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 ∈T 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 ∈ ∈

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 | ∈ } 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

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

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 = ? \F 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

σ∞(1) is called the Fibonacci word σ∞(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

σ∞(1) is called the Fibonacci word

σ∞(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. a∼ finite number of prototiles T1, T2,..., Tm The rule taking T to the union of tiles in{φ(T ) is called an} inflate-and-subdivide rule because it inflates using thean expanding ∈expansiveT map transformationφ and then decomposesQ (the inflation the image factor into ) the union of tiles on the original scale. If ais ruleφ-subdividing, that allows one then to it divide will be each invariantQT into under copies this rule, of the therefore we show the T i inflate-and-subdivideT1, T rule2,..., ratherTm than the tiling itself. The rule given in Figure 1 is an inflate-and- subdivide rule with φ(z) = 3z. However, the rule given in Figure 3 is not an inflate-and-subdivide rule. A substitution is a simple production method that allows one to construct infinite tilings using a finite number of tiles Example 5. The “L-triomino” or “chair” substitution uses four prototiles, each being an L formed by three unit squares. We have chosen to color the prototiles since they are inequivalent up to translation. TheExample expansion map is φ(z) = 2z and in Figure 5 we show the substitution of the four prototiles.

Figure 5. The “chair” or “L-triomino” substitution. A PRIMER ON SUBSTITUTION TILINGS OF THE EUCLIDEAN PLANE 5

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) T and T ! are equivalent tiles 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 has finite localT complexity. If φ 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. The rule taking T to the union of tiles in φ(T ) is called an inflate-and-subdivide rule because it inflates using the expanding∈ T map φ and then decomposes the image into the union of tiles on the original scale. If is φ-subdividing, then it will be invariant under this rule, therefore we show the inflate-and-subdivideT rule rather than the tiling itself. The rule given in Figure 1 is an inflate-and- subdivide rule with φ(z) = 3z. However, the rule given in Figure 3 is not an inflate-and-subdivide rule. ExampleThe 5. The chair “L-triomino” tiling or “chair” substitution uses four prototiles, each being an L formed by three unit squares. We have chosen to color the prototiles since they are inequivalent up to translation. The expansion map is φ(z) = 2z and in Figure 5 we show the substitution of the four prototiles.

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 subdividedFiguren times a level- 5. nThetile. In “chair” Figure 6 orwe show “L-triomino” level-n tiles for substitution.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 expansion constant λ C of a self-similar tiling of C. ∈ Theorem 2.1. (Thurston [59], Kenyon [27]) A complex number λ is the expansion constant for some self-similar tiling if and only if λ is an algebraic integer which is strictly larger than all its Galois conjugates other than its complex conjugate. The forward direction was proved by Thurston and the reverse direction by Kenyon. In [28], d Kenyon extends the result to self-affine tilings of R 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 called the “transition”, “composition”, “subdivision”, or even “abelianization” matrix). Suppose that the prototile set (or alphabet) has m elements labeled by 1, 2, ..., m . The substitution matrix M is the 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. 91 For example, the substitution in Example 3 has substitution matrix M = when we 08 label a = 1 and b = 2. If an initial configuration of tiles has n white tiles and!m blue" tiles, then M[nm]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 be n irreducible: for every i, j 1, 2, ..., m there exists an n such that (M )ij > 0. Among other things, the Perron-Frobenius∈ { theorem states} that if M is irreducible, then the largest eigenvalue will be a positive real number that is larger in modulus than any of the other eigenvalues of the matrix. This eigenvalue is unique, has multiplicity one, and is called the Perron eigenvalue of the matrix. Primitivity, a special case of irreducibility, is particularly important. A matrix M is primitive if there is an n>0 such that M n has strictly positive entries. Primitivity of M means if one substitutes any tile (or letter) a fixed number of times, one will see all of the other tiles (or letters). Substitutions and tilings

Let φ : Rd Rd be an expanding linear map → Principle One takes

a ﬁnite numer of prototiles T1, T2,..., Tm { } an expansive transformation φ (the inﬂation factor )

a rule that allows one to divide each φTi into copies of the T1, T2,..., Tm

A tile-substitution s with expansion φ is a map Ti s(Ti ), where s(Ti ) is a patch made of translates of the prototiles and7→

