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Combinatorics of Tilings Combinatorics of tilings V. Berthé LIAFA-CNRS-Paris-France [email protected] http ://www.liafa.jussieu.fr/~berthe Substitutive tilings and fractal geometry-Guangzhou-July 2010 Tilings Definition Covering of the ambiant space without overlaps using some tiles Tile of Rd a compact set which is the closure of its interior Tiles are also often assumed to be homeomorphic to the ball Tilings of Rd A tiling is a set of tiles such that T d R = T 2T T[ where distinct tiles have non-intersecting interiors each compact set intersects a finite number of tiles in T Labeling Rigid motion Euclidean isometry preserving orientation (composition of translations and rotations) One might distinguish between congruent tiles by labeling them Two tiles are equivalent if they differ by a rigid motion and carry the same label A prototile set is a finite set of inequivalent tiles P Each tile of the tiling is equivalent to a prototile of T P We will work here mainly with translations Bibliography R. Robinson Symbolic Dynamics and Tilings of Rd , AMS Proceedings of Symposia in Applied Mathematics. N. Priebe Frank A primer on substitution tilings of the Euclidean plane – Expo. Math. 26 (2008) 295-326. B. Solomyak Dynamics of self-similar tilings, Ergodic Theory and Dynam. Sys. 17 (1997), 695-738 C. Goodman-Strauss http ://comp.uark.edu/ strauss/ The tiling ListServe Casey Mann [email protected] Tilings Encyclopedia http ://tilings.math.uni-bielefeld.de/E. Harriss, D. Frettlöh R. M. Robinson Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae (1971) Grünbaum et Shephard Tilings and patterns Toward symbolic dynamics We assume that the tiling has finitely many tiles types up translations (‘finite alphabet’) Any two tiles of the same type must be translations of each other Let be a set of prototiles P -patch Finite union of tiles with nonintersecting interiors covering a connectedP set Equivalence Two patches are equivalent if there is a translation between them that matches up equivalent tiles Finite local complexity For any R > 0, there are finitely many patches of diameter less than R up to translation finite number of local patterns Continuous action of Rd t d T ( ) := t; for XT and t R S S − S 2 2 Compacity Suppose XT is a Finite Local Complexity tiling space. Then XT is compact in the tiling metric d d The action of R is continuous dynamical system Tiling spaces as dynamical systems Let be a tiling T d XT := t + t R {− T j 2 g with respect to the ‘local’ topology Two tilings are close if they agree on a large ball around the origin after a small translation local Rd / rigid Zd The tiling metric is complete Tiling spaces as dynamical systems Let be a tiling T d XT := t + t R {− T j 2 g with respect to the ‘local’ topology Two tilings are close if they agree on a large ball around the origin after a small translation local Rd / rigid Zd The tiling metric is complete Continuous action of Rd t d T ( ) := t; for XT and t R S S − S 2 2 Compacity Suppose XT is a Finite Local Complexity tiling space. Then XT is compact in the tiling metric d d The action of R is continuous dynamical system Repetitivity Repetitivity For any patch P, there exists R > 0 such that every ball of radius R contains a translated copy of P cf. almost periodicity, local isomorphism prop., uniform recurrence... Minimality Repetitivity is equivalent with minimality Local isomorphism Two repetitive tilings and 0 are said to be LI if T T ( ) = ( 0) O T O T Construction methods for tilings Local matching rules Tiling substitutions and hierarchical structures Cut-and-project schemes Construction methods for tilings Local matching rules Tiling substitutions and hierarchical structures Cut-and-project schemes E. Harriss [http ://www.mathematicians.org.uk/eoh/] Construction methods for tilings Local matching rules Tiling substitutions and hierarchical structures Cut-and-project schemes E. Harriss [http ://www.mathematicians.org.uk/eoh/] Wang tiles We are given a finite set of unit square tiles located along Z2 with colored vertices common vertices must have the same color we are allowed only translations Domino problem [Wang’61] Can we decide whether a set of Wang tiles tiles the entire plane ? d = 1 Yes More generally, we are given a set of prototiles and a set ∗ of forbidden patches P F ⊂ P Tiling problem Is X = ? nF 6 ; NO Existence of an aperiodic tiling : its tiling space contains no periodic tiling Completion and domino problems Completion problem We are given a set of tiles and a configuration. Is the completion problem decidable? NO [Wang’61] for every Turing machine, one constructs a set of tiles and a configuration for which the configuration can be made into a tiling iff the machine does