V. Berthé - IRISA · Combinatorics of tilings V. Berthé LIAFA-CNRS-Paris-France...

$: V. Berthé - IRISA · Combinatorics of tilings V. Berthé LIAFA-CNRS-Paris-France berthe@liafa.jussieu.fr http ://berthe Substitutive tilings and fractal geometry ...$
Combinatorics of tilings

V. Berthé

[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 usingsome 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 T is a set of tiles such that

Rd =⋃

T∈T

T

wheredistinct tiles have non-intersecting interiorseach compact set intersects a finite number of tiles in T

Labeling

Rigid motion Euclidean isometry preserving orientation (compositionof translations and rotations)

One might distinguish between congruent tiles by labeling themTwo tiles are equivalent if they differ by a rigid motion and carrythe same labelA prototile set is a finite set P of inequivalent tilesEach tile of the tiling T is equivalent to a prototile of P

We will work here mainly with translations

Bibliography

R. Robinson Symbolic Dynamics and Tilings of Rd , AMSProceedings of Symposia in Applied Mathematics.N. Priebe Frank A primer on substitution tilings of the Euclideanplane – Expo. Math. 26 (2008) 295-326.B. Solomyak Dynamics of self-similar tilings, Ergodic Theory andDynam. Sys. 17 (1997), 695-738C. Goodman-Strauss http ://comp.uark.edu/ strauss/The tiling ListServe Casey Mann [email protected] Encyclopedia http ://tilings.math.uni-bielefeld.de/ E.Harriss, D. FrettlöhR. M. Robinson Undecidability and nonperiodicity for tilings of theplane, Inventiones Mathematicae (1971)Grünbaum et Shephard Tilings and patterns

Toward symbolic dynamics

We assume that the tiling has finitely many tiles types uptranslations (‘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 aconnected set

Equivalence Two patches are equivalent if there is a translationbetween them that matches up equivalent tiles

Finite local complexity For any R > 0, there are finitely many patchesof diameter less than R up to translation

finite number of local patterns

Tiling spaces as dynamical systems

Let T be a tilingXT := −t + T | t ∈ Rd

with respect to the ‘local’ topologyTwo tilings are close if they agree on a large ball around theorigin after a small translation

local Rd / rigid Zd

The tiling metric is complete

Continuous action of Rd

T t (S) := S − t , for S ∈ XT and t ∈ Rd

Compacity Suppose XT is a Finite Local Complexity tiling space.Then XT is compact in the tiling metric d

The action of Rd is continuous dynamical system

Tiling spaces as dynamical systems

Let T be a tilingXT := −t + T | t ∈ Rd

with respect to the ‘local’ topologyTwo tilings are close if they agree on a large ball around theorigin after a small translation

local Rd / rigid Zd

The tiling metric is completeContinuous action of Rd

T t (S) := S − t , for S ∈ XT and t ∈ Rd

Compacity Suppose XT is a Finite Local Complexity tiling space.Then XT is compact in the tiling metric d

The action of Rd is continuous dynamical system

Repetitivity

Repetitivity For any patch P, there exists R > 0 such that every ball ofradius 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 ′)

Construction methods for tilings

Local matching rulesTiling substitutions and hierarchical structuresCut-and-project schemes

Construction methods for tilingsLocal matching rulesTiling substitutions and hierarchical structuresCut-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 withcolored verticescommon vertices must have the same colorwe are allowed only translations

Domino problem [Wang’61] Can we decide whether a set of Wangtiles tiles the entire plane ?

d = 1 Yes

More generally, we are given a set P of prototiles and a set F ⊂ P∗ offorbidden patches

Tiling problem Is X\F 6= ∅?

NO Existence of an aperiodic tiling : its tiling space contains noperiodic tiling

Completion and domino problems

Completion problem We are given a set of tiles and a configuration. Isthe completion problem decidable ?

NO [Wang’61] for every Turing machine, one constructs a set of tilesand a configuration for which the configuration can be made into atiling 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 analgorithm 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 tilesBerger ’66 : 20426 tilesRobinson’7 : 6 tilesRobinson’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 consistingof 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 forthe size of configurations that can be formed ? Heesch numberIf there exists a periodic tiling : what is the dimension of theperiod lattice ? what is the lowest bound on the number oforbits=equivalence classes of tiles ? isohedral numberIf there exists no periodic tiling : is it possible to produce themalgorithmically ?