φ(Ti ) = Tj Tj ∈[s(Ti) Substitutions and tilings Let φ : Rd Rd be an expanding linear map → Principle One takes

a ﬁnite numer of prototiles T1, T2,..., Tm { } an expansive transformation φ (the inﬂation factor ) a rule that allows one to divide each φTi into copies of the T1, T2,..., Tm A tile-substitution s with expansion φ is a map Ti s(Ti ), where s(Ti ) is a patch made of translates of the prototiles and7→

φ(Ti ) = Tj Tj ∈[s(Ti) 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 of the tile of type j Substitutions and tilings

Let φ : Rd Rd be an expanding linear map → Principle One takes

a ﬁnite numer of prototiles T1, T2,..., Tm { } an expansive transformation φ (the inﬂation factor )

a rule that allows one to divide each φTi into copies of the T1, T2,..., Tm

A tile-substitution s with expansion φ is a map Ti s(Ti ), where s(Ti ) is a patch made of translates of the prototiles and7→

φ(Ti ) = Tj Tj ∈[s(Ti) cf. Self-afﬁne tiles. See the lectures by C.-K. Lai, T.-M. Tang Self-afﬁne tilings

The tiling is said φ-subdividing if T for each tile T XT , φ(T ) is a union of -tiles ∈ T T and T 0 are equivalent tiles iff φ(T ) and φ(T 0) form equivalent patches of tiles in T Self-afﬁne tiling is φ-subdividing, repetitive and has ﬁnite local complexity T Substitution matrix Mij := 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] 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 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 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 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) 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 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’2010] Let φ be a diagonalizable (over C) expanding linear map of Rd and let be a self-afﬁne tiling of Rd with expansion φ. Then T 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 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 Cut and project scheme in Z2 A cut-and-project scheme consists of

a direct product Rk H, k 1 × ≥ where H is a locally compact abelian group and a lattice L in Rk H × such that the canonical projections

k k k π0 : R H H, π1 : R H R × → × → satisfy

π0(L) is dense in H

π1 restricted to L is one-to-one onto its image π1(L)

π1 π0 Rk Rk H H ←− × −→ ∪Λ ∪L Ω ∪ A cut-and-project scheme consists of

a direct product Rk H, k 1 × ≥ where H is a locally compact abelian group and a lattice L in Rk H × such that the canonical projections

k k k π0 : R H H, π1 : R H R × → × → satisfy

π0(L) is dense in H

π1 restricted to L is one-to-one onto its image π1(L)

π1 π0 Rk Rk H 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 A cut-and-project scheme consists of a direct product Rk H, k 1 × ≥ where H is a locally compact abelian group and a lattice L in Rk H × such that the canonical projections

k k k π0 : R H H, π1 : R H R × → × → satisfy

π0(L) is dense in H

π1 restricted to L is one-to-one onto its image π1(L)

π1 π0 Rk Rk H H ←− × −→ ∪Λ ∪L Ω ∪

Choose some compact Ω H with Ω = Ω◦ with zero-measure boundary ⊂ Λ := π1(x) x L, π0(x) Ω { | ∈ ∈ } Then Λ is a regular cut-and-project set (or model set) Cut-and-project set

π1 π0 Rk Rk H H ←− × −→ ∪Λ ∪L Ω ∪

Λ := π1(x) x L, π0(x) Ω { | ∈ ∈ } Λ 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 is both relatively dense : there exists R > 0 such that any Euclidean ball of Rk 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 Examples

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

π1 one-to-one π0 dense Rk Rk H H ←−−−−−−−−−− × −−−−−−→ ∪Λ ∪L Ω ∪ k = 1, H = R Examples

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

k = 2, H = R Examples

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

2 k = 1, H = R ∼= C, Tribonacci substitution

πe πc V = R R3 W = R2 ∼ ←− −→ ∼ ∪Λ ∪L Ω Rauzy ∪ fractal

V : expanding space of Mσ, W : contracting space, L : lattice of the eigenvectors, Λ : vertex set of the 1D substitution tiling Pisot conjecture The vertex set of a Pisot substitution tiling is a regular 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 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] 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 map A repetitive→ FLC tiling is a pseudo-self-afﬁne tiling if

