Combinatorics of tilings

Transcription

1 Combinatorics of tilings V. Berthé LIAFA-CNRS-Paris-France http :// Substitutive tilings and fractal geometry-guangzhou-july 200

2 Tilings Definition Covering of the ambiant space without overlaps using some tiles Tile of R d a compact set which is the closure of its interior Tiles are also often assumed to be homeomorphic to the ball Tilings of R d A tiling T is a set of tiles such that R d = T where T T distinct tiles have non-intersecting interiors each compact set intersects a finite number of tiles in T

3 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 P of inequivalent tiles Each tile of the tiling T is equivalent to a prototile of P We will work here mainly with translations

4 Bibliography R. Robinson Symbolic Dynamics and Tilings of R d, AMS Proceedings of Symposia in Applied Mathematics. N. Priebe Frank A primer on substitution tilings of the Euclidean plane Expo. Math. 26 (2008) B. Solomyak Dynamics of self-similar tilings, Ergodic Theory and Dynam. Sys. 7 (997), C. Goodman-Strauss http ://comp.uark.edu/ strauss/ The tiling ListServe Casey Mann cmann@uttyler.edu 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 (97) Grünbaum et Shephard Tilings and patterns

5 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 P be a set of prototiles P-patch Finite union of tiles with nonintersecting interiors covering a connected 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

6 Tiling spaces as dynamical systems Let T be a tiling X T := { t + T t R d } with respect to the local topology Two tilings are close if they agree on a large ball around the origin after a small translation The tiling metric is complete local R d / rigid Z d

7 Tiling spaces as dynamical systems Let T be a tiling X T := { t + T t R d } with respect to the local topology Two tilings are close if they agree on a large ball around the origin after a small translation The tiling metric is complete Continuous action of R d local R d / rigid Z d T t (S) := S t, for S X T and t R d Compacity Suppose X T is a Finite Local Complexity tiling space. Then X T is compact in the tiling metric d The action of R d is continuous dynamical system

8 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 T and T are said to be LI if O(T ) = O(T )

9 Construction methods for tilings Local matching rules Tiling substitutions and hierarchical structures Cut-and-project schemes

10 Construction methods for tilings Local matching rules Tiling substitutions and hierarchical structures Cut-and-project schemes E. Harriss [http ://

11 Construction methods for tilings Local matching rules Tiling substitutions and hierarchical structures Cut-and-project schemes E. Harriss [http ://

12 Wang tiles We are given a finite set of unit square tiles located along Z 2 with colored vertices common vertices must have the same color we are allowed only translations Domino problem [Wang 6] Can we decide whether a set of Wang tiles tiles the entire plane? d = Yes More generally, we are given a set P of prototiles and a set F P of forbidden patches Tiling problem Is X \F? NO Existence of an aperiodic tiling : its tiling space contains no periodic tiling

13 Completion and domino problems Completion problem We are given a set of tiles and a configuration. Is the completion problem decidable? NO [Wang 6] 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]

14 Aperiodic tiles Undecidability of the tiling problem [Berger 66] There exist aperiodic sets of tiles Berger 66 : tiles Robinson 7 : 6 tiles Robinson 74 : Penrose tiling (2 tiles, reflections and rotations) Culik-Kari 95 : 3, translations and Wang tiles einstein problem Does there exist an aperiodic prototile set consisting of a single tile?

15 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]

16 How to construct aperiodic tilings?

17 Substitutions Substitutions on words and symbolic dynamical systems Substitutions on tiles : inflation/subdivision rules, tilings and point sets

18 An example of a substitution on words : Fibonacci substitution Definition A substitution σ is a morphism of the free monoid Example σ : 2, σ () is called the Fibonacci word σ () =

19 An example of a substitution on words : Fibonacci substitution Definition A substitution σ is a morphism of the free monoid Example σ : 2, σ () is called the Fibonacci word σ () = Substitutions produce tilings of the line