[C. Goodmann-Strauss, Open questions in tilings]

How to construct aperiodic tilings ?

Substitutions

Substitutions on words and symbolic dynamical systemsSubstitutions on tiles : inflation/subdivision rules, tilings andpoint sets

An example of a substitution on words : Fibonaccisubstitution

Definition A substitution σ is a morphism of the free monoid

Example

σ : 1 7→ 12, 2 7→ 1

1121211211212112121

σ∞(1) is called the Fibonacci word

σ∞(1) = 121121211211212 · · ·

An example of a substitution on words : Fibonaccisubstitution

Definition A substitution σ is a morphism of the free monoid

Example

σ : 1 7→ 12, 2 7→ 1

1121211211212112121

σ∞(1) is called the Fibonacci word

σ∞(1) = 121121211211212 · · ·Substitutions produce tilings of the line

Substitutions and tilings

Principle One takesa finite number of prototiles T1,T2, . . . ,Tman expansive transformation Q (the inflation factor )a rule that allows one to divide each QTi into copies of theT1,T2, . . . ,Tm

A substitution is a simple production method that allows one toconstruct infinite tilings using a finite number of tiles

Example

A PRIMER ON SUBSTITUTION TILINGS OF THE EUCLIDEAN PLANE 5

1.6. Outline of the paper. Substitutions of constant length have a natural generalization totilings in higher dimensions, which we introduce in Section 2. These generalizations, which includethe well-studied self-similar tilings, rely upon the use of linear expansion maps and are thereforerigidly geometric. We present examples in varying degrees of generality and include a selection ofthe major results in the field.

Extending substitutions of non-constant length to higher dimensions seems to be more di!cult,and is the topic of Section 3. To even define what this class contains has been problematic andthere is not yet a consensus on the subject. For lack of existing terminology we have decided to callthis type of substitution combinatorial as tiles are combined to create the substitutions withoutany geometric restriction save that they can be iterated without gaps or overlaps, and because incertain 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 througha limit process. In Section 4, we will discuss how to do this and what the e"ects are to the extentthat they are known. We conclude the paper by discussing several of the di"erent ways substitutiontilings 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 aformal definition and introduced to the wider public by Thurston in a series of four AMS Colloquiumlectures, with lecture notes appearing thereafter [59]. Throughout the literature one finds varyingdegrees of generality and some commonly used restrictions. We make an e"ort to give precisedefinitions here, adding remarks which point out some of the di"erences in usage and in terminology.

2.1. Self-similar tilings: proper inflate-and-subdivide rules. For the moment we assumethat 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.

Let ! : Rd ! Rd be a linear transformation, diagonalizable over C, that is expanding in the sensethat all of its eigenvalues are greater than one in modulus. A tiling T is called !-subdividing if

(1) for each tile T " T , !(T ) is a union of T -tiles, and(2) T and T ! are equivalent tiles if and only if !(T ) and !(T !) form equivalent patches of tiles

in T .

A tiling T will be called self-a!ne with expansion map ! if it is !-subdividing, repetitive, andhas finite local complexity. If ! is a similarity the tiling will be called self-similar. For self-similartilings of R or R2 #= C there is an expansion constant " for which !(z) = "z.

The rule taking T " T to the union of tiles in !(T ) is called an inflate-and-subdivide rule becauseit inflates using the expanding map ! and then decomposes the image into the union of tiles on theoriginal scale. If T is !-subdividing, then it will be invariant under this rule, therefore we show theinflate-and-subdivide rule rather than the tiling itself. The rule given in Figure 1 is an inflate-and-subdivide rule with !(z) = 3z. However, the rule given in Figure 3 is not an inflate-and-subdividerule.

Example 5. The “L-triomino” or “chair” substitution uses four prototiles, each being an L formedby three unit squares. We have chosen to color the prototiles since they are inequivalent up totranslation. The expansion map is !(z) = 2z and in Figure 5 we show the substitution of the fourprototiles.

Figure 5. The “chair” or “L-triomino” substitution.

