Duality,sections and projections of certain Euclidean tilings

Pegged tilings localize the defining property of or Laguerre tilings, and, like them, admit a natural duality (corresponding to the Delaunay tilings of tilings). It can thus be shown that the projection method, which is generally used to construct quasi-periodic tilings related to tilings of higher dimensional lattices, applies to this wider class of tilings. Of further importance is that pegged tilings are just those which can be lifted to the graphs of convex functions with a certain strong locally polyhedrality property. The context of convex functions also provides a direct way of viewing the projection method, and leads to alternative pictures of special cases such as various grid methods.     

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.     

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.     

Some 2-isohedral tilings of the plane

In this paper we describe and classify, using adjacency symbols, the 2-isohedral tilings of the plane such that all tiles have four edges and four tiles meet at each vertex. There are 69 such tilings. Since many of these can be constructed by dissecting isohedral tilings appropriately, we show which isohedral tilings are related in this way to these 2-isohedral tilings.     

Monotone labelings in polygonal tilings

Labeling the vertices of a finite sequence of polygonal tilings with fewest monotonicity violations enables to represent the tilings by merely specifying sets of vertices—the sequences of their appearance results from the labels. Eventually, this allows a lossless data compression for the sequence of tilings.The existence and computation of suitable labelings is derived from matching and graph colorings which induce an order on the tilings. This order is series-parallel on each individual tiling.     

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.     

Spotlight Tiling

This article introduces spotlight tiling, a type of covering which is similar to tiling. The distinguishing aspects of spotlight tiling are that the “tiles” have elastic size, and that the order of placement is significant. Spotlight tilings are decompositions, or coverings, and can be considered dynamic as compared to typical static tiling methods. A thorough examination of spotlight tilings of rectangles is presented, including the distribution of such tilings according to size, and how the directions of the spotlights themselves are distributed. The spotlight tilings of several other regions are studied, and suggest that further analysis of spotlight tilings will continue to yield elegant results and enumerations.     

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.     

Domino Tiling Congruence Modulo 4

The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino tilings, modulo 2, and leads to a variety of corollaries, including that the number of domino tilings of a k × 2k rectangle is congruent to 1 mod 4.     

Linear Repetitivity, I. Uniform Subadditive Ergodic Theorems and Applications

This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive ergodic theorems on tilings. The results of this paper can be applied in the study of both random operators and lattice gas models on tilings. Received July 19, 2000, and in revised form January 30, 2001. Online publication July 25, 2001.     

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.     

A Framework for the Construction of Self-replicating Tilings

Dekking (Adv. Math. 44:78–104, 1982; J. Comb. Theory Ser. A 32:315–320, 1982) provided an important method to compute the boundaries of lattice rep-tiles as a ‘recurrent set’ on a free group of a finite alphabet. That is, those tilings are generated by lattice translations of a single tile, and there is an expanding linear map that carries tiles to unions of tiles. The boundary of the tile is identified with a sequence of words in the alphabet obtained from an expanding endomorphism (substitution) on the alphabet. In this paper, Dekking’s construction is generalized to address tilings with more than one tile, and to have the elements of the tilings be generated by both translation and rotations. Examples that fall within the scope of our main result include self-replicating multi-tiles, self-replicating tiles for crystallographic tilings and aperiodic tilings.     

Duality properties between spectra and tilings

Spectra and tilings play an important role in analysis and geometry respectively.The relations between spectra and tilings have bafied the mathematicians for a long time.Many conjectures,such as the Fuglede conjecture,are placed on the establishment of relations between spectra and tilings,although there are no desired results.In the present paper we derive some characteristic properties of spectra and tilings which highlight certain duality properties between them.     

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.     

A primer of substitution tilings of the Euclidean plane

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to represent the field as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.     

Isohedral tilings of a ribbon

The isohedral tilings of a ribbon or infinite strip are classified. There are 24 of them, 5 of which must be realized as either marked tilings or tilings of a ribbon with edges which are not straight lines.     

Detecting Combinatorial Hierarchy in Tilings Using Derived Voronoi Tessellations

Abstract. Tilings of R ² can display hierarchy similar to that seen in the limit sequences of substitutions. Self-similarity for tilings has been used as the standard generalization, but this viewpoint is limited because such tilings are analogous to limit points of constant-length substitutions. To generalize limit points of non-constant-length substitutions, we define hierarchy for infinite, labelled graphs, then extend this definition to tilings via their dual graphs. Examples of combinatorially substitutive tilings that are not self-similar are given. We then find a sufficient condition for detecting combinatorial hierarchy that is motivated by the characterization by Durand of substitutive sequences. That characterization relies upon the construction of the ``derived sequence'—a recoding in terms of reappearances of an initial block. Following this, we define the ``derived Vorono? tiling'—a retiling in terms of reappearances of an initial patch of tiles. Using derived Vorono? tilings, we obtain a sufficient condition for a tiling to be combinatorially substitutive.     

Parallelogram Tilings, Worms, and Finite Orientations

This paper studies properties of tilings of the plane by parallelograms. In particular, it is established that in parallelogram tilings using a finite number of shapes all tiles occur in only finitely many orientations.     

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.