A note on the structure of spaces of domino tilings

We study spaces of tilings, formed by tilings which are on a geodesic between two fixed tilings of the same domain (the distance is defined using local flips). We prove that each space of tilings is homeomorphic to an interval of tilings of a domain when flips are classically directed by height functions.     

Rhombus Tilings: Decomposition and Space Structure

We study the spaces of rhombus tilings, i.e. the graphs whose vertices are tilings of a fixed zonotope. Two tilings are linked if one can pass from one to the other by a local transformation, called a flip. We first use a decomposition method to encode rhombus tilings and give a useful characterization for a sequence of bits to encode a tiling. We use the previous coding to get a canonical representation of tilings, and two order structures on the space of tilings. In codimension 2 we prove that the two order structures are equal. In larger codimensions we study the lexicographic case, and get an order regularity result.     

On the enumeration of Archimedean polyhedra in the Lobachevsky space

We describe the class of Archimedean polyhedra in the three-dimensional Lobachevsky space, which technically reduces to studying Archimedean tilings of the Lobachevsky plane. We analyze the possibility of obtaining Archimedean tilings by methods that are usually applied on the sphere and in the Euclidean plane. It is pointed out that such tilings can be constructed by using certain types of Fedorov groups in the Lobachevsky plane. We propose a general approach to the problem of classifying Archimedean tilings of the Lobachevsky plane.     

Cubist algebras

We construct algebras from rhombohedral tilings of Euclidean space obtained as projections of certain cubical complexes. We show that these ‘Cubist algebras’ satisfy strong homological properties, such as Koszulity and quasi-heredity, reflecting the combinatorics of the tilings. We construct derived equivalences between Cubist algebras associated to local mutations in tilings. We recover as a special case the Rhombal algebras of Michael Peach and make a precise connection to weight 2 blocks of symmetric groups.     

Fusion: a general framework for hierarchical tilings of \mathbb{R }^d

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli–Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing.     

Ammann bars and quasicrystals

In 1988 Danzer [3] constructed a family of four tetrahedra which allows—with certain matching conditions—only aperiodic tilings. By analogy with the Ammann bars of planar Penrose tilings we define Ammann bars in space in the form of planar Penrose tilings we define Ammann bars in space in the form of plane sections of the four tetrahedra. If we require that the plane sections continue as planes across the faces of the tilings, we obtain an alternative matching condition, thus answering a question of Danzer.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Automatic generation of nonperiodic patterns from dynamical systems

A new and fast algorithm is presented in this paper for the automatic generation of aesthetic patterns on nonperiodic tilings by means of dynamical systems. The Chair and the Sphinx tilings are used as illustrations. Invariant mappings are constructed for the creation of striking patterns on these tilings. A modified convergence time scheme is described to enhance the artistic appeal of generated images. This algorithm can be used to create a great variety of exotic patterns with various nonperiodicities.     

Convex pentagons that admit <Emphasis Type="Italic">i</Emphasis>-block transitive tilings

The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit i-block transitive tilings for \(i \in \mathbb {N}\). These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings for each \(i \in \mathbb {N}\). We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting i-block transitive tilings for \(i \le 4\), among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling.     

Self-Similar Lattice Tilings

We study the general question of the existence of self-similar lattice tilings of Euclidean space. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In dimension two we further prove the existence of connected self-similar lattice tilings for parabolic and elliptic dilations. These results apply to produce Haar wavelet bases and certain canonical number systems.     

Characterization of Planar Pseudo-Self-Similar Tilings

A pseudo-self-similar tiling is a hierarchical tiling of Euclidean space which obeys a nonexact substitution rule: the substitution for a tile is not geometrically similar to itself. An example is the Penrose tiling drawn with rhombi. We prove that a nonperiodic repetitive tiling of the plane is pseudo-self-similar if and only if it has a finite number of derived Vorono\"{\i} tilings up to similarity. To establish this characterization, we settle (in the planar case) a conjecture of E. A. Robinson by providing an algorithm which converts any pseudo-self-similar tiling of R ² into a self-similar tiling of R ² in such a way that the translation dynamics associated to the two tilings are topologically conjugate. Received June 20, 2000, and in revised form January 25, 2001. Online publication July 25, 2001.     

Tiling a Polygon with Two Kinds of Rectangles

We fix two rectangles with integer dimensions. We give a quadratic time algorithm which, given a polygon F as input, produces a tiling of F with translated copies of our rectangles (or indicates that there is no tiling). Moreover, we prove that any pair of tilings can be linked by a sequence of local transformations of tilings, called flips. This study is based on the use of Conway’s tiling groups and extends the results of Kenyon and Kenyon (limited to the case when each rectangle has a side of length 1).     

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).     

Quasicrystallic Tilings and Projection Method

Basic notions related to quasiperiodic tilings and Delone sets in Eucledean space are discussed. It is shown how the cut and project method of constructing them is used to calculate their spectra. Special attention is paid to self-similar tilings and the way one can obtain one-dimensional substitutional tilings by the projection scheme. Bibliography: 18 titles.     

On the Number of Tilings of a Square by Rectangles

We develop a recursive formula for counting the number of rectangulations of a square, i.e., the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was calculated and extend to the non-generic case. An interesting feature of the number of rectangulations is that it appears to have an 8-fold periodicity modulo 2. We verify this periodicity for small values of n, but the general result remains elusive, perhaps hinting at some unseen structure on the space of rectangulations, analogous to Reading’s discovery that generic rectangulations are in 1–1 correspondence with a certain class of permutations. Finally, we use discrete Morse theory to show that the space of tilings by ≤ n rectangles is homotopy-equivalent to a wedge of some number of (n?1)-dimensional spheres. Combining this result with the formulae for the number of tilings, the exact homotopy type is computed for n ≤ 28.     

Tiling polygons with parallelograms

Under what conditions can a simple polygon be tiled by parallelograms? In this paper we give matching necessary and sufficient conditions on the polygon to be tilable and characterize the set of possible tilings. We also provide an efficient algorithm for constructing a tiling.     

An Algebraic Invariant for Substitution Tiling Systems

We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.     

On tile-k-transitive tilings of the space

General methods for finding tile-k-transitive tilings of the three-dimensional Euclidean space with polyhedral bodies are discussed. Analogous methods for enumerating k-isohedral tilings of a two-dimensional plane of constant curvature have been obtained previously.     

Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ<Superscript>2</Superscript>

Given a tiling T, one may form a related tiling, called the derived Voronoi tiling of T, based on a patch of tiles in T. Similarly, for a tiling space X, one can identify a patch which appears regularly in all tilings in X, and form a derived Voronoi space of tilings, based on that patch.     

Quasiperiodic tilings and quasicrystals

One gets quasiperiodic tilings by projecting a periodic lattice from a space of a larger number of dimensions. One can choose a fundamental domain of the lattice in various ways — this leads to different quasi-periodic tilings. Thus, one can generalize Penrose's nonperiodic tiling of the plane and the same for space filling.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 116–135, 1986.Finally, I would like to express my gratitude to M. M. Skriganov for discussions.