Limit Theorems for Self-Similar Tilings

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.     

Generation of aperiodic tilings with fivefold symmetry by the method of intersecting decagons and diffraction from finite size tilings

A method for generating aperiodic tilings with five fold symmetry is discussed here. Basic patterns formed within decagons can be used to fill two dimensional space, by matching such suitable patterns. It appears to be possible to generate perfect tilings without retracing already established coordinates imposing conditions at the initial stages of generating them. Various possible ways to generate tilings, when perfectness is not required, are discussed. The calculated diffraction patterns for some representative finite size tilings are shown. There are subtle differences in the intensities of peaks in the diffraction patterns corresponding to different finite size tilings constructed using intersecting decagons. These effects persist for a larger number of scatterers in weak peaks than in strong peaks. They are unaffected by an introduction of systematic disorder. These effects could be termed as the finite size boundary effects. There are also small shifts in the peak positions owing to the finite size effects. The possibility of formation of large approximate square cells in large tilings is shown.     

Symmetry of quasicrystals

The definition of an aperiodic crystal (quasicrystal) as a solid that is characterized by the forbidden symmetry suggests the existence of an unsolved problem, because, in a mutually exclusive manner, it appeals to the fundamental theorem of classical crystallography. Using the Penrose tiling as an example, we have investigated the symmetry properties of aperiodic tilings for the purpose to establish the allowed symmetry groups of quasicrystals. The filling of the Euclidean space according to an aperiodic law is considered as the action of an infinite number of group elements on a fundamental domain in the non-Euclidean space. It is concluded that all locally equivalent tilings have a common “parent” structure and, consequently, the same symmetry group. An idealized object, namely, an infinitely refined tiling, is introduced. It is shown that the symmetry operations of this object are operations of the similarity (rotational homothety). A positive answer is given to the question about a possible composition of operations of the similarity with different singular points. It is demonstrated that the transformations of orientation-preserving aperiodic crystals are isomorphic to a discrete subgroup of the Möbius group PSL(2, ?); i.e., they can be realized as discrete subgroups of the full group of motions in the Lobachevsky space. The problem of classification of the allowed types of aperiodic tilings is reduced to the procedure of enumeration of the aforementioned discrete subgroups.     

Minimum perimeter partitions of the plane into equal numbers of regions of two different areas

We identify the minimum-perimeter periodic tilings of the plane by equal numbers of regions (cells) of areas 1 and λ (minimal tilings), with at most two cells of each area per period and for which all cells of the same area are topologically equivalent. For λ close to 1 the minimal tiling is hexagonal. For smaller values of λ the minimal tilings contain pairs of 5/7, 4/8 and 3/9 cells, the cells with fewer sides having smaller area. The correlation between area fraction and number of sides in the minimal tilings is approximately linear and consistent with Lewis' law. Received 27 June 2001 and Received in final form 29 August 2001     

Quasi-periodicity,global stability and scaling in a model of Hamiltonian round-off

We investigate the effects of round-off errors on the orbits of a linear symplectic map of the plane, with rational rotation number nu=p/q. Uniform discretization transforms this map into a permutation of the integer lattice Z(2). We study in detail the case q=5, exploiting the correspondence between Z and a suitable domain of algebraic integers. We completely classify the orbits, proving that all of them are periodic. Using higher-dimensional embedding, we establish the quasi-periodicity of the phase portrait. We show that the model exhibits asymptotic scaling of the periodic orbits and a long-range clustering property similar to that found in repetitive tilings of the plane. (c) 1997 American Institute of Physics.     

Quasiperiodicity and randomness in tilings of the plane

We define new tilings of the plane with Robinson triangles, by means of generalized inflation rules, and study their Fourier spectrum. Penrose's matching rules are not obeyed; hence the tilings exhibit new local environments, such as three different bond lengths, as well as new patterns at all length scales. Several kinds of such generalized tilings are considered. A large class of deterministic tilings, including chiral tilings, is strictly quasiperiodic, with a tenfold rotationally symmetric Fourier spectrum. Random tilings, either locally (with extensive entropy) or globally random (without extensive entropy), exhibit a mixed (discrete+continuous) diffraction spectrum, implying a partial perfect long-range order.     

When Periodicities Enforce Aperiodicity

Non-periodic tilings and local rules are commonly used to model the long range aperiodic order of quasicrystals and the finite-range energetic interactions that stabilize them. This paper focuses on planar rhombus tilings, which are tilings of the Euclidean plane, which can be seen as an approximation of a real plane embedded in a higher dimensional space. Our main result is a characterization of the existence of local rules for such tilings when the embedding space is four-dimensional. The proof is an interplay of algebra and geometry that makes use of the rational dependencies between the coordinates of the embedded plane. We also apply this result to some cases in a higher dimensional embedding space, notably tilings with n-fold rotational symmetry.     

