Spectral and topological properties of a family of generalised Thue-Morse sequences

Baake, Michael; Gähler, Franz; Grimm, Uwe

doi:10.1063/1.3688337

Cited by 28 publications

(49 citation statements)

References 63 publications

(133 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The recursion follows directly from equation (19), with coefficients ðk; '; rÞ ¼ k þ ' À r À 2 minðk; '; r; k þ ' À rÞ. This system has a unique solution, and properties of the solution show that the corresponding diffraction measure b is purely singular continuous; see Baake, Gä hler & Grimm (2012) for the mathematical details of the argument. The diffraction measure is periodic with period 1 (due to the fact that the Dirac comb is supported on the integer lattice Z) feature articles Acta Cryst.…”

Section: Order and Singular Continuous Diffractionmentioning

confidence: 99%

Aperiodic crystals and beyond

Grimm

2015

Acta Crystallogr B Struct Sci Cryst Eng Mater

Self Cite

View full text Add to dashboard Cite

Crystals are paradigms of ordered structures. While order was once seen as synonymous with lattice periodic arrangements, the discoveries of incommensurate crystals and quasicrystals led to a more general perception of crystalline order, encompassing both periodic and aperiodic crystals. The current definition of crystals rests on their essentially point-like diffraction. Considering a number of recently investigated toy systems, with particular emphasis on non-crystalline ordered structures, the limits of the current definition are explored.

show abstract

Section: Order and Singular Continuous Diffractionmentioning

confidence: 99%

Aperiodic crystals and beyond

Grimm

2015

Acta Crystallogr B Struct Sci Cryst Eng Mater

Self Cite

View full text Add to dashboard Cite

show abstract

“…Since the densities f N become increasingly spiky (and do not converge as a sequence of functions), one uses the distribution functions F N to illustrate the resulting measure. Note that the sequence (F N ) N∈N converges uniformly, 8 but not absolutely. This is in line with the fact that µ is singular continuous, and thus cannot be approximated by a norm-converging sequence of absolutely continuous measures.…”

Section: Singular Continuous Diffractionmentioning

confidence: 99%

“…This exact renormalisation-type structure is the golden key to prove the spectral type and to calculate the measure explicitly. The diffraction measure is 1-periodic, 5,8 and hence of the form γ = µ * δ Z with a positive, singular continuous measure µ. To describe the latter explicitly, one defines the distribution function F(x) = µ([0, x]) on the unit interval.…”

Section: Singular Continuous Diffractionmentioning

confidence: 99%

Mathematical diffraction of aperiodic structures

Baake

Grimm

2012

Chem. Soc. Rev.

Self Cite

View full text Add to dashboard Cite

Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details. IntroductionDiffraction techniques have dominated the structure analysis of solids for the last century, ever since von Laue and Bragg employed X-ray diffraction to determine the atomic structure of crystalline materials. Despite the availability of direct imaging techniques such as electron and atomic force microscopy, diffraction by X-rays, electrons and neutrons continues to be the method of choice to detect order in the atomic arrangements of a substance; see Cowley's book 35 and references therein for background.In its full generality, the diffraction of a beam of X-rays, electrons or neutrons from a macroscopic piece of solid is a complicated physical process. It is the presence of inelastic and multiple scattering, prevalent particularly in electron diffraction, which makes it essentially impossible to arrive at a complete mathematical description of the process. Here, we restrict to kinematic diffraction in the far-field or Fraunhofer limit. In this case, powerful tools of harmonic analysis are available to attack the direct problem of calculating the (kinematic) diffraction pattern of a given structure.In contrast, the inverse problem of determining a structure from its diffraction intensities is extremely involved. A diffraction pattern rarely determines a structure uniquely, as there can be homometric structures sharing the same autocorrelation (and hence the same diffraction). 9,54,57,103 We are far away from a complete understanding of the homometry classes of structures, in particular if the diffraction spectrum contains continuous components. At present, a picture is emerging, based on the analysis of explicit examples, which highlight how large the homometry classes may be.Originally, much of the effort concentrated on the pure point part of diffraction, also called the Bragg diffraction, for † Part of a themed issue on Quasicrystals in honour of the 2011 Nobel Prize in Chemistry winner, Professor Dan Shechtman.

show abstract

“…Correspondingly, theČech cohomology of the hull is the direct limit of the inflation action on the cohomology of the approximant complex. For further details, we refer to [27,28]; compare also [20] for some explicitly worked-out examples. For our four tiling spaces, the following results are obtained.…”

Section: Corollary 3 the Hull Of The Half-hex Tiling Viewed As A Dynmentioning

confidence: 99%

“…They turn out to be particularly useful for deriving the structure of the hull. Since several examples of a similar nature have recently been investigated in full detail, compare [20][21][22], we felt that this short account is adequate (in particular, as the explicit results are the outcome of a computer algebra program).…”

Section: Introductionmentioning

confidence: 99%

Hexagonal Inflation Tilings and Planar Monotiles

2012

Self Cite

View full text Add to dashboard Cite

Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces.

show abstract

Spectral and topological properties of a family of generalised Thue-Morse sequences

Cited by 28 publications

References 63 publications

Aperiodic crystals and beyond

Aperiodic crystals and beyond

Mathematical diffraction of aperiodic structures

Hexagonal Inflation Tilings and Planar Monotiles

Contact Info

Product

Resources

About