The chair tiling


1.6. Outline of the paper. Substitutions of constant length have a natural generalization totilings in higher dimensions, which we introduce in Section 2. These generalizations, which includethe well-studied self-similar tilings, rely upon the use of linear expansion maps and are thereforerigidly geometric. We present examples in varying degrees of generality and include a selection ofthe major results in the field.

Extending substitutions of non-constant length to higher dimensions seems to be more di!cult,and is the topic of Section 3. To even define what this class contains has been problematic andthere is not yet a consensus on the subject. For lack of existing terminology we have decided to callthis type of substitution combinatorial as tiles are combined to create the substitutions withoutany geometric restriction save that they can be iterated without gaps or overlaps, and because incertain 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 througha limit process. In Section 4, we will discuss how to do this and what the e"ects are to the extentthat they are known. We conclude the paper by discussing several of the di"erent ways substitutiontilings 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 aformal definition and introduced to the wider public by Thurston in a series of four AMS Colloquiumlectures, with lecture notes appearing thereafter [59]. Throughout the literature one finds varyingdegrees of generality and some commonly used restrictions. We make an e"ort to give precisedefinitions here, adding remarks which point out some of the di"erences in usage and in terminology.

2.1. Self-similar tilings: proper inflate-and-subdivide rules. For the moment we assumethat 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.

Let ! : Rd ! Rd be a linear transformation, diagonalizable over C, that is expanding in the sensethat all of its eigenvalues are greater than one in modulus. A tiling T is called !-subdividing if

(1) for each tile T " T , !(T ) is a union of T -tiles, and(2) T and T ! are equivalent tiles if and only if !(T ) and !(T !) form equivalent patches of tiles

in T .

A tiling T will be called self-a!ne with expansion map ! if it is !-subdividing, repetitive, andhas finite local complexity. If ! is a similarity the tiling will be called self-similar. For self-similartilings of R or R2 #= C there is an expansion constant " for which !(z) = "z.

The rule taking T " T to the union of tiles in !(T ) is called an inflate-and-subdivide rule becauseit inflates using the expanding map ! and then decomposes the image into the union of tiles on theoriginal scale. If T is !-subdividing, then it will be invariant under this rule, therefore we show theinflate-and-subdivide rule rather than the tiling itself. The rule given in Figure 1 is an inflate-and-subdivide rule with !(z) = 3z. However, the rule given in Figure 3 is not an inflate-and-subdividerule.

Example 5. The “L-triomino” or “chair” substitution uses four prototiles, each being an L formedby three unit squares. We have chosen to color the prototiles since they are inequivalent up totranslation. The expansion map is !(z) = 2z and in Figure 5 we show the substitution of the fourprototiles.

Figure 5. The “chair” or “L-triomino” substitution.

6 NATALIE PRIEBE FRANK

This geometric substitution can be iterated simply by repeated application of ! followed by theappropriate subdivision. Parallel to the symbolic case, we call a tile that has been inflated andsubdivided n times a level-n tile. In Figure 6 we show level-n tiles for n = 2, 3, and 4.

Figure 6. Level-2, level-3, and level-4 tiles.

2.2. A few important results. One of the earliest results was a characterization of the expansionconstant " ! C of a self-similar tiling of C.

Theorem 2.1. (Thurston [59], Kenyon [27]) A complex number " is the expansion constant forsome self-similar tiling if and only if " is an algebraic integer which is strictly larger than all itsGalois conjugates other than its complex conjugate.

The forward direction was proved by Thurston and the reverse direction by Kenyon. In [28],

Kenyon extends the result to self-a!ne tilings of Rd in terms of eigenvalues of the expansion map.In the study of substitutions, from one-dimensional symbolic substitutions to very general tiling

substitutions, the substitution matrix is an indispensable tool. (This matrix has also been calledthe “transition”, “composition”, “subdivision”, or even “abelianization” matrix). Suppose that theprototile set (or alphabet) has m elements labeled by 1, 2, ...,m . The substitution matrix M isthe m"m matrix with entries given by

(1) Mij = the number of tiles of type i in the substitution of the tile of type j.