Brane tilings and their applications

We review recent developments in the theory of brane tilings and four‐dimensional 𝒩 = 1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of brane tilings, emphasizing the physical interpretation of brane tilings as fivebrane systems. In part II, we discuss application of brane tilings to AdS/CFT correspondence and homological mirror symmetry. More topics, such as orientifold of brane tilings, phenomenological model building, similarities with BPS solitons in supersymmetric gauge theories, are also briefly discussed. This paper is a revised version of the author's master's thesis submitted to Department of Physics, Faculty of Science, the University of Tokyo on January 2008, and is based on his several papers and some works in progress [1–7].     

A remark on Schrödinger operators on aperiodic tilings

We prove that for a large class of Schrödinger operators on aperiodic tilings the spectrum and the integrated density of states are the same for all tilings in the local isomorphism class, i.e., for all tilings in the orbit closure of one of the tilings. This generalizes the argument in earlier work from discrete strictly ergodic operators onl ²( ^d ) to operators on thel ²-spaces of sets of vertices of strictly ergodic tilings.     

Interlaced Particle Systems and Tilings of the Aztec Diamond

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N, and particles on successive lines are subject to an interlacing constraint. It is shown that this particle system is exactly solvable, to the extent that not only can the partition function be computed exactly, but so too can the marginal distributions. These results in turn are used to give new derivations within the particle picture of a number of known fundamental properties of the tiling problem, for example that the number of distinct configurations is 2 ^N(N+1)/2, and that there is a limit to the GUE minor process, which we show at the level of the joint PDFs. It is shown too that the study of tilings of the half Aztec diamond—not known from earlier literature—also leads to an interlaced particle system, now with successive lines 2n−1 and 2n (n=1,…,N/2−1) having n particles. Its exact solution allows for an analysis of the half Aztec diamond tilings analogous to that given for the Aztec diamond tilings.     

Archimedean-like colloidal tilings on substrates with decagonal and tetradecagonal symmetry

Two-dimensional colloidal suspensions subjected to laser interference patterns with decagonal symmetry can form an Archimedean-like tiling phase where rows of squares and triangles order aperiodically along one direction (J. Mikhael et al., Nature 454, 501 (2008)). In experiments as well as in Monte Carlo and Brownian dynamics simulations, we identify a similar phase when the laser field possesses tetradecagonal symmetry. We characterize the structure of both Archimedean-like tilings in detail and point out how the tilings differ from each other. Furthermore, we also estimate specific particle densities where the Archimedean-like tiling phases occur. Finally, using Brownian dynamics simulations we demonstrate how phasonic distortions of the decagonal laser field influence the Archimedean-like tiling. In particular, the domain size of the tiling can be enlarged by phasonic drifts and constant gradients in the phasonic displacement. We demonstrate that the latter occurs when the interfering laser beams are not ideally adjusted.     

Diffraction of Random Tilings: Some Rigorous Results

The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diffraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diffraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous, and absolutely continuous parts.     

Quadratic-Like Dynamics of Cubic Polynomials

A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.     

Non-periodic tilings in 2-dimensions with 4, 6, 8, 10 and 12-fold symmetries

The two dimensional plane can be filled with rhombuses, so as to generate non-periodic tilings with 4, 6, 8, 10 and 12-fold symmetries. Some representative tilings constructed using the rule of inflation are shown. The numerically computed diffraction patterns for the corresponding tilings are also shown to facilitate a comparison with possible X-ray or electron diffraction pictures.     

On the analytic structure of the Henon-Heiles system

Solutions of the Henon-Heiles hamiltonian that are analytically continued into the complex time domain are found to possess a neutral boundary with a self-similar structure.     

On the Self-Similar Solutions of the 3D Euler and the Related Equations

We generalize and localize the previous results by the author on the study of self-similar singularities for the 3D Euler equations. More specifically we extend the restriction theorem for the representation for the vorticity of the Euler equations in a bounded domain, and localize the results on asymptotically self-similar singularities. We also present progress towards relaxation of the decay condition near infinity for the vorticity of the blow-up profile to exclude self-similar blow-ups. The case of the generalized Navier-Stokes equations having the laplacian with fractional powers is also studied. We apply the similar arguments to the other incompressible flows, e.g. the surface quasi-geostrophic equations and the 2D Boussinesq system both in the inviscid and supercritical viscous cases.     

Ermakov System and Plane Curve Motions in Affine Geometries

It is shown that the Pinney equation, Ermakov systems, and their higher-order generalizations describe self-similar solutions of plane curve motions in centro-affine and affine geometries.     

On the stability of Archimedean tilings formed by patchy particles

We have investigated the possibility of decorating, using a bottom-up strategy, patchy particles in such a way that they self-assemble in (two-dimensional) Archimedean tilings. Except for the trihexagonal tiling, we have identified conditions under which this is indeed possible. The more compact tilings, i.e., the elongated triangular and the snub square tilings (which are built up by triangles and squares only) are found to be stable up to intermediate pressure values in the vertex representation, i.e., where the tiling is decorated with particles at its vertices. The other tilings, which are built up by rather large hexagons, octagons and dodecagons, are stable over a relatively large pressure range in the centre representation where the particles occupy the centres of the polygonal units.     

Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies

We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in the limit of continuous symmetry. Tilings with phason strain appear to share the same entropy as unstrained tilings, as predicted by mean-field theory. We consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy, although these differ for finite rotational symmetry.