submodules and coincidence site modules

本文将格的相似与重合等距理论推广到一般自由 Z-模及 S-模，证明商群 OS(M)/OC(M) 为 Abel 群，并给出 CSL 与相似子格指数间的关系。

Colourings of lattices and coincidence site lattices

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.