For example, the substitution in Example 3 has substitution matrix M =

!9 10 8

"when we

label a = 1 and b = 2. If an initial configuration of tiles has n white tiles and m blue tiles, thenM [n m]T is the number of white and blue tiles after one application of the substitution.

Since the substitution matrix is always an integer matrix with nonnegative entries, Perron-Frobenius theory is relevant (see for example [29, 57]). The results we need require M to beirreducible: for every i, j ! 1, 2, ...,m there exists an n such that (Mn)ij > 0. Among otherthings, the Perron-Frobenius theorem states that if M is irreducible, then the largest eigenvaluewill be a positive real number that is larger in modulus than any of the other eigenvalues of thematrix. This eigenvalue is unique, has multiplicity one, and is called the Perron eigenvalue of thematrix.

Primitivity, a special case of irreducibility, is particularly important. A matrix M is primitiveif there is an n > 0 such that Mn has strictly positive entries. Primitivity of M means if onesubstitutes any tile (or letter) a fixed number of times, one will see all of the other tiles (or letters).


Let φ : Rd → Rd be an expanding linear map

Principle One takesa finite numer of prototiles T1,T2, . . . ,Tman expansive transformation φ (the inflation factor )a rule that allows one to divide each φTi into copies of theT1,T2, . . . ,Tm

A tile-substitution s with expansion φ is a map Ti 7→ s(Ti ), where s(Ti )is a patch made of translates of the prototiles and

φ(Ti ) =⋃

Tj∈s(Ti)

Tj

Substitutions and tilingsLet φ : Rd → Rd be an expanding linear map



φ(Ti ) =⋃

Tj∈s(Ti)

Tj

The substitution is extended to translated of prototiles by

s(x + Ti ) = φ(x) + s(Ti )

and to patches and tilings

s(P) =⋃s(T ) | T ∈ P

Substitution matrix Mij := number of tiles of type i in the subdivision ofthe tile of type j


Let φ : Rd → Rd be an expanding linear map



φ(Ti ) =⋃

Tj∈s(Ti)

Tj

cf. Self-affine tiles. See the lectures by C.-K. Lai, T.-M. Tang

Self-affine tilings

The tiling T is said φ-subdividing iffor each tile T ∈ XT , φ(T ) is a union of T -tilesT and T ′ are equivalent tiles iff φ(T ) and φ(T ′) form equivalentpatches of tiles in T

Self-affine tiling T is φ-subdividing, repetitive and has finite localcomplexitySubstitution matrix Mij := number of tiles of type i in the subdivision ofthe tile of type j

Primitivity implies repetitivity

Primitivity+ Invertible φ implies aperiodicity

For any tile of a self-affine tiling, Vol(δT ) = 0 [Praggastis]

Which expansions are possible ?

The expansion map φ must have algebraic integers as eigenvalues

Substitutive case d = 1 A positive real number λ is the expansion fora 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 itsother conjugates in modulus






A self-affine tiling is said self-similar if φ is a similarity

Theorem [Thurston-Kenyon] - Complex case d = 2If a complex number λ is the expansion factor for some self-similartiling then λ is a complex Perron number (i.e., an algebraic integerwhich is strictly larger than all its Galois conjugates other than itscomplex conjugate)






Theorem [Thurston-Kenyon’2010] Let φ be a diagonalizable (over C)expanding linear map of Rd and let T be a self-affine tiling of Rd withexpansion φ. Then

every eigenvalue of φ is an algebraic integerif λ is an eigenvalue of φ of multiplicity k and γ is an algebraicconjugate of λ, then either |γ| < |λ|, or γ is also an eigenvalue ofφ of multiplicity greater than or equal to k

What are quasicrystals ?

Quasicrystals are atomic structures discovered in 84 that are bothordered and nonperiodic [Shechtman-Blech-Gratias-Cahn]

Like crystals, quasicrystals produce Bragg diffractionDiffraction comes from regular spacing and long-range order

A large family of models of quasicrystals is produced by cut andproject schemes :

projection of a slice of a higher dimensional lattice

The order comes from the lattice structureThe nonperiodicity comes from the irrationality of the parametersfor the slice