φ LD T → T Theorem [Priebe-Solomyak, Solomyak] Let be a pseudo-self-affine T tiling of Rd with expansion φ Then for any k sufficiently large, there exists a tiling 0 T which is self-affine with expansion φk such that is MLD with 0 T 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 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 → → → → → → MLD

0 σ1 : a cb, b c, c cab σ : a bc, b c, c cba → → → 1 → → → Let −1 ρ1 : a bab , b b, c c] → → → −1 0 σ1 = ρ σ ρ1 1 ◦ 1 ◦

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 tiling of 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 with ﬁnite matching rules

effective tilings : forbidden patterns are given by a Turing machine A PRIMER ON SUBSTITUTION TILINGS OF THE EUCLIDEAN PLANE 13 them in the tiling. For example, consider two horizontally adjacent tiles of type 1. That adjacency appears twice in the level-3 tile of type 1 and we have circled them on the left side of Figure 20; what happens under substitution is shown on the right of the same ﬁgure. For this graph substitution, the problem can be handled by relabeling the edges and facets of the graph in terms of the immediate conﬁguration of tiles the edge or facet is contained in.

312 1 3 2 31 4 2 1 3 1 1 4 2 132 3 1 2 3 1 421 1 3 3 1 1 4 2 1 423 1 3 1 2 4 2 1 13 1 1 3 424 2 1 3 1 3 1 2 2 3 1 2 3 1 1 3 4 2 4 2 1 1 4 2 1 4 2 3 1 1 1 3 1 1 3 4 2

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

2 141 NATALIE2 PRIEBE3 FRANK3 1 1 2 2 2 2 1 2 1 1 2 1 1 3 1 1 Figure 21. A two-dimensional substitution based3 on the Rauzy one-dimensional1 substitution. not enough information to iterate the substitution,2 so we specify how to substitute the “important 1 2 1 3 1 1 adjacencies” in Figure 22. This is enough [1]: there are3 no ambiguities1 when substituting other adjacencies, and facet substitutions do not include1 any new information. We show a few iterates of the tile of type 1 in Figure 23, starting with the level-2 tile of type 1. The fact that this substitution rule can be extendedFigure to 22. anHow infinite to substitute tiling of important the plane adjacencies is proved for usingthe Rauzy noncombinatorial substitution. methods in [1]; a combinatorial proof of existence would2 be welcomed.2 2 2 2 2 2 2 2 2 2 1 3 1 2 1 3 1 2 1 3 1 2 1 2 1 3.2. Non-constructive3 1 tiling3 1 substitutions.3 1 When3 1 trying to3 1 make up new3 1 examples3 1 of combi- 1 1 1 2 1 2 1 1 natorial tiling substitutions it is easy to create examples that3 fail1 to be constructive.3 1 The problem arises in the substitution of adjacencies: it may happen that no1 finite label1 set can be chosen to describe all adjacencies sufficiently to know how to substitute them. There is evidence to suggest that this sort of exampleFigure can 23. ariseA few when iterates the of the constant Rauzy two-dimensional which best approximates substitution. the linear growth of blocks isDefinition not a Pisot 3.1. number.A (non-constructive) The author tiling is substitution not awareon of a any finite formal prototile definition set is a containing set of this group andnonempty, so proposes connected the patchesfollowing= definition,Sn(p):p whichand n works1, 2, ... directlysatisfying with the following: theP tiling and does not S { ∈ P ∈ } involve dual(1) graphs.For each prototile p and tile t S1(p), and for each integer n 2, 3, ..., there are rigid ∈d P d ∈ n n ∈1 motions g(p, n, t):R R such that S (p)= g(p, n, t) S − (t) , where → t S1(p) 1 n∈!1 " n# 1 (2) for any t = t# in S (p), the patches g(p, n, t) S − (t) and g(p, n, t#) S − (t#) intersect at most along# their boundaries. " # " # We say a tiling is admitted by the substitution if every patch in appears as a subpatch of some element ofT . This very general definition isS satisfied by every substitutionT appearing in this paper except theS Penrose substitution on rhombi and those generalized pinwheel tilings that do not allow a finite number of tile sizes. The Rauzy substitution of Example 11 has a particularly efficient representation by this definition. The patches S1(p) are given to the right of the arrows in Figure 21, with all lower right n n 1 n n 1 corners at the origin. Now, S (2) = S − (3) and S (3) = S − (1), so g(2, n, 3) and g(3, n, 1) are n n 1 n 1 the identity map. We find S (1) = S − (1) g(1, n, 2) S − (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 translation by a vector !v that can be computed recursively. Let" !v = (0# , 0),!v = (0, 1) and !v =( 1, 0); n 0 1 2 − 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 Definition 3.1. 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)− are computable recursively as well. The next example is also a DPV, but it cannot be defined in terms of dual graphs and is non-constructive. We encourage the reader to think about how to write up Definition 3.1 in this case. Example 12. Consider a DPV arising from a one-dimensional substitution a abbb, b a. From the direct product of this substitution with itself, we choose only to rearrange→ the substitution→ of the type-1 tile as in Figure 24. The substitution matrix of this one-dimensional substitution has Perron eigenvalue θ = (1 + √13/2), which is not a Pisot number: its algebraic conjugate θ2 = (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 A PRIMER ON SUBSTITUTION TILINGS OF THE EUCLIDEAN PLANE 13 them in the tiling. For example, consider two horizontally adjacent tiles of type 1. That adjacency appears twice in the level-3 tile of type 1 and we have circled them on the left side of Figure 20; what happens under substitution is shown on the right of the same figure. For this graph substitution, the problem can be handled by relabeling the edges and facets of the graph in terms of the immediate configuration of tiles the edge or facet is contained in.