20 for each tile T T, φ(t ) is a union of T -tiles, and T and T Substitutions are equivalent tiles if and only if φ(t ) and φ(t and tilings ) form equivalent patches o in T. ling T will be called self-affine with expansion map φ if it is φ-subdividing, repetitive ite local Principle complexity. OneIf takes φ is a similarity the tiling will be called self-similar. For self-s of R or R 2 a = finite C there number is an of expansion prototiles constant {T, T λ 2,. for.., T which m } φ(z) =λz. rule taking T T to the union of tiles in φ(t ) is called an inflate-and-subdivide rule be tes using the anexpanding expansivemap transformation φ and then decomposes Q (the inflation the image factor into ) the union of tiles o l scale. If Ta is rule φ-subdividing, that allows one thentoitdivide will beeach invariant QT i into under copies this rule, of the therefore we sho -and-subdivide T, Trule 2,.. rather., T m than the tiling itself. The rule given in Figure is an inflate ide rule with φ(z) = 3z. However, the rule given in Figure 3 is not an inflate-and-subd A substitution is a simple production method that allows one to construct infinite tilings using a finite number of tiles ple 5. The L-triomino or chair substitution uses four prototiles, each being an L fo ee unit squares. We have chosen to color the prototiles since they are inequivalent tion. The expansion map is φ(z) = 2z and in Figure 5 we show the substitution of th Example iles. Figure 5. The chair or L-triomino substitution.

21 ple 5. The L-triomino or chair substitution uses four prototiles, each being an L fo The chair tiling ee unit squares. We have chosen to color the prototiles since they are inequivalent tion. The expansion map is φ(z) = 2z and in Figure 5 we show the substitution of th iles. 6 NATALIE PRIEBE FRANK This geometric substitution can be iterated simply by repeated application of φ followed by the appropriate subdivision. Parallel to the symbolic case, we call a tile that has been inflated and subdivided n times a level-n tile. In Figure 6 we show level-n tiles for n =2, 3, and 4. Figure 5. The chair or L-triomino substitution. Figure 6. Level-2, level-3, and level-4 tiles A few important results. One of the earliest results was a characterization of the expansion constant λ C of a self-similar tiling of C.

22 Substitutions and tilings Let φ : R d R d be an expanding linear map Principle One takes a finite numer of prototiles {T, T 2,..., T m } an expansive transformation φ (the inflation factor ) a rule that allows one to divide each φt i into copies of the T, T 2,..., T m A tile-substitution s with expansion φ is a map T i s(t i ), where s(t i ) is a patch made of translates of the prototiles and φ(t i ) = T j s(ti) T j

23 Substitutions and tilings Let φ : R d R d be an expanding linear map Principle One takes a finite numer of prototiles {T, T 2,..., T m } an expansive transformation φ (the inflation factor ) a rule that allows one to divide each φt i into copies of the T, T 2,..., T m A tile-substitution s with expansion φ is a map T i s(t i ), where s(t i ) is a patch made of translates of the prototiles and φ(t i ) = T j s(ti) The substitution is extended to translated of prototiles by and to patches and tilings T j s(x + T i ) = φ(x) + s(t i ) s(p) = {s(t ) T P} Substitution matrix M ij := number of tiles of type i in the subdivision of the tile of type j

24 Substitutions and tilings Let φ : R d R d be an expanding linear map Principle One takes a finite numer of prototiles {T, T 2,..., T m } an expansive transformation φ (the inflation factor ) a rule that allows one to divide each φt i into copies of the T, T 2,..., T m A tile-substitution s with expansion φ is a map T i s(t i ), where s(t i ) is a patch made of translates of the prototiles and φ(t i ) = T j s(ti) cf. Self-affine tiles. See the lectures by C.-K. Lai, T.-M. Tang T j

25 Self-affine tilings The tiling T is said φ-subdividing if for each tile T X T, φ(t ) is a union of T -tiles T and T are equivalent tiles iff φ(t ) and φ(t ) form equivalent patches of tiles in T Self-affine tiling T is φ-subdividing, repetitive and has finite local complexity Substitution matrix M ij := number of tiles of type i in the subdivision of the tile of type j Primitivity implies repetitivity Primitivity+ Invertible φ implies aperiodicity For any tile of a self-affine tiling, Vol(δT ) = 0 [Praggastis]

26 Which expansions are possible? The expansion map φ must have algebraic integers as eigenvalues Substitutive case d = A positive real number λ is the expansion for a self-similar tiling of R iff it is a Perron number Perron-Frobenius theorem only if direction [Lind] if direction Perron number Real algebraic integer which is strictly larger than its other conjugates in modulus

27 Which expansions are possible? The expansion map φ must have algebraic integers as eigenvalues Substitutive case d = A positive real number λ is the expansion for a self-similar tiling of R iff it is a Perron number Perron-Frobenius theorem only if direction [Lind] if direction Perron number Real algebraic integer which is strictly larger than its other conjugates in modulus A self-affine tiling is said self-similar if φ is a similarity Theorem [Thurston-Kenyon] - Complex case d = 2 If a complex number λ is the expansion factor for some self-similar tiling then λ is a complex Perron number (i.e., an algebraic integer which is strictly larger than all its Galois conjugates other than its complex conjugate)