Cut and project scheme in Z2

A cut-and-project scheme consists of

a direct product Rk × H, k ≥ 1where H is a locally compact abelian groupand a lattice L in Rk × H

such that the canonical projections

π0 : Rk × H → H, π1 : Rk × H → Rk

satisfyπ0(L) is dense in Hπ1 restricted to L is one-to-one onto its image π1(L)

Rk π1←− Rk × H π0−→ H∪ ∪ ∪Λ L Ω




π0 : Rk × H → H, π1 : Rk × H → Rk


Rk π1←− Rk × H π0−→ H∪ ∪ ∪Λ L Ω

G is called the direct space, H is called the internal spaceH can be a p-adic space as in the case of the chair tiling




π0 : Rk × H → H, π1 : Rk × H → Rk


Rk π1←− Rk × H π0−→ H∪ ∪ ∪Λ L Ω

Choose some compact Ω ⊂ H with Ω = Ω with zero-measureboundary

Λ := π1(x) | x ∈ L, π0(x) ∈ ΩThen Λ is a regular cut-and-project set (or model set)

Cut-and-project set

Rk π1←− Rk × H π0−→ H∪ ∪ ∪Λ L Ω

Λ := π1(x) | x ∈ L, π0(x) ∈ ΩΛ is a discrete point set in Rk , it induces a tiling

One gets a set of points of Rk which is a Delone set, i.e., a set that isboth

relatively dense : there exists R > 0 such that any Euclidean ballof Rk of radius R contains a point of this set,uniformly discrete : there exists r > 0 such that any ball of radiusr contains at most one point of this set.

Uniform discreteness comes from the compactness of W and relativedenseness comes from its non-empty interior

Tilings ∼= Delone multisets

Examples

Rk π1 one-to-one←−−−−−−−−−− Rk × H π0 dense−−−−−−→ H∪ ∪ ∪Λ L Ω

Examples


k = 1, H = R

Examples


k = 2, H = R

Examples


k = 1, H = R2 ∼= C, Tribonacci substitution

V ∼= R πe←− R3 πc−→ W ∼= R2

∪ ∪ ∪Λ L Ω Rauzy fractal

V : expanding space of Mσ, W : contracting space, L : lattice of theeigenvectors, Λ : vertex set of the 1D substitution tiling

Pisot conjecture The vertex set of a Pisot substitution tiling is aregular model set

cf. S. Akiyama’s lecture

Local mappings

Local mapping= sliding block code in symbolic dynamics

A continuous mapping between two tiling spaces Q : X → Y is alocal 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 locallyderivableMLD implies topological conjugacy between the tiling spacesThe converse is false [Petersen, Radin-Sadun]

Penrose substitutions

A pseudo-self-affine version/ Rhombic penrose tiles

A self-affine version/Triangular Penrose tiles

Pseudo-self-affine tilings

Let φ : Rd → Rd be an expanding linear mapA repetitive FLC tiling is a pseudo-self-affine tiling if

φT →LD T

Theorem [Priebe-Solomyak, Solomyak] Let T be a pseudo-self-affinetiling of Rd with expansion φ

Then for any k sufficiently large, there exists a tiling T ′which is self-affine with expansion φk such thatT is MLD with T ′

Penrose substitutions

A pseudo-self-affine version/ Rhombic penrose tiles

A self-affine version/Triangular Penrose tiles

MLD [F. Gähler MLD relations of Pisot substitution Tilings (ArXiv)]

Two tilings are MLD (mutual local derivability) if one can bereconstructed from the other one in a local way, and vice versaTwo model set tilings are MLD if and only if the window of onecan be constructed by finite unions and intersections of latticetranslates of the window of the other, and vice versa

a→ cb, b → c, c → cab a→ bc, b → c, c → cba

MLD

σ1 : a→ cb, b → c, c → cab σ′1 : a→ bc, b → c, c → cba

Letρ1 : a→ bab−1, b → b, c → c]

σ1 = ρ−11 σ′1 ρ1

ab ba

a=red, b=green, c= blue

Rauzy fractal

It is a solution of a GIFS

Back to matching rules and soficity

Theorem [Mozes–Goodman-Strauss] Every ‘good’ substitution tilingof Rd , d > 1, can be enforced with matching rules

See E. Harriss, X. Bressaud, M. Sablik’s lectures

Theorem [Bressaud-Sablik] The Rauzy fractal can be enforced withfinite matching rules

effective tilings : forbidden patterns are given by a Turing machine

Combinatorial substitutions


them in the tiling. For example, consider two horizontally adjacent tiles of type 1. That adjacencyappears twice in the level-3 tile of type 1 and we have circled them on the left side of Figure20; what happens under substitution is shown on the right of the same figure. For this graphsubstitution, the problem can be handled by relabeling the edges and facets of the graph in termsof the immediate configuration of tiles the edge or facet is contained in.