312 1 3 2 31 4 2 1 3 1 1 4 2 132 3 1 2 3 1 421 1 3 3 1 1 4 2 1 423 1 3 1 2 4 2 1 13 1 1 3 424 2 1 3 1 3 1 2 2 3 1 2 3 1 1 3 4 2 4 2 1 1 4 2 1 4 2 3 1 1 1 3 1 1 3 4 2

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

14 NATALIE PRIEBE FRANK 2 141 NATALIE2 PRIEBE3 FRANK3 1 1 2 2 2 2 1 2 1 1 2 1 1 3 1 2 1 2 2 2 3 1 1 2 1 1 2 1 1 3 1 1 Figure 21. A two-dimensional substitution based3 on the Rauzy one-dimensional1 substitution. 2 1 2 1 not enough information to iterate3 the1 substitution,1 2 so we specify how to substitute the “important 3 1 12 1 3 1 1 1 adjacencies” in Figure 22. This is enough [1]: there are3 no ambiguities1 when substituting other adjacencies, and facet substitutions do not include1 any new information. We show a few iterates of the tile of typeFigure 1 in Figure 22. How23, to starting substitute with important the level-2 adjacencies tile of for type the 1.Rauzy The substitution. fact that this substitution Figure 22. How to substitute important adjacencies for the Rauzy substitution. rule can be extended to an infinite tiling2 of the plane2 2 is proved2 using2 noncombinatorial2 2 2 methods in [1]; a combinatorial2 proof2 of existence2 1 would2 be3 1 welcomed.22 1 2 3 122 12 3 12 2 12 2 12 3 1 2 3 1 2 3 1 2 1 33 11 2 1 33112 1 3 31 12 31 12 1 3.2. Non-constructive3 1 tiling13 1 substitutions.13 1 When31 1 trying2 to311 make up new23 11 examples3 1 of combi- 1 1 1 3 12 1 3 12 1 1 natorial tiling substitutions it is easy to create examples that3 fail11 to be constructive.3 11 The problem arises in the substitution of adjacencies: it may happen that no1 finite label1 set can be chosen to describe all adjacenciesFigure su 23.fficientlyA few iterates to know of the how Rauzy to substitute two-dimensional them. substitution. There is evidence to suggest that this sort of exampleFigure can 23. ariseA few when iterates the of the constant Rauzy two-dimensional which best approximates substitution. the linear growth Definition 3.1. A (non-constructive) tiling substitution on a finite prototile set is a set of of blocks isDefinition not a Pisot 3.1. number.A (non-constructive) Then author tiling is substitution not awareon of a any finite formal prototile definition set P is a containing set of this nonempty, connected patches = S (pn):p and n 1, 2, ... satisfying the following:P group andnonempty, so proposes connected the patchesfollowingS ={ definition,S (p):∈p1P whichand n∈ works1, 2, ...} directlysatisfying with the following: the tiling and does not (1) For each prototile p Sand{ tile t S ∈(1pP), and for∈ each integer} n 2, 3, ..., there are rigid involve dual(1) graphs.For each prototile∈dpP dand tile∈t Sn(p), and for each integer nn∈1 2, 3, ..., there are rigid motions g(p, n, t):R ∈d PR suchd that∈ S (np)= g(p, n, t) S n−∈1(t) , where motions g(p, n, t):R→ R such that S (p)= g(p, n, t) S − (t) , where → t S1(p) ∈t!S1(p) " # (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 (2) for any t =# t in S1(p), the patches g(p, n, t) S−n 1(t) and g(p, n,# t ) Sn− 1(t# ) intersect at most along# their# boundaries. − # − # most along# their boundaries. " " ## "" ## WeWe say say a tiling a tiling is admittedis admitted by by the the substitution substitution ifif every every patch patch in in appearsappears as as a a subpatch subpatch of some element ofT .T This very general definition isS satisfiedS by every substitutionTT appearing in of some elementS of . This very general definition is satisfied by every substitution appearing in thisthis paper paper except except the the PenroseS Penrose substitution substitution on on rhombi rhombi and and those those generalized generalized pinwheel pinwheel tilings tilings that do notdo not allow allow a finite a finite number number of oftile