Two-dimensional Tiling Research Articles

Quasiperiodic packings of fibres with icosahedral symmetry.

Different icosahedral packings of fibres have been experimentally realized. A packing construction with straight fibres of the same circular cross section, only parallel to fivefold icosahedral axes and respecting the closest packing condition, is reported. Its characteristics of point-group symmetry and related two-dimensional tilings are analysed. But for determining unambiguously all the fibre positions it appears that a mathematical construction has to be made from the cut and projection of a five-dimensional space. Through such a method, the volume fraction of fibrous reinforcement in a composite material can be calculated. The related two-dimensional tiling can be proved to be different from a Penrose tiling. Finally, the characteristics of other icosahedral packings where fibres are parallel to threefold axes or to both threefold and fivefold axes are briefly discussed and a few further experiments on their elasticity properties and photonic band-gap structure are suggested.

Real-space renormalization group approach to the Potts model on the two-dimensional Penrose tiling

A one-step real-space renormalization group (RSRG) transformation is used to study the q-state Potts model on the two-dimensional (2D) Penrose tiling (PT). The critical exponents of the correlation length ν( q) in the region between q=1 and q=4 are obtained. The bond percolation (BP) threshold and Ising critical temperature in the isotropic case are also obtained. The comparison of the results with previous results on the PT and the exact results on square lattice (SQL) indicates that the universal classes of q-state Potts model on PT and SQL may be the same.

Self-assembly of novel co-ordination polymers containing polycatenated molecular ladders and intertwined two-dimensional tilings

Four novel co-ordination polymers have been obtained from the reactions of MII(NO3)2 salts (MII = Zn, Co or Cd) and the bidentate spacer ligands 1,2-bis(4-pyridyl)ethyne (bpethy) and trans-4,4′-azobis(pyridine) (azpy), all exhibiting the same metal-to-ligand molar ratio of 2∶3. Crystal structure analyses, however, have revealed two quite different polymeric motifs, in spite of the presence of similar co-ordination geometries of the metal ions, that are bound to three pyridyl groups, with a T-shaped disposition, and to the oxygen atoms of two η2-nitrates, thus resulting in a distorted pentagonal-bipyramidal seven-co-ordination. The species [M2(bpethy)3(NO3)4] (M = Zn or Co) consist of ladder-like polymers that interpenetrate in an unprecedented fashion to produce an overall three-dimensional array. On the other hand, the species [Cd2(bpethy)3(NO3)4]·CH2Cl2 and [Cd2(azpy)3(NO3)4] contain undulated two-dimensional layers with an unprecedented tiled pattern, that give threefold interpenetration in a parallel fashion.

A real-space renormalization group approach to the ferromagnetic Potts model on the two-dimensional octagonal quasi-periodic tiling

A one-step real-space renormalization group (RSRG) transformation is used to study the ferromagnetic (FM) q-state Potts model on the two-dimensional (2D) octagonal quasi-periodic tiling (OQT). The critical exponents of the correlation length for different values of q and the critical temperature of the Ising model are obtained. The results are shown to be not sensitive to the choice of parameters. The comparison of the results with previous results for the OQT and the square lattice (SQL) seems to show that the universal classes of the q-state Potts models on the OQT and the SQL are the same for the range from q = 1 to q = 3, in accordance with previous research.

Topological invariants for substitution tilings and their associated $C^\ast$-algebras

We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

Monte Carlo investigation of the eight-state Potts model on quasiperiodic tilings

A Monte Carlo investigation of the eight-state Potts model on the two-dimensional (2D) quasiperiodic octagonal tiling with free boundary conditions is performed in order to determine the nature of a temperature-driven transition. It is shown that numerical data suffer from drastic free boundary effects that strongly disturb the probability distributions of the internal energy and, consequently, the scaling behavior of the specific heat. An alternative way consisting in analysing the core of the tilings is applied to pass over free boundary effects. This analysis combined with the Lee–Kosterlitz method allows one to evidence that the system undergoes a first-order transition as in 2D periodic lattices. The first-order type of scaling is observed for the maximum in the susceptibility of the core of the tilings but not for the maximum in the specific heat.