28 Which expansions are possible? The expansion map φ must have algebraic integers as eigenvalues Substitutive case d = A positive real number λ is the expansion for a self-similar tiling of R iff it is a Perron number Perron-Frobenius theorem only if direction [Lind] if direction Perron number Real algebraic integer which is strictly larger than its other conjugates in modulus Theorem [Thurston-Kenyon 200] Let φ be a diagonalizable (over C) expanding linear map of R d and let T be a self-affine tiling of R d with expansion φ. Then every eigenvalue of φ is an algebraic integer if λ is an eigenvalue of φ of multiplicity k and γ is an algebraic conjugate of λ, then either γ < λ, or γ is also an eigenvalue of φ of multiplicity greater than or equal to k

29 What are quasicrystals? Quasicrystals are atomic structures discovered in 84 that are both ordered and nonperiodic [Shechtman-Blech-Gratias-Cahn] Like crystals, quasicrystals produce Bragg diffraction Diffraction comes from regular spacing and long-range order A large family of models of quasicrystals is produced by cut and project schemes : projection of a slice of a higher dimensional lattice The order comes from the lattice structure The nonperiodicity comes from the irrationality of the parameters for the slice

30 Cut and project scheme in Z2

31 A cut-and-project scheme consists of a direct product R k H, k where H is a locally compact abelian group and a lattice L in R k H such that the canonical projections satisfy π 0 (L) is dense in H π 0 : R k H H, π : R k H R k π restricted to L is one-to-one onto its image π (L) R k π R k H π 0 H Λ L Ω

32 A cut-and-project scheme consists of a direct product R k H, k where H is a locally compact abelian group and a lattice L in R k H such that the canonical projections satisfy π 0 (L) is dense in H π 0 : R k H H, π : R k H R k π restricted to L is one-to-one onto its image π (L) R k π R k H π 0 H Λ L Ω G is called the direct space, H is called the internal space H can be a p-adic space as in the case of the chair tiling

33 A cut-and-project scheme consists of a direct product R k H, k where H is a locally compact abelian group and a lattice L in R k H such that the canonical projections π 0 : R k H H, π : R k H R k satisfy π 0 (L) is dense in H π restricted to L is one-to-one onto its image π (L) R k π R k H π 0 H Λ L Ω Choose some compact Ω H with Ω = Ω with zero-measure boundary Λ := {π (x) x L, π 0 (x) Ω} Then Λ is a regular cut-and-project set (or model set)

34 Cut-and-project set R k π R k H π 0 H Λ L Ω Λ := {π (x) x L, π 0 (x) Ω} Λ is a discrete point set in R k, it induces a tiling One gets a set of points of R k which is a Delone set, i.e., a set that is both relatively dense : there exists R > 0 such that any Euclidean ball of R k of radius R contains a point of this set, uniformly discrete : there exists r > 0 such that any ball of radius r contains at most one point of this set. Uniform discreteness comes from the compactness of W and relative denseness comes from its non-empty interior Tilings = Delone multisets

35 Examples R k π one-to-one R k H π 0 dense H Λ L Ω

36 Examples k =, H = R R k π one-to-one R k H π 0 dense H Λ L Ω

37 Examples k = 2, H = R R k π one-to-one R k H π 0 dense H Λ L Ω

38 Examples R k π one-to-one R k H π 0 dense H Λ L Ω k =, H = R 2 = C, Tribonacci substitution V = R πe R 3 π c W = R 2 Λ L Ω Rauzy fractal V : expanding space of M σ, W : contracting space, L : lattice of the eigenvectors, Λ : vertex set of the D substitution tiling Pisot conjecture The vertex set of a Pisot substitution tiling is a regular model set cf. S. Akiyama s lecture

39 Local mappings Local mapping= sliding block code in symbolic dynamics A continuous mapping between two tiling spaces Q : X Y is a local mapping if there is an r > 0 such that for all x X, Q(x)[{0}] depends only on x[b(0, r)] If Q is invertible, x and Q(x) are said to be mutually locally derivable MLD implies topological conjugacy between the tiling spaces The converse is false [Petersen, Radin-Sadun]

40 Penrose substitutions A pseudo-self-affine version/ Rhombic penrose tiles A self-affine version/triangular Penrose tiles