tile sizes. sizes. TheThe Rauzy Rauzy substitution substitution of ofExample Example1111hashas a a particularly particularly effi efficientcient representation representation by by this this defi- 1 nition.nition. The The patches patchesS (Sp1)(p are) are given given to to the the right right of of the the arrows arrows in in Figure Figure2121,, with with all all lower lower right n n n n1 1 n n n n1 1 cornerscorners at the at the origin. origin. Now, Now,S S(2)(2) = =S S− −(3)(3) and andS S(3)(3) = =SS−−(1),(1), so sogg(2(2,, n, n,3)3) and and gg(3(3,, n, n,1) are n n n n1 1 n n1 1 thethe identity identity map. map. We We find findS S(1)(1) = =S S− (1)(1) g(1g(1, n,, n,2)2)SS− (2)(2),, so sogg(1(1,, n, n,1)1) is is the the identity identity map − ∪ − andand all we all wehave have left left to figureto figure out out is theis the formula formula∪ for forg(1g(1, n,, n,2).2). It It turns turns out out that thatgg(1(1,, n, n,2)2) is is transla- " " ## tiontion by a by vector a vector!vn !vthatthat can can be be computed computed recursively. recursively. Let Let!v0!v == (0 (0, 0), 0),!,v!v1 == (0 (0,,1)1) and and!v!v2 =( 1, 0); n 0 1 2 − for nfor n3 we3 wehave have that that!vn !v=n !v=n !vn3 3 !vn!vn2. 2. ≥ − − − − − The≥The Fibonacci Fibonacci DPV DPV of Exampleof Example−1010alsoalso has has a relativelya relatively simple simple formuation formuation in in terms terms of of Definition Definition 3.1.3.1 The. The side side lengths lengths of level-of level-n tilesn tiles are are given given recursively, recursively, and and the the placement placement of of the the level-( level-(n 1) − tilestiles to create to create level- level-n tilesn tiles depends depends only only on on these these side side lengths. lengths. Thus Thus the the translations translations gg((p,p, n, t)− are computablecomputable recursively recursively as aswell. well. The The next next example example is is also also a a DPV, DPV, but but it it cannot cannot be be defined defined in in terms of dualof dual graphs graphs and and is non-constructive. is non-constructive. We We encourage encourage the the reader reader to to think think about about how how to to write write up DefinitionDefinition3.13.1in thisin this case. case. Example 12. Consider a DPV arising from a one-dimensional substitution a abbb, b a. From Example 12. Consider a DPV arising from a one-dimensional substitution a →abbb, b → a. From thethe direct direct product product of this of this substitution substitution with with itself, itself, we we choose choose only only to to rearrange rearrange→ the the substitution substitution→ of thethe type-1 type-1 tile tile as in as Figure in Figure24.24. TheThe substitution substitution matrix matrix of ofthis this one-dimensional one-dimensional substitution substitution has has Perron Perron eigenvalue eigenvalue θθ = (1 + √13/2), which is not a Pisot number: its algebraic conjugate θ = (1 √13/2) is larger than one √13/2), which is not a Pisot number: its algebraic conjugate θ 2= (1 √13/2) is larger than one in modulus. Using analysis similar to what we will see in Section2 4.1, we− find that constants times in modulus. Using analysis similar to what we will see in Section 4.1, we− find that constants times A PRIMER ON SUBSTITUTION TILINGS OF THE EUCLIDEAN PLANE 13 them in the tiling. For example, consider two horizontally adjacent tiles of type 1. That adjacency appears twice in the level-3 tile of type 1 and we have circled them on the left side of Figure 20; what happens under substitution is shown on the right of the same figure. For this graph substitution, the problem can be handled by relabeling the edges and facets of the graph in terms of the immediate configuration of tiles the edge or facet is contained in.