Finite-size behavior of the three-state Potts model on the quasiperiodic octagonal tiling

The static critical behavior of the three-state Potts model on the two-dimensional (2D) quasiperiodic octagonal tiling with free boundary conditions is investigated by means of the importance-sampling Monte Carlo method and the single histogram technique. The results strongly suggest that the static critical exponents \ensuremath{\nu} and \ensuremath{\gamma} are the same as in 2D periodic lattices whereas the different estimates of \ensuremath{\alpha} are not really consistent with the 2D periodic value. The infinite tiling critical temperature, ${\mathrm{kT}}_{c}/J\ensuremath{\approx}1.557,$ is slightly higher than the critical temperature of the three-state Potts model on the square lattice, in agreement with previous studies on the Ising model on quasiperiodic tilings.

A Model of Quasicrystal Growth

Entropically stabilized quasicrystals are usually modeled as equilibrium ensembles of random tilings. In several models, such as the two-dimensional square-triangle tiling studied here, the corresponding kinetics may be very slow because a large number of tiles must be rearranged at each step through the ensemble. Here we consider a simple growth model that generates a single element of the square-triangle tiling ensemble. Even though tile rearrangements occur only at the growth surface, in the limit of slow growth one obtains a structure that is representative of the equilibrium ensemble.

Magnetic states induced by electron-electron interactions in a plane quasiperiodic tiling

We consider the Hubbard model for electrons in a two-dimensional quasiperiodic tiling using the Hartree--Fock approximation. Numerical solutions are obtained for the first three square approximants of the perfect octagonal tiling. At half-filling, the magnetic state is antiferromagnetic. We calculate the distributions of local magnetizations and their dependence on the local environments as U (Hubbard repulsion strength) is varied. The inflation symmetry of quasicrystals results in a corresponding inflation symmetry of the magnetic configurations when passing from one tiling to the next in the progression towards the infinite quasicrystal.

New perspectives on forbidden symmetries, quasicrystals, and Penrose tilings.

Quasicrystals are solids with quasiperiodic atomic structures and symmetries forbidden to ordinary periodic crystals-e.g., 5-fold symmetry axes. A powerful model for understanding their structure and properties has been the two-dimensional Penrose tiling. Recently discovered properties of Penrose tilings suggest a simple picture of the structure of quasicrystals and shed new light on why they form. The results show that quasicrystals can be constructed from a single repeating cluster of atoms and that the rigid matching rules of Penrose tilings can be replaced by more physically plausible cluster energetics. The new concepts make the conditions for forming quasicrystals appear to be closely related to the conditions for forming periodic crystals.

Exact solution of an octagonal random tiling model.

We consider the two-dimensional random tiling model introduced by Cockayne, i.e. the ensemble of all possible coverings of the plane without gaps or overlaps with squares and various hexagons. At the appropriate relative densities the correlations have eight-fold rotational symmetry. We reformulate the model in terms of a random tiling ensemble with identical rectangles and isosceles triangles. The partition function of this model can be calculated by diagonalizing a transfer matrix using the Bethe Ansatz (BA). The BA equations can be solved providing {\em exact} values of the entropy and elastic constants.

Two-dimensional groups, orbifolds and tilings

Given the triangulation of a 2-dimensional orbifold in terms of the Delaney-Dress symbol of a periodic tiling, we discuss how to compute its orbifold symbol, as defined by J. Conway. It is shown that the number of types of equivariant tilings depends only on the ‘form’ of the corresponding orbifold symbols. The method is applied to obtain a refined classification of equivariant tilings for certain hyperbolic symmetry groups.

Static critical behavior of a weakly frustrated Ising model on the octagonal tiling.

The static critical behavior of a weakly frustrated ferromagnetic Ising model on the two-dimensional (2D) quasiperiodic octagonal tiling is studied by means of Monte Carlo simulations and finite-size scaling analysis. Our results strongly suggest that this frustrated Ising model on the octagonal tiling belongs to the same universality class as the ferromagnetic Ising model on 2D periodic lattices. The infinite tiling critical temperature, ${\mathit{kT}}_{\mathit{c}}$/J=1.49\ifmmode\pm\else\textpm\fi{}0.02, agrees with previous studies indicating that tendency to ferromagnetic ordering is higher in quasiperiodic tilings than in periodic lattices. \textcopyright{} 1996 The American Physical Society.

