Colourings of lattices and coincidence site lattices

Loquias, Manuel Joseph C.; Zeiner, Peter

doi:10.1080/14786435.2010.524900

Cited by 4 publications

(5 citation statements)

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“…Aperiodic tilings are now studied and classified using advanced topological tools, as explained by Franz Ga¨hler [24]; and new methods are being devised for the generation of icosahedral tilings, as described by Nobuhisa Fujita [25]. Traditional theories of color symmetry [26] and magnetic order [27,28] have been generalized to deal with imperfect and partial coloring, as we saw in the presentations of Peter Zeiner [29] and Louise de Las Pen˜as [30], and in a new model of frustrated antiferromagnetic order on the Penrose tiling that was presented by Anu Jagannathan [31]. But probably most exciting is the fact that diffraction theory has gained much progress, bringing us closer than ever to the point where we can understand what constitutes a crystal [32,33], namely, what sets of points in space give rise to Bragg peaks in their diffraction [34].…”

Section: Advances In Theorymentioning

confidence: 99%

Quasicrystals: diversity and complexity

Dubois

Lifshitz

2011

Philosophical Magazine

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a variety of stimulating new experimental data and novel theoretical results. New aperiodic crystals were presented; new theoretical ideas were described; exciting experimental results were revealed; and potential applications of quasicrystals were reviewed, showing an unprecedented level of development. ICQ11 was a great success thanks to the high standard of its scientific content and to the efficiency of its organization. ICQ11 proved that quasicrystal research is sure to continue offering diverse challenges and profound insights into the complexity of matter.Keywords: quasicrystals; aperiodic crystals; complex metallic alloys (CMA); electronic and magnetic properties; soft matter; surface science; metamaterials; applications The 11th international conference on quasicrystalsThe International Conference on Quasicrystals was convened in Sapporo, on June 13-18, 2010, for the second time in Japan, and the 11th since its inception in Les Houches, in 1985. The program was exceptional in its diversity, with 145 contributions, covering topics like formation, growth and stability; structure and modeling; mathematics; physical properties; surfaces and overlayers; applications; and metamaterials. Such a variety of subjects can be taken as a promising fingerprint for the future of the field. These notes do not provide an accurate summary of the conference, nor do they reflect an exact assessment of the current status of quasicrystal research. They are merely a selection of highlights from the conference -reflecting our own personal tastes and biases -whose purpose is to demonstrate that quasicrystals are still very interesting and challenging almost three decades after their discovery [1]. New materials, better samples, improved experimental resultsIt is quite common in conferences on quasicrystals to learn of newly discovered quasicrystal-forming alloy systems. ICQ11 was no exception. Only a mere decade has

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Section: Advances In Theorymentioning

confidence: 99%

Quasicrystals: diversity and complexity

Dubois

Lifshitz

2011

Philosophical Magazine

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“…2.2] (with u = 1 in the notation used there). Note that this result is the basis of the concept of colour coincidences; compare [65,63,67].…”

Section: Similar Submodulesmentioning

confidence: 76%

“…Whereas the second statement of Lemma 7.9 can be generalised immediately, the first claim of Theorem 7.25 requires a different approach, as we generally lack the notion of a dual module. The proof is algebraic in nature and can be found in [98]; compare also [65], where a similar approach for lattices is described. 7.6.…”

Section: 5mentioning

confidence: 98%

“…Lemma 7.9 provides us with some useful bounds on the coincidence indices of a sublattice. In certain cases, we can even get sharper bounds [65,98]. As an example, we mention the following result, which is a special case of [98, Thm.…”

Section: Similar Submodulesmentioning

confidence: 99%

“…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].…”

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confidence: 99%

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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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