312 1 3 2 31 4 2 1 3 1 1 4 2 132 3 1 2 3 1 421 1 3 3 1 1 4 2 1 423 1 3 1 2 4 2 1 13 1 1 3 424 2 1 3 1 3 1 2 2 3 1 2 3 1 1 3 4 2 4 2 1 1 4 2 1 4 2 3 1 1 1 3 1 1 3 4 2

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] and in a similar form in [1] is this. Given a labeled vertex set V representing the prototile types, a map from V to the set of nonempty labeled graphs on V is the basis for the substitution rule. The edges and facets of these graphs are labeled to give information about the types of adjacencies they A PRIMER ON SUBSTITUTION TILINGS OF THE EUCLIDEAN PLANE 17 represent in a tiling. The substitution rule also specifies how to substitute the labeled edges and substitution are supported on rectangles with side lengths given by either the nth or the (n 1)st − facets so that we knowFibonacci how numbers. to connect We rescale the the volumes vertex by 1/γ substitutions2n to obtain prototilescontained for our self-similar in tiling. certain labeled graphs. 2 2 The “right” way to see this process is to consider a linear map φ : R R that expands with (The need to specifythe graph Perron eigenvalue substitutions of the substitution onfacets matrix M. and (How not to find just this→ map edges in general is illustratedis quite in an example γ 0 in [11]). Defining a tilingunclear.) substitution In this example φ is givenrule by this the matrix way is quite. Denoting tricky the support since of mostthe level-n labeled graphs do not 0 γ n ! " represent the dual graphtile of type oft as asupp tiling.(S (t)), we This can find interplay the support of betweenthe prototile t! combinatoricsfor the inflate-and-subdivide and geometry is where rule that corresponds to t by setting the technicalities come in to the formal definitionsn inn the literature. t! = lim φ− (supp(S (t)). n Combinatorial substitutions→∞ Example 11. TheIn tiling Figure 28 substitutionwe compare level-5 tiles of from Figure the DPV21 (left), introduced and the self-similar in tiling [1 (right).], is based on a variation of the one-dimensional “Rauzy substitution” σ(1) = 1 2, σ(2) = 3, σ(3) = 1. Figure 21 is obviously