Critical temperature of the non-frustrated ferromagnetic ising model on the quasiperiodic octagonal tiling

The critical temperature of some non-frustrated ferromagnetic Ising spin systems on the two-dimensional octagonal tiling is calculated by Monte Carlo simulations using the simulated annealing method. The ferromagnetic interactions are limited to the third neighbours ( J 1, J 2, J 3) where only J 2 is a percolating interaction. The infinite-tiling critical temperature is estimated at kT c/ J = 2.39 ± 0.02 ( J 2 = J and J 1 = J 3 = 0), kT c/ J = 3.05 ± 0.03 ( J 2 = J 1 = J and J 3 = 0), kT c/ J = 3.60 ± 0.04 ( J 2 = J 3 = J and J 1 = 0) and kT c/〈 z〉 J is generally slightly higher in the octagonal tiling than in the square and triangular lattices indicating that the tendency to ferromagnetic ordering is higher in quasiperiodic tilings. It is found that the critical temperature, T c, varies linearly with the different exchange integrals.

Static critical behavior of the ferromagnetic Ising model on the quasiperiodic octagonal tiling

The static critical behavior of the nonfrustrated ferromagnetic Ising model on the two-dimensional (2D) quasiperiodic octagonal tiling with free boundary conditions is studied by means of Monte Carlo simulations and finite-size scaling analysis. Several estimates of the critical temperature are clearly consistent and provide the final value ${\mathit{kT}}_{\mathit{c}}$/J=2.39\ifmmode\pm\else\textpm\fi{}0.01. This result shows that tendency to ferromagnetic ordering is higher in the octagonal quasilattice than in the square lattice (the mean number of interacting neighbors is equal to 4 in the two lattices). The estimates of the static critical exponents \ensuremath{\nu}, \ensuremath{\beta}, and \ensuremath{\gamma} are in agreement with previous studies on the Penrose tiling and evidence that the nonfrustrated ferromagnetic Ising model on 2D quasilattices belongs to the same universality class as the ferromagnetic Ising model on 2D periodic lattices.

On the grey Penrose tiling

The construction of a two-dimensional and two-colour Penrose tiling known as a grey Penrose tiling is accomplished by using an algorithm comprising the geometrical operations akin to normal crystallographic operations. This method rules out the involvement of selection-making steps based on empirical matching rules as proposed by Li, Dubois and Kuo in a recent publication.

The structure of an Al-Pd decagonal quasicrystal studied by high-resolution electron microscopy

Abstract The structure of a decagonal quasicrystal with 1.6nm periodicity, formed in a rapidly solidified Al3Pd alloy, was examined by high-resolution electron microscopy, with the aid of the structure of an Al3Pd crystalline phase. Its structure is characterized as a two-dimensional arrangement formed by sharing a side of decagonal atom columns. A two-dimensional tiling constructed by connecting the atom columns by lines is compared with the Penrose tilings obtained from the projection of hypercubic lattices in the five-dimensional space to the two-dimensional space.

Approaches to an icosahedral glass model for the icosahedral phase formed by transition from the amorphous state

Structures of icosahedral phases formed by transition from the amorphous state in Al 75Cu 15V 10 and Al 53Si 27Mn 20 alloys have been studied. X-ray peak widths of the icosahedral phase in both alloys exhibit a linear dependence on phason momentum. A high resolution lattice image of the icosahedral AlCuV (i-AlCuV) phase reveals a two-dimensional tiling pattern composed mainly of pentagons stuck together side by side randomly and filled with unique tear- and crack-like defects exhibited in the icosahedral glass model. Convergent beam electron diffraction reveals the isotropic randomness in the i-AlCuV. These two icosahedral phases are well described by the icosahedral glass model.

Two-colour Penrose tiling

Abstract Penrose tiling is a two-dimensional quasiperiodic pattern, which is composed of two tiles, namely thick and thin rhombi. In this letter, white and black colours are given to the tiles to distinguish their top and bottom surfaces. Then, a two-colour Penrose tiling is yielded by the white black rhombi. The new pattern is a three-dimensional array and its projection is considered as a two-dimensional grey Penrose tiling. These new concepts can be applied to analyse the structure of the decagonal quasicrystal.

The real-space renormalization group and generating function for Penrose lattices

We present a real-space renormalization group (RG) approach to study the electronic properties of the two-dimensional Penrose tilings. This approach is based on a recursive evaluation of the generating function which gives the density of states (DOS) of the system. The RG equations are derived in terms of the inflation rules of the Penrose tilings and the calculated DOS matches the results derived by purely numerical methods.