1

3

4 24

3

1

12

1

32

1

2

11 3

41

1

2

2

42

1

12

1

3

3

1

1

22

4

1

13

2

13

1

1

1 3

2

4

4

1

3

2

3

34 2

1

1

3 1

24 1

2

3

3

31

4

1

2

2

13

31

3 1

24

1

1

2

31

3

4 2

4

1

Figure 20. The substitution of an adjacent pair of tiles depends on its context.

The basic idea of a constructive combinatorial substitution for tilings as it appears in [11] andin a similar form in [1] is this. Given a labeled vertex set V representing the prototile types, amap from V to the set of nonempty labeled graphs on V is the basis for the substitution rule. Theedges and facets of these graphs are labeled to give information about the types of adjacencies theyrepresent in a tiling. The substitution rule also specifies how to substitute the labeled edges andfacets so that we know how to connect the vertex substitutions contained in certain labeled graphs.(The need to specify graph substitutions on facets and not just edges is illustrated in an examplein [11]). Defining a tiling substitution rule this way is quite tricky since most labeled graphs do notrepresent the dual graph of a tiling. This interplay between combinatorics and geometry is wherethe technicalities come in to the formal definitions in the literature.

Example 11. The tiling substitution of Figure 21, introduced in [1], is based on a variation ofthe one-dimensional “Rauzy substitution” !(1) = 1 2, !(2) = 3, !(3) = 1. Figure 21 is obviously

111

22 3 3

Figure 21. A two-dimensional substitution based on the Rauzy one-dimensional substitution.

not enough information to iterate the substitution, so we specify how to substitute the “importantadjacencies” in Figure 22. This is enough [1]: there are no ambiguities when substituting otheradjacencies, and facet substitutions do not include any new information. We show a few iterates ofthe tile of type 1 in Figure 23, starting with the level-2 tile of type 1. The fact that this substitutionrule can be extended to an infinite tiling of the plane is proved using noncombinatorial methods in[1]; a combinatorial proof of existence would be welcomed.

3.2. Non-constructive tiling substitutions. When trying to make up new examples of combi-natorial tiling substitutions it is easy to create examples that fail to be constructive. The problemarises in the substitution of adjacencies: it may happen that no finite label set can be chosen todescribe all adjacencies su!ciently to know how to substitute them. There is evidence to suggestthat this sort of example can arise when the constant which best approximates the linear growthof blocks is not a Pisot number. The author is not aware of any formal definition containing thisgroup and so proposes the following definition, which works directly with the tiling and does notinvolve dual graphs.


3

1

1

2

1

2

31

2 1

2

1

1

3 1

1

1

21

1

2

3

1

2

2 1

Figure 22. How to substitute important adjacencies for the Rauzy substitution.

1

2

3

1

1

2

1

2

3

1

2

3

1

1

2

3

1

1

2

1

2

3

1

1

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1

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3 1

2

3

1

1

2

3

1

1

2

1

2

3

1

2

3

1

1

2

1

2

3

1

2

3

1

Figure 23. A few iterates of the Rauzy two-dimensional substitution.

Definition 3.1. A (non-constructive) tiling substitution on a finite prototile set P is a set ofnonempty, connected patches S = Sn(p) : p ! P and n ! 1, 2, ... satisfying the following:

(1) For each prototile p ! P and tile t ! S1(p), and for each integer n ! 2, 3, ..., there are rigid

motions g(p, n, t) : Rd " Rd such that Sn(p) =!

t!S1(p)

g(p, n, t)"Sn"1(t)

#, where

(2) for any t #= t# in S1(p), the patches g(p, n, t)"Sn"1(t)

#and g(p, n, t#)

"Sn"1(t#)

#intersect at

most along their boundaries.