2 141 NATALIE2 PRIEBE3 FRANK3 1 1 2 2 2 2 1 2 1 1 2 1 1 3 1 1 Figure 21. A two-dimensional substitution based3 on the Rauzy one-dimensional1 substitution. not enough information to iterate the substitution,2 so we specify how to substitute the “important Figure 28. Comparing the DPV with the1 SST of Example2 1 15. 3 1 1 adjacencies” in Figure 22. This is enough [1]: there are3 no ambiguities1 when substituting other adjacencies, and facetExample substitutions 16. The self-similar do tiling not associated include1 with any the Rauzy new two-dimensional information. substitution We of show a few iterates of Example 11 has as its volume expansion the largest root of the polynomial x3 x2 1. The three the tile of type 1 in Figuretile types obtained23, starting by the replace-and-rescale with the level-2 method are tile shown of in type Figure 29 1.−, compared The− fact with a that this substitution rule can be extendedFigurelarge to iteration 22. anHow infinite of the to substituton. substitute tiling of important the plane adjacencies is proved for usingthe Rauzy noncombinatorial substitution. methods in [1]; a combinatorial proof of existence would2 2 2 be welcomed.2 2 2 2 2 2 2 3 1 2 1 2 2 2 1 3 1 3 1 2 1 3 1 2 1 3 1 2 1 2 1 3 1 3 1 3 1 2 1 2 2 3 1 3 1 3 1 3 1 3.2. Non-constructive tiling2 substitutions.2 3 1 2 1 2 1 When trying to make up new examples of combi- 1 3 1 2 1 21 2 3 1 3 1 1 2 1 2 1 1 natorial tiling substitutions it is3 1 easy3 1 to2 1 create2 1 1 examples that3 fail1 to be constructive.3 1 The problem 2 1 2 2 3 1 3 1 1 1 arises in the substitution of adjacencies:3 1 2 1 2 1 2 1 2 1 it2 may happen that no finite label set can be chosen to 2 2 3 1 3 1 3 1 2 1 2 1 describe all adjacencies su3ffi1 ciently2 1 2 1 2 1 to2 3 know1 3 1 how to substitute them. There is evidence to suggest Figure3 23.1 3 1A3 few1 2 1 iterates2 1 2 1 2 of2 the Rauzy two-dimensional substitution. that this sort of example can2 1 arise1 3 1 when3 1 3 1 2 the1 2 1 constant which best approximates the linear growth 3 1 2 1 1 3 1 3 1 of blocks is not a Pisot number.1 3 1 The author2 1 1 is not aware of any formal definition containing this Definition 3.1. A (non-constructive)1 3 1 tiling substitution on a finite prototile set is a set of n P group andnonempty, so proposes connected the patchesfollowing= definition,S 1(p):p whichand n works1, 2, ... directlysatisfying with the following: the tiling and does not S { ∈ P ∈ } involve dual(1) graphs.For each prototile p and tile t S1(p), and for each integer n 2, 3, ..., there are rigid Figure 29.∈d PA comparisond of∈ an iteraten with the limiting self-similar tiles.n ∈1 motions g(p, n, t):R R such that S (p)= g(p, n, t) S − (t) , where → t S1(p) 1 n∈!1 " n# 1 (2) for any t = t# in S (p), the patches g(p, n, t) S − (t) and g(p, n, t#) S − (t#) intersect at most along# their boundaries. " # " # We say a tiling is admitted by the substitution if every patch in appears as a subpatch of some element ofT . This very general definition isS satisfied by every substitutionT appearing in this paper except theS Penrose substitution on rhombi and those generalized pinwheel tilings that do not allow a finite number of tile sizes. The Rauzy substitution of Example 11 has a particularly efficient representation by this definition. The patches S1(p) are given to the right of the arrows in Figure 21, with all lower right n n 1 n n 1 corners at the origin. Now, S (2) = S − (3) and S (3) = S − (1), so g(2, n, 3) and g(3, n, 1) are n n 1 n 1 the identity map. We find S (1) = S − (1) g(1, n, 2) S − (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 translation by a vector !v that can be computed recursively. Let" !v = (0# , 0),!v = (0, 1) and !v =( 1, 0); n 0 1 2 − 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 Definition 3.1. 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)− are computable recursively as well. The next example is also a DPV, but it cannot be defined in terms of dual graphs and is non-constructive. We encourage the reader to think about how to write up Definition 3.1 in this case. Example 12. Consider a DPV arising from a one-dimensional substitution a abbb, b a. From the direct product of this substitution with itself, we choose only to rearrange→ the substitution→ of the type-1 tile as in Figure 24. The substitution matrix of this one-dimensional substitution has Perron eigenvalue θ = (1 + √13/2), which is not a Pisot number: its algebraic conjugate θ2 = (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