Planar coincidences for N-fold symmetry

The problem of computing the index of a coincidence isometry of the hyper cubic lattice $\mathbb{Z}^{n}$ is considered. The normal form of a rational orthogonal matrix is analyzed in detail, and explicit formulas for the index of certain coincidence isometries of $\mathbb{Z}^{n}$ are obtained. These formulas generalize the known results for $n\le 4$.

In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in cr...

Consider the following problem: how many collinear triples of points must a transversal of (Z/nZ)^2 have? This question is connected with venerable issues in discrete geometry. We show that the answer, for n prime, is between (n-1)/4 and (n-1)/2, and consider an analogous question for collinear quadruples. We conjecture that the upper bound is the truth and suggest several other interesting problems in this area.

The groups of (linear) similarity and coincidence isometries of certain modules in d-dimensional Euclidean space, which naturally occur in quasicrystallography, are considered. It is shown that the structure of the factor group of similarity modulo coincidence isometries is the direct sum of cyclic groups of prime power orders that divide d. In particular, if the dimension d is a prime number p, the factor group is an elementary Abelian p-group. This generalizes previous resu...

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 s...

Planar ornaments, a.k.a. wallpapers, are regular repetitive patterns which exhibit translational symmetry in two independent directions. There are exactly $17$ distinct planar symmetry groups. We present a fully automatic method for complete analysis of planar ornaments in $13$ of these groups, specifically, the groups called $p6m, \, p6, \, p4g, \,p4m, \,p4, \, p31m, \,p3m, \, p3, \, cmm, \, pgg, \, pg, \, p2$ and $p1$. Given the image of an ornament fragment, we present a m...

Discrete point sets $\mathcal{S}$ such as lattices or quasiperiodic Delone sets may permit, beyond their symmetries, certain isometries $R$ such that $\mathcal{S}\cap R\mathcal{S}$ is a subset of $\mathcal{S}$ of finite density. These are the so-called coincidence isometrie. They are important in understanding and classifying grain boundaries and twins in crystals and quasicrystals. It is the purpose of this contribution to introduce the corresponding coincidence problem in a...

This paper is a continuation of an earlier one, and completes a classification of the configurations of points in a plane lattice that determine angles that are rational multiples of ${\pi}$. We give a complete and explicit description of lattices according to which of these configurations can be found among their points.

The structure of the coincidence symmetry group of an arbitrary $n$-dimensional lattice in the $n$-dimensional Euclidean space is considered by describing a set of generators. Particular attention is given to the coincidence isometry subgroup (the subgroup formed by those coincidence symmetries which are elements of the orthogonal group). Conditions under which the coincidence isometry group can be generated by reflections defined by vectors of the lattice will be discussed, ...

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.Bulletin of the American Mathema...