41 Pseudo-self-affine tilings Let φ : R d R d be an expanding linear map A repetitive FLC tiling is a pseudo-self-affine tiling if φt LD T Theorem [Priebe-Solomyak, Solomyak] Let T be a pseudo-self-affine tiling of R d with expansion φ Then for any k sufficiently large, there exists a tiling T which is self-affine with expansion φ k such that T is MLD with T

42 Penrose substitutions A pseudo-self-affine version/ Rhombic penrose tiles A self-affine version/triangular Penrose tiles

43 MLD [F. Gähler MLD relations of Pisot substitution Tilings (ArXiv)] Two tilings are MLD (mutual local derivability) if one can be reconstructed from the other one in a local way, and vice versa Two model set tilings are MLD if and only if the window of one can be constructed by finite unions and intersections of lattice translates of the window of the other, and vice versa a cb, b c, c cab a bc, b c, c cba

44 MLD σ : a cb, b c, c cab σ : a bc, b c, c cba Let ρ : a bab, b b, c c] σ = ρ σ ρ ab ba a=red, b=green, c= blue

45 Rauzy fractal It is a solution of a GIFS

46 Back to matching rules and soficity Theorem [Mozes Goodman-Strauss] Every good substitution tiling of R d, d >, can be enforced with matching rules See E. Harriss, X. Bressaud, M. Sablik s lectures Theorem [Bressaud-Sablik] The Rauzy fractal can be enforced with finite matching rules effective tilings : forbidden patterns are given by a Turing machine

47 represent in a tiling. The substitution rule also specifies how to substitute the labeled edges and facets so that we know how to connect the vertex substitutions contained in certain labeled graphs. (The Combinatorial need to specify graphsubstitutions facets and not just edges is illustrated in an example in []). Defining a tiling substitution rule this way is quite tricky since most labeled graphs do not represent the dual graph of a tiling. This interplay between combinatorics and geometry is where the technicalities come in to the formal definitions in the literature. Example. The tiling substitution of Figure 2, introduced in [], is based on a variation of the one-dimensional Rauzy substitution σ() = 2, σ(2) = 3, σ(3) =. Figure 2 is obviously NATALIE PRIEBE FRANK Figure 2. A two-dimensional substitution 3 based on the Rauzy 3 one-dimensional substitution. not enough information to iterate the substitution, 2 so we specify how to substitute the important 2 adjacencies in Figure 22. This is3enough []: there are 3 no ambiguities when substituting other adjacencies, and facet substitutions do not include any new information. We show a few iterates of the tile of type in Figure 23, starting with the level-2 tile of type. The fact that this substitution Figure 22. How to substitute important adjacencies for the Rauzy substitution. rule can be extended to an infinite tiling of the plane is proved using noncombinatorial methods in []; a combinatorial proof of existence would 2 be welcomed Non-constructive 3 tiling 3 substitutions. 3 When 3 trying to 3 make up new3 examples 3 of combinatorial tiling substitutions it is easy to create examples that3fail to be constructive. 3 The problem 2 2 arises in the substitution of adjacencies: it may happen that no finite labelset can be chosen to describe all adjacencies sufficiently to know how to substitute them. There is evidence to suggest that this sort of example Figure can 23. arise A fewwhen iterates the of the constant Rauzy two-dimensional which best approximates substitution. the linear growth of blocks is Definition not a Pisot 3.. number. A (non-constructive) The author tilingissubstitution not awareonofaany finiteformal prototile definition set P is acontaining set of this n 2 2 2

