Pseudo-self-affine tilings in ℝ<Superscript>d</Superscript>

It is proved that every pseudo-self-affine tiling in ?^d is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronoï tessellations is a corollary. Previously, these results were obtained in the planar case, jointly with Priebe Frank. The new approach is based on the theory of graph-directed iterated function systems and substitution Delone sets developed by Lagarias and Wang. Bibliography: 18 titles.     

Tiling R 5 by Crosses

An n-dimensional cross comprises 2n+1 unit cubes: the center cube and reflections in all its faces. It is well known that there is a tiling of R ⁿ by crosses for all n. AlBdaiwi and the first author proved that if 2n+1 is not a prime then there are $2^{\aleph_{0}}$ non-congruent regular (= face-to-face) tilings of R ⁿ by crosses, while there is a unique tiling of R ⁿ by crosses for n=2,3. They conjectured that this is always the case if 2n+1 is a prime. To support the conjecture we prove in this paper that also for R ⁵ there is a unique regular, and no non-regular, tiling by crosses. So there is a unique tiling of R ³ by crosses, there are $2^{\aleph_{0}}$ tilings of R ⁴, but for R ⁵ there is again only one tiling by crosses. We guess that this result goes against our intuition that suggests ‘the higher the dimension of the space, the more freedom we get’.     

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.     

Lattice substitution systems and model sets

This paper studies ways in which the sets of a partition of a lattice in \Bbb R ⁿ become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in \Bbb R ⁿ gives rise to regular model sets (based on p -adic-like internal spaces), and hence to pure point diffractive sets. The methods developed here are used to show that the n -dimensional chair tiling and the sphinx tiling are pure point diffractive. Received January 13, 2000, and in revised form May 30, 2000. Online publication January 17, 2001.     

Bernuau Spline Wavelets and Sturmian Sequences

We present spline wavelets of class Cⁿ(R) supported by sequences of aperiodic discretizations of R. The construction is based on multiresolution analysis recently elaborated by G. Bernuau. At a given scale, we consider discretizations that are sets of left-hand ends of tiles in a self-similar tiling of the real line with finite local complexity. Corresponding tilings are determined by two-letter Sturmian substitution sequences. We illustrate the construction with examples having quadratic Pisot–Vijayaraghavan units (like = (1+\sqr{5})/2 or ² = (3+\sqr{5})/2) as scaling factor. In particular, we present a comprehensive analysis of the Fibonacci chain and give the analytic form of related scaling functions and wavelets. We also give some hints for the construction of multidimensional spline wavelets based on stone-inflation tilings in arbitrary dimension.     

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.     

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.     

Tiling Groupoids and Bratteli Diagrams II: Structure of the Orbit Equivalence Relation

In this second paper, we study the case of substitution tilings of \mathbb R^d{{\mathbb R}^d} . The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j = 0, . . . , d − 1. We reconstruct the tiling’s equivalence relation in a purely combinatorial way using the AF-relations given by the lower dimensional substitutions. We define a Bratteli multi-diagram B{{\mathcal B}} which is made of the Bratteli diagrams B^j, j=0, ?d{{\mathcal B}^j, j=0, \ldots d} , of all those substitutions. The set of infinite paths in B^d{{\mathcal B}^d} is identified with the canonical transversal Ξ of the tiling. Any such path has a “border”, which is a set of tails in B^j{{\mathcal B}^j} for some j ≤ d, and this corresponds to a natural notion of border for its associated tiling. We define an étale equivalence relation R_B{{\mathcal R}_{\mathcal B}} on B{{\mathcal B}} by saying that two infinite paths are equivalent if they have borders which are tail equivalent in B^j{{\mathcal B}^j} for some j ≤ d. We show that R_B{{\mathcal R}_{\mathcal B}} is homeomorphic to the tiling’s equivalence relation R_X{{\mathcal R}_\Xi} .     

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.     