not halt Conjecture [Wang] Every set of Wang tiles admits a periodic tiling FALSE Tiling problem We are given a set of tiles . Does there exist an algorithm that decides if this set of prototiles tiles the plane ? NO [Berger’66,Robinson’64] Aperiodic tiles Undecidability of the tiling problem [Berger’66] There exist aperiodic sets of tiles Berger ’66 : 20426 tiles Robinson’7 : 6 tiles Robinson’74 : Penrose tiling (2 tiles, reflections and rotations) Culik-Kari’95 : 13, translations and Wang tiles einstein problem Does there exist an aperiodic prototile set consisting of a single tile ? Combinatorial complexity How to measure the complexity of a set of tiles ? If one cannot tile the entire plane : what is the upper bound for the size of configurations that can be formed ? Heesch number If there exists a periodic tiling : what is the dimension of the period lattice ? what is the lowest bound on the number of orbits=equivalence classes of tiles ? isohedral number If there exists no periodic tiling : is it possible to produce them algorithmically ? [C. Goodmann-Strauss, Open questions in tilings] How to construct aperiodic tilings ? Substitutions Substitutions on words and symbolic dynamical systems Substitutions on tiles : inflation/subdivision rules, tilings and point sets An example of a substitution on words : Fibonacci substitution Definition A substitution σ is a morphism of the free monoid Example σ : 1 12; 2 1 7! 7! 1 12 121 12112 12112121 σ1(1) is called the Fibonacci word σ1(1) = 121121211211212 ··· An example of a substitution on words : Fibonacci substitution Definition A substitution σ is a morphism of the free monoid Example σ : 1 12; 2 1 7! 7! 1 12 121 12112 12112121 σ1(1) is called the Fibonacci word σ1(1) = 121121211211212 ··· Substitutions produce tilings of the line A PRIMER ON SUBSTITUTION TILINGS OF THE EUCLIDEAN PLANE 5 1.6. Outline of the paper. Substitutions of constant length have a natural generalization to tilings in higher dimensions, which we introduce in Section 2. These generalizations, which include the well-studied self-similar tilings, rely upon the use of linear expansion maps and are therefore rigidly geometric. We present examples in varying degrees of generality and include a selection of the major results in the field. Extending substitutions of non-constant length to higher dimensions seems to be more difficult, and is the topic of Section 3. To even define what this class contains has been problematic and there is not yet a consensus on the subject. For lack of existing terminology we have decided to call this type of substitution combinatorial as tiles are combined to create the substitutions without any geometric restriction save that they can be iterated without gaps or overlaps, and because in certain cases it is possible to define them in terms of their graph-theoretic structure. In many cases one can transform combinatorial tiling substitutions into geometric ones through a limit process. In Section 4, we will discuss how to do this and what the effects are to the extent that they are known. We conclude the paper by discussing several of the different ways substitution tilings can be studied, and what sorts of questions are of interest. 2. Geometric tiling substitutions Although the idea had been around for several years, self-similar tilings of the plane were given a formal definition and introduced to the wider public by Thurston in a series of four AMS Colloquium lectures, with lecture notes appearing thereafter [59]. Throughout the literature one finds varying degrees of generality and some commonly used restrictions. We make an effort to give precise definitions here, adding remarks which point out some of the differences in usage and in terminology. 2.1. Self-similar tilings: proper inflate-and-subdivide rules. For the moment we assume that the only rigid motions allowed for equivalence of tiles are translations; this follows [59] and [57]. We give the definitions as they appear in [57], which includes that of [59] as a special case. d d Let φ : R R be a linear transformation, diagonalizable over C, that is expanding in the sense that all of its→ eigenvalues are greater than one in modulus. A tiling is called φ-subdividing if T (1) for each tile T , φ(T ) is a union of -tiles, and ∈ T T (2) TSubstitutionsand T ! are equivalent and tiles tilings if and only if φ(T ) and φ(T !) form equivalent patches of tiles in . T A tiling will be called self-affine with expansion map φ if it is φ-subdividing, repetitive, and T has finite localPrinciple complexity. One If takesφ is a similarity the tiling will be called self-similar. For self-similar 2 tilings of R or R = C there is an expansion constant λ for which φ(z)=λz.
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