48 in []). Defining a tiling rule this way is quite tricky since most labeled graphs do not represent Combinatorial the dual graph ofsubstitutions a tiling. This interplay between combinatorics and geometry is where the technicalities come in to the formal definitions in the literature. Example. The tiling substitution of Figure 2, introduced in [], is based on a variation of the one-dimensional Rauzy substitution σ() = 2, σ(2) = 3, σ(3) =. Figure 2 is obviously 4 NATALIE PRIEBE FRANK 2 4 NATALIE 2 PRIEBE3 FRANK Figure 2. A two-dimensional substitution 3 based on the Rauzy 3 one-dimensional substitution. 2 2 not enough information to iterate3 the substitution, 2 so 3we specify how to substitute the important adjacencies in Figure 22. This is3enough 2 []: there are 3 no ambiguities when substituting other adjacencies, and facet substitutions do not include any new information. We show a few iterates of the tile of typefigure in Figure 22. How 23, tostarting substitute with important the level-2 adjacencies tile offor type the. Rauzy Thesubstitution. fact that this substitution Figure 22. How substitute important adjacencies for the Rauzy substitution. rule can be extended to an infinite tiling 2 of the plane 2 2 is proved2 using 2 noncombinatorial methods in []; a combinatorial 2 proof2of existence 2 would 2 be 3 welcomed Non-constructive 3 tiling 3 substitutions. 3 When 3 trying 2to 3make up new 23 examples 3 of combinatorial tiling substitutions it is easy to create examples that3fail to be constructive. 3 The problem arises in the substitution of adjacencies: it may happen that no finite labelset can be chosen to describe all adjacencies Figuresufficiently 23. A few iterates to know of the howrauzy to substitute two-dimensional them. substitution. There is evidence to suggest that this sort of example Figure can 23. arise A fewwhen iterates the of the constant Rauzy two-dimensional which best approximates substitution. the linear growth of blocks Definition is Definition not a3.. Pisot A 3.. number. (non-constructive) A (non-constructive) The author tiling substitution tilingissubstitution not aware on onof a finite aany finiteformal prototile prototile definition set P is a set P is acontaining set of set of this nonempty, connected patches S = {S group andnonempty, so proposes connected the patches following S = definition, n (p) {S n :p P and n, 2,...} satisfying the following: (p) :p Pwhich and n works, 2,...} directly satisfying with the following: the tiling and does not () For each prototile p P and tile t S involve dual () graphs. (p), For each prototile p P and tile t and for each integer n 2, 3,..., there are rigid (p), and for each integer n 2, 3,..., there are rigid motions g(p, n, t) :R d motions g(p, n, t) :R d R d such R d that S n (p) such that S n = (p) = g(p, n, t) ( g(p, n, t) ( S n S n (t) ) (t) ), where, where t S (p)

49 rule that corresponds to t by setting the technicalities come in to the formal definitions in the literature. t Combinatorial substitutions = lim Example. The Intiling Figure 28substitution we compare level-5 tiles of from Figure the DPV 2, (left) introduced and the self-similarin tiling [], (right). is based on a variation of the one-dimensional Rauzy substitution σ() = 2, σ(2) = 3, σ(3) =. Figure 2 is obviously 2 n φ n (supp(s n (t)) NATALIE PRIEBE FRANK Figure 2. A two-dimensional substitution 3 based on the Rauzy 3 one-dimensional substitution. not enough information to iterate the substitution, so we specify how to substitute the important Figure 28. Comparing the 2 DPV with the SST of Example 5. 2 adjacencies in Figure 22. This is3enough []: there are 3 no ambiguities when substituting other adjacencies, and facet Example substitutions 6. The self-similar do tiling notassociated include withany the Rauzy newtwo-dimensional information. substitution We of show a few iterates of Example has as its volume expansion the largest root of the polynomial x the tile of type in Figure 23, starting with the level-2 tile of type 3 x. 2. The three tile types obtained by the replace-and-rescale method are shown in Figure 29, compared The fact with athat this substitution Figure large iteration 22. How of the tosubstituton. substitute important adjacencies for the Rauzy substitution. rule can be extended to an infinite tiling of the plane is proved using noncombinatorial methods in []; a combinatorial proof of existence would be welcomed Non-constructive 3 tiling 3 substitutions When 3 trying to 3 make up new3 examples 3 of combinatorial tiling substitutions it is easy3 to2 create 2 examples that3fail to be constructive. 3 The problem arises in the substitution of adjacencies: it may happen that no finite label set can be chosen to describe all adjacencies sufficiently to 2 3 know 3 how to substitute them. There is evidence to suggest that this sort of example Figure3 can arise A3 few 2 when iterates the of 2the constant Rauzy two-dimensional which best approximates substitution the linear growth of blocks is Definition not a Pisot 3.. number. 3 A (non-constructive) The author 2 tilingissubstitution not awareonofaany finiteformal prototile definition set P is acontaining set of this 3 group andnonempty, so proposes connected the patches following S = definition, {S n (p) :p Pwhich and n works, 2,...} directly satisfying with the following: the tiling and does not involve dual () graphs. For each prototile p P and tile t S (p), and for each integer n 2, 3,..., there are rigid Figure 29. motions g(p, n, t) :R d A comparison R d of an iterate such that S n with the (p) = limiting self-similar g(p, n, t) ( tiles. S n (t) ), where t S (p) (2) for any t t in S (p), the patches g(p, n, t) ( S n (t) ) and g(p, n, t ) ( S n (t ) ) intersect at 2 2 2