Non-periodic Tilings of ? n by Crosses

An n-dimensional cross consists of 2n+1 unit cubes: the “central” cube and reflections in all its faces. A tiling by crosses is called a Z-tiling if each cross is centered at a point with integer coordinates. Periodic tilings of ℝ ⁿ by crosses have been constructed by several authors for all n∈N. No non-periodic tiling of ℝ ⁿ by crosses has been found so far. We prove that if 2n+1 is not a prime, then the total number of non-periodic Z-tilings of ℝ ⁿ by crosses is 2^à₀2^{\aleph _{0}} while the total number of periodic Z-tilings is only ℵ₀. In a sharp contrast to this result we show that any two tilings of ℝ ⁿ ,n=2,3, by crosses are congruent. We conjecture that this is the case not only for n=2,3, but for all n where 2n+1 is a prime.     

Self similar symmetric planar tilings

A new variant of the projection method yields aperiodic tilings of the plane with some rotational symmetry. In particular we display three tilings E _s with full D ₇-symmetry. Each of them is self similar. Further, there is an uncountable number of tilings E without any symmetry, but being almost equivalent to each of the symmetric tiling E _s , i.e. for each R > 0 there is a translation T(E) of E which is equal to E _s in all vertices but a set of error points which are distributed all over the plane but have mutual distance greater than R.      

On the K-theory of the stable C-algebras from substitution tilings

We describe a method to compute the K-theory of the C^?-algebra arising from the stable equivalence relation in the Smale space associated to a substitution tiling, and give detailed computations for one- and two-dimensional examples. We prove that for one-dimensional tilings the group K₀ is always torsion free and give an example of a two-dimensional tiling such that K₀ has torsion.     

Alternating-Sign Matrices and Domino Tilings (Part I)

We introduce a family of planar regions, called Aztec diamonds, and study tilings of these regions by dominoes. Our main result is that the Aztec diamond of order n has exactly 2 ^n(n+1)/2 domino tilings. In this, the first half of a two-part paper, we give two proofs of this formula. The first proof exploits a connection between domino tilings and the alternating-sign matrices of Mills, Robbins, and Rumsey. In particular, a domino tiling of an Aztec diamond corresponds to a compatible pair of alternating-sign matrices. The second proof of our formula uses monotone triangles, which constitute another form taken by alternating-sign matrices; by assigning each monotone triangle a suitable weight, we can count domino tilings of an Aztec diamond.     

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

   Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.     

Integer Cech cohomology of a class of n‐dimensional substitutions

A class of non‐periodic tilings in n‐dimensions is considered. They are based on one‐dimensional substitution tilings that force the border, a property preserved in the construction for higher dimensions. This fact allows to compute the integer?ech cohomology of the tiling spaces in an efficient way. Several examples are analyzed, some of them with PV numbers as inflation factors, and they have finitely or infinitely generated torsion‐free cohomologies. Copyright © 2010 John Wiley & Sons, Ltd.     

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.     

4-Regular Vertex-Transitive Tilings of E 3

{There exist precisely 149 topological types of semipolytopal tile-transitive tilings of E ³ by ``extetrahedra' (obtained from tetrahedra by introducing certain new vertices of degree 2 ). Dualization gives rise to 149 types of 4-regular vertex-transitive tilings. The 4-coordinated networks carried by these tilings are closely related to crystal structures such as zeolites or diamond. These results are obtained using ``combinatorial tiling theory.'} Received February 12, 1999, and in revised form September 21, 1999. Online publication May 15, 2000.     

On the Characterization of Expansion Maps for Self-Affine Tilings

We consider self-affine tilings in ℝ ⁿ with expansion matrix φ and address the question which matrices φ can arise this way. In one dimension, λ is an expansion factor of a self-affine tiling if and only if |λ| is a Perron number, by a result of Lind. In two dimensions, when φ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex λ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for φ to be an expansion matrix for any n, assuming only that φ is diagonalizable over ℂ. We conjecture that this condition on φ is also sufficient for the existence of a self-affine tiling.     

Some Generalizations of the Pinwheel Tiling

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but can be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable number of cases both the number of sizes and the number of orientations is infinite. Received April 9, 1996, and in revised form September 16, 1996.     

Variations on Tilings in the Manhattan Metric

We investigate tilings of the integer lattice in the Euclidean n-dimensional space. The tiles considered here are the union of spheres defined by the Manhattan metric. We give a necessary condition for the existence of such a tiling for Z ⁿ when n 2. We prove that this condition is sufficient when n=2. Finally, we give some tilings of Z ⁿ when n 3.