We say a tiling T is admitted by the substitution S if every patch in T appears as a subpatchof some element of S. This very general definition is satisfied by every substitution appearing inthis paper except the Penrose substitution on rhombi and those generalized pinwheel tilings thatdo not allow a finite number of tile sizes.

The Rauzy substitution of Example 11 has a particularly e!cient representation by this defi-nition. The patches S1(p) are given to the right of the arrows in Figure 21, with all lower rightcorners at the origin. Now, Sn(2) = Sn"1(3) and Sn(3) = Sn"1(1), so g(2, n, 3) and g(3, n, 1) arethe identity map. We find Sn(1) = Sn"1(1) $ g(1, n, 2)

"Sn"1(2)

#, so g(1, n, 1) is the identity map

and all we have left to figure out is the formula for g(1, n, 2). It turns out that g(1, n, 2) is transla-tion by a vector !vn that can be computed recursively. Let !v0 = (0, 0),!v1 = (0, 1) and !v2 = (%1, 0);for n & 3 we have that !vn = !vn"3 % !vn"2.

The Fibonacci DPV of Example 10 also has a relatively simple formuation in terms of Definition3.1. The side lengths of level-n tiles are given recursively, and the placement of the level-(n % 1)tiles to create level-n tiles depends only on these side lengths. Thus the translations g(p, n, t) arecomputable recursively as well. The next example is also a DPV, but it cannot be defined in termsof dual graphs and is non-constructive. We encourage the reader to think about how to write upDefinition 3.1 in this case.

Example 12. Consider a DPV arising from a one-dimensional substitution a" abbb, b" a. Fromthe direct product of this substitution with itself, we choose only to rearrange the substitution ofthe type-1 tile as in Figure 24.

The substitution matrix of this one-dimensional substitution has Perron eigenvalue " = (1 +'13/2), which is not a Pisot number: its algebraic conjugate "2 = (1%

'13/2) is larger than one

in modulus. Using analysis similar to what we will see in Section 4.1, we find that constants times




1

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111

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t!S1(p)

g(p, n, t)"Sn"1(t)

#, where


#and g(p, n, t#)

"Sn"1(t#)

#intersect at




"Sn"1(2)









3

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t!S1(p)

g(p, n, t)"Sn"1(t)

#, where


#and g(p, n, t#)

"Sn"1(t#)

#intersect at




"Sn"1(2)











1

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4 24

3

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1

32

1

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11 3

41

1

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42

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1 3

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4

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3

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34 2

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3 1

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31

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2

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31

3 1

24

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31

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1




111

22 3 3





3

1

1

2

1

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31

2 1

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2 1


1

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t!S1(p)

g(p, n, t)"Sn"1(t)

#, where


#and g(p, n, t#)

"Sn"1(t#)

#intersect at




"Sn"1(2)









substitution are supported on rectangles with side lengths given by either the nth or the (n! 1)stFibonacci numbers. We rescale the volumes by 1/!2n to obtain prototiles for our self-similar tiling.

The “right” way to see this process is to consider a linear map " : R2 " R2 that expands withthe Perron eigenvalue of the substitution matrix M . (How to find this map in general is quite

unclear.) In this example " is given by the matrix

!! 00 !

". Denoting the support of the level-n

tile of type t as supp(Sn(t)), we can find the support of the prototile t! for the inflate-and-subdividerule that corresponds to t by setting

t! = limn"#

"$n(supp(Sn(t)).

In Figure 28 we compare level-5 tiles from the DPV (left) and the self-similar tiling (right).

Figure 28. Comparing the DPV with the SST of Example 15.

Example 16. The self-similar tiling associated with the Rauzy two-dimensional substitution ofExample 11 has as its volume expansion the largest root of the polynomial x3 ! x2 ! 1. The threetile types obtained by the replace-and-rescale method are shown in Figure 29, compared with alarge iteration of the substituton.

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Figure 29. A comparison of an iterate with the limiting self-similar tiles.

V. Berthé - IRISA · Combinatorics of tilings V. Berthé LIAFA-CNRS-Paris-France...

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Transcript of V. Berthé - IRISA · Combinatorics of tilings V. Berthé LIAFA-CNRS-Paris-France...