Multiple coincidences in dimensions
            <b>
              <i>d</i>
              ≤ 3
            </b>

Baake, Michael; Zeiner, Peter

doi:10.1080/14786430701264186

Cited by 8 publications

(10 citation statements)

References 17 publications

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“…The equality of the two Dirichlet series to the left is non-trivial, and was proved in [8] with an argument involving Eichler orders. The same formula also applies to the other cubic lattices in 3-space [1,9]. The simple coincidence spectrum is thus the set of odd integers, which is again a monoid.…”

Section: Related Results and Outlookmentioning

confidence: 83%

“…The simple coincidence spectrum is thus the set of odd integers, which is again a monoid. The multiple analogues have recently been derived in [29,30], see also [4,9] for related results. Several of these results are also included by now in [25].…”

Section: Related Results and Outlookmentioning

confidence: 99%

“…Consequently, the simple coincidence spectrum Σ(OC(A 4 )) is the set of all positive integers. Although multiple coincidences may produce further lattices, compare [4,29,9], the total spectrum Σ A 4 cannot be larger than the elementary one, so that the following consequence is clear.…”

Section: Coincidence Indices and Generating Functionsmentioning

confidence: 99%

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Coincidence rotations of the root lattice A4

Baake

Grimm

Heuer

et al. 2008

European Journal of Combinatorics

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The coincidence site lattices of the root lattice A4 are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of A4 into the icosian ring with its rich arithmetic structure, which recently [6] led to the classification of the similar sublattices of A4. Dedicated to Ludwig Danzer on the occasion of his 80th birthday

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Section: Related Results and Outlookmentioning

confidence: 83%

Section: Related Results and Outlookmentioning

confidence: 99%

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Coincidence rotations of the root lattice A4

Baake

Grimm

Heuer

et al. 2008

European Journal of Combinatorics

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“…In fact, intersections of more than two isometric commensurate copies of a lattice have already been discussed in [8,95,18,98]. Let us first recall the corresponding definitions.…”

Section: Similar Submodulesmentioning

confidence: 99%

“…However, not much is known about lattices in dimensions d > 5, although there are some partial results for rational lattices [99,100,57].The original concept of CSLs has been generalised in several ways. In particular, one may study the intersection of several rotated copies of a lattice, which are known as multiple CSLs; compare [8,95,18]. They have applications to so-called multiple junctions [40,41,42], which are multiple crystal grains meeting at some common manifold.…”

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confidence: 99%

Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Baake¹,

Zeiner²

Aperiodic Order

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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 crystallography. As many results are algebraic in nature, we also generalise them to Z-modules embedded in R d . R d , which led to the notion of coincidence site modules (CSMs). They are used to describe grain boundaries in quasicrystals; compare [15,72,88] and references therein.This new development also triggered a more detailed study of lattices in dimensions d > 3, as they are used to generate aperiodic point sets by the now common cut and project technique; compare [9, Ch. 7]. In particular, lattices in dimension d = 4 such as the hypercubic lattices [4,94] and the A 4 -lattice [11,56] were studied.Further applications of CSLs can be found in coding theory in connection with so-called lattice quantisers, where lattices in large dimensions and with high packing densities are important; compare [34,85] for general background, as well as [1] for concrete applications of the A 4 -lattice and [2] for the hexagonal lattice. However, not much is known about lattices in dimensions d > 5, although there are some partial results for rational lattices [99,100,57].The original concept of CSLs has been generalised in several ways. In particular, one may study the intersection of several rotated copies of a lattice, which are known as multiple CSLs; compare [8,95,18]. They have applications to so-called multiple junctions [40,41,42], which are multiple crystal grains meeting at some common manifold. Whereas classical CSLs involve only linear isometries, one may consider affine isometries as well, which is directly related to the question of coincidences of crystallographic point packings; compare [64,66,63]. The latter are connected to the problem of coincidences of coloured lattices and colour coincidences [65,63,67].The planar case is certainly the best studied. Here, also a connection between CSLs and well-rounded sublattices has been established [17]. Moreover, even some results for the hyperbolic plane [78] have been found.Naturally, CSLs are not the only sublattices that are of interest in crystallography and coding theory. Classifying sublattices with certain symmetry constraints has a long tradition in mathematics and in crystallography; compare [79,80] and references therein. An interesting question is the number of sublattices that are similar to its parent lattice. It has been answered in detail for a considerable collection of lattices [7,14,12] in dimensions d ≤ 4. For higher dimensions, some existence results have been obtained by Conway, Rains and Sloane, who were motivated by problems in coding theory [26].Actually, some years ago, a cl...

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Coincidences in four dimensions

Baake

Zeiner

2008

Philosophical Magazine

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The coincidence site lattices (CSLs) of prominent 4-dimensional lattices are considered. CSLs in 3 dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest to also look at CSLs in dimensions d > 3. Here, we discuss the CSLs of the A 4-lattice and the hypercubic lattices, which are of particular interest both from the mathematical and the crystallographic viewpoint. Quaternion algebras are used to derive their coincidence rotations and the CSLs. We make use of the fact that the CSLs can be directly linked to certain ideals and compute their indices, their multiplicities and encapsulate all this in generating functions in terms of Dirichlet series. In addition, we sketch how these results can be generalized for 4-dimensional Z-modules by discussing the icosian ring.

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Multiple coincidences in dimensions d ≤ 3

Cited by 8 publications

References 17 publications

Coincidence rotations of the root lattice A4

Coincidence rotations of the root lattice A4

Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Coincidences in four dimensions

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