Multiple planar coincidences with N-fold symmetry

Abstract

Abstract. Planar coincidence site lattices and modules with N -fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions. Key Words:Lattices, Coincidence Ideals, Planar Modules, Cyclotomic Fields, Dirichlet Series, Asymptotic Properties IntroductionGiven a lattice Γ ⊂ R d , with d ≥ 2, it is interesting to know its coincidence site lattices (CSLs), which originate from intersections of Γ with a rotated copy. In fact,is a group, whose structure in general (i.e., for d > 2) is not well understood. The coincidence indexof a rotation R is defined as the number of cosets of Γ ∩ RΓ in Γ (which can be ∞), and the spectrum of finite Σ-values, Σ SOC(Γ ) , is often the first quantity considered, followed by counting all CSLs of a given index. Note that one can also consider the obvious extension to general orthogonal transformations R ∈ O(d) and the corresponding group OC(Γ ).Coincidence lattices play an important role in the theory of grain boundaries [9,13], and small indices can be determined in suitable experiments. The classification of these elementary or simple coincidences is partly done, in particular for low-dimensional lattices with irreducible symmetries, compare [7,2] and references given there, but also for certain generalizations to quasicrystals [17,2]. The situation for various related problems More recent is the problem of optimal lattice quantizers [19], and connected with it is the question for multiple coincidences. Here, one would like to classify all lattices that can be obtained as multiple intersections of the formwith R ℓ ∈ SO(d). One defines the corresponding index aswhich is finite if and only if each R ℓ ∈ SOC(Γ ), due to the mutual commensurability of the lattices Γ ∩ R ℓ Γ (we shall explain this in more detail below). Consequently, one attaches the group SOC(Γ ) m to the setting of m-fold coincidences. The corresponding spectrum is its image under Σ, while the full (or complete) coincidence spectrum of Γ isThis is an inductive limit, withwhich often stabilizes: from a certain m on, the spectra are stable, i.e., they do not grow any more [26]. In our examples below, this actually happens at m = 1, so that the spectrum is Σ Γ = Σ SOC(Γ ) . Clearly, multiple coincidences are also relevant in crystallography, as they are the basis for triple or multiple junctions, in obvious generalization of twinning, compare [11].The problem of multiple coincidences is considerably more involved than that of the simple ones, in particular in dimensions d ≥ 3. However, for d = 2, one can rather easily extend the treatment of elementary coincidences to multiple ones, building on the powerful and well understood connection to the ...

Multiple coincidences in dimensions d ≤ 3

“…This is of interest in connection with triple and multiple junctions [11,12] in solid state physics and in connection with quantizing procedures [13,14] in information theory. So far, multiple coincidences are well understood only for some highly symmetric lattices and modules in the plane, see [15]. Since so far only some preliminary results on the cubic case have been published [16], we present here the main results for the cubic case, whose details will be published elsewhere [17].…”

“…In particular, we exploit the fact that the Gaussian integers possess unique prime factorization (up to units). Rotations can then be represented by unimodular complex numbers e i and it turns out that e i corresponds to a coincidence rotation if, and only if, e i 2 QðiÞ (for details, see [7,15]). Hence, we may write e i ¼ = with ¼ "…”

“…. , Àn ðkÞ p Þ, see [15] for details. This implies that the spectrum stays the same, a fact that alternatively can be deduced from our general considerations in section 1.…”

Coincidences of a Shifted Hexagonal Lattice and the Hexagonal Packing

“…The coincidence problem for the hexagonal lattice has been solved in [9] (see also [10]). Any coincidence rotation R of Γ by an angle of ϕ corresponds to multiplication by…”

Coincidence Site Modules in 3-Space

“…In several important examples, compare [3,27], these spectra are equal or stabilize, in the sense that the total spectrum is reached after finitely many lattice intersections. Usually, there are more multiple CSLs than simple ones with a given index, though this quantity also stabilizes in the examples mentioned.…”