Root lattices and quasicrystals

Baake, Michael; Joseph, Dieter; Krämer, Peter; Schlottmann, M.

doi:10.1088/0305-4470/23/19/004

Cited by 45 publications

(37 citation statements)

References 23 publications

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“…There, the main structure is derived with the aid of the algebraic number theory of cyclotomic fields, followed by various explicitly worked out cases in Section 4. They include 8-, 10-and 12-fold symmetry, the most important cases for quasicrystalline T -phases, and thus cover all cases linked to quadratic irrationalities [16]. In Section 5 we then show, in an illustrative way, how to use the method for the eightfold symmetric Ammann-Beenker rhombus pattern and the tenfold symmetric Tübingen triangle tiling.…”

Section: Introductionmentioning

confidence: 99%

Planar coincidences for N-fold symmetry

Pleasants

Baake

Roth

1996

Journal of Mathematical Physics

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The coincidence problem for planar patterns with N -fold symmetry is considered. For the N -fold symmetric module with N < 46, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case N ≥ 46 is briefly discussed and N = 46 is described explicitly.The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.

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Section: Introductionmentioning

confidence: 99%

Planar coincidences for N-fold symmetry

Pleasants

Baake

Roth

1996

Journal of Mathematical Physics

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“…It is also known that not all lattice points k in Z 5 can be mapped onto vertices of a Penrose tiling; only those points in a particular 'cut' slice whose projections into the 3-dimensional orthogonal space W are inside the window of acceptance, 11,16 contribute. The window has been shown 11 to be the projection of the 5-d unit cell Cu(5) with 2 5 vertices into this 3-d space W.…”

Section: Grids and The 'Cut And Projection Method'mentioning

confidence: 99%

Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions

Au-Yang¹,

Perk²

2006

Differential Geometry and Physics

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We show that generalized Penrose tilings can be obtained by the projection of a cut plane of a 5-dimensional lattice into two dimensions, while 3-d quasiperiodic lattices with overlapping unit cells are its projections into 3d. The frequencies of all possible vertex types in the generalized Penrose tilings, and the frequencies of all possible types of overlapping 3-d unit cells are also given here. The generalized Penrose tilings are found to be nonconvertable to kite and dart patterns, nor can they be described by the overlapping decagons of Gummelt.

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“…(5) This method, which is known as the cut-and-project method, has been applied and generalized by many authors. (7,66,67,68,69,70,8,71,72,73,74,75,76) The equivalence of the projection method and a generalized grid method has been demonstrated. (66,67,68) The positions of the Bragg peaks have been worked out, (7,69,70,8,71,72) together with the values of probabilities of local configurations.…”

Section: Conclusion and Final Remarksmentioning

confidence: 98%

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

Au-Yang

Perk

2007

J Stat Phys

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We study the q-dependent susceptibility χ(q) of a Z-invariant ferromagnetic Ising model on a Penrose tiling, as first introduced by Korepin using de Bruijn's pentagrid for the rapidity lines. The pair-correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers. Its Fourier transform χ(q) is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins, reducing this calculation to linear programming. Since the lattice is quasiperiodic, we find that χ(q) is aperiodic and has everywhere dense peaks, which are not all visible at very low or high temperatures. More and more peaks become visible as the correlation length increases-that is, as the temperature approaches the critical temperature.

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Root lattices and quasicrystals

Abstract: It is shown how root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models. All non-periodic symmetries observed so far are covered in minimal embedding with maximal symmetry.

Cited by 45 publications

References 23 publications

Planar coincidences for N-fold symmetry

Planar coincidences for N-fold symmetry

Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

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