Local and Matching Rules for Canonical Tilings

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Local and Matching Rules for Canonical Tilings

Thomas Fernique

LIF – CNRS & Univ. de Provence

FRAC, 12/01/2008

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Introduction

This talk is mostly a survey of the two papers:

L. Levitov,Local rules for quasicrystals (1988)

J. Socolar, Weak matching rules for quasicrystals (1990) With some guest contributions.

Interesting results, but proofs are probably not in theBook.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules

1 Some words about crystals

2 Canonical tilings

3 Strong Local Rules

4 Weak Local Rules

5 Matching Rules

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2 Canonical tilings

4 Weak Local Rules

5 Matching Rules

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Crystals

Definition

Crystal: lattice decorated with an atomic pattern.

i.e., periodic structure. Experimentally, crystals “diffract”:

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Diffraction

Principle

XRay diffract on atoms; if those show some long range order, then phases can sum up or cancel out (according to the outgoing angle).

A material “diffracts” if one get sharp bright spots (Bragg peaks).

Diffraction reveals long-range order. .

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules The crystallographic restriction

A⊂R² hasD-fold symmetry ifR^2π

D(A) =A.

Ais periodicif there is a rank 2 lattice Λ s.t. A+ Λ =A.

Crystallographic restriction

Only 2,3,4,6-fold symmetries can be periodicity-compatible.

Proof:

A-preserving rotation in a base of Λ: integer matrix;

trace ofR_θ in any base: 2 cos(θ).

2 cos(θ)∈Z⇔cos(θ)∈ {0,±1

2,±1} ⇔θ∈ {0,2π 2 ,2π

3 ,2π 4 ,2π

6 }.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Beyond the crystallographic restriction

In 1984, the following diffraction pattern was observed:

10-fold symmetry not a crystal.

Other non-crystallographic symmetries observed: 5,8,10,12-fold.

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2 Canonical tilings

4 Weak Local Rules

5 Matching Rules

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Canonical tiling

~v1, . . . , ~vD non-colinear vectors of R^d ^D_d

parallelepipedal tiles:

T_i₁_,...,i_d ={λ_i₁~v_i₁+. . .+λ_i_d~v_i_d |0≤λ_i_k ≤1}.

T₂₃

T₁₂ T₁₃

v₁ v₃

v₂

Definition

AD →d canonical tiling is a tiling of R^d with these tiles.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Canonical tiling

~v₁, . . . , ~v_D non-colinear vectors of R^d ^D_d

parallelepipedal tiles:

T_i₁_,...,i_d ={λ_i₁~v_i₁+. . .+λ_i_d~v_i_d |0≤λ_i_k ≤1}.

v₁ v₃

v₂

Definition

AD→d canonical tiling is a tiling of R^d with these tiles.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Lift of a canonical tiling

Let (~e₁, . . . , ~e_D) be the canonical basis ofR^D.

Mapφon vertices and edges of aD →d canonical tiling T: φ(~x₀) =~y₀ ∈Z^D, φ([~x, ~x+~v_i]) = [φ(~x), φ(~x) +~e_i].

Linearity over tiles map φfromT toR^D, calledlift.

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v1

v₃

v2

(1,1,0)

φ(~x0) = (1,1,0).

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v1

v₃

e

3 (1,1,1)

(1,1,0)

v2

φ([~x0, ~x0+~v3]) = [φ(~x0), φ(~x0) +~e3]) = [(1,1,0),(1,1,1)].

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v₁ v3

v2

(2,0,1)

(2,0,0)

(2,1,0)

(2,2,0) (1,2,−1)

(0,2,−1) (0,2,1)

(0,2,0) (0,0,2)

(1,1,1) (0,0,1) (2,0,2)

(1,0,2)

(0,1,2)

(1,0,0) (1,0,1)

(1,1,0) (0,1,0)

(1,2,0) (0,1,1)

φ(T): surface made ofd-dim. facets ofD-dim. unit cubes.

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v₁ v3

v2

(2,0,1)

(2,1,0)

(2,2,0) (1,2,−1)

(0,2,−1) (0,2,1)

(0,2,0) (0,0,2)

(1,1,1) (0,0,1) (2,0,2)

(1,0,2)

(0,1,2)

(1,0,0) (1,0,1)

(1,1,0) (0,1,0)

(1,2,0) (0,1,1)

(2,0,0)

Easily seen in the particular 3→2 case, by shadowing faces.

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v₁ v3

v2

(2,0,1)

(2,1,0)

(2,2,0) (1,2,−1)

(0,2,−1) (0,2,1)

(0,2,0) (0,0,2)

(1,1,1) (0,0,1) (2,0,2)

(1,0,2)

(0,1,2)

(1,0,0) (1,0,1)

(1,1,0) (0,1,0)

(1,2,0) (0,1,1)

(2,0,0)

Note: the lift is unique up to a translation inZ^D.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules C-tiling

LetC be a d-dim. affine subspace ofR^D. Definition

AC-tiling is a canonical tiling with a lift in the slice C + [0,1)^D. One shows that, givenC, there is a unique C-tiling.

Such tilings diffract model for long-range order of quasi-crystals.

Example

AD →2 C-tiling has D-fold symmetry when ~C is spanned by: (cos(2kπ/D))_0≤k<D and (sin(2kπ/D))_0≤k<D. ForD = 5 andC =C~, we get the celebrated Penrose tiling.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules C-tiling

LetC be a d-dim. affine subspace ofR^D. Definition

AC-tiling is a canonical tiling with a lift in the slice C + [0,1)^D. One shows that, givenC, there is a unique C-tiling.

Such tilings diffract model for long-range order of quasi-crystals.

Example

AD→2 C-tiling has D-fold symmetry when ~C is spanned by:

(cos(2kπ/D))0≤k<D and (sin(2kπ/D))0≤k<D. ForD = 5 andC =C~, we get the celebrated Penrose tiling.

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2 Canonical tilings

4 Weak Local Rules

5 Matching Rules

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Definition

Forr >0:

Definition (r-atlas)

Ar-pattern of a tilingT is a union of tiles of T appearing in an open ball of radiusr. The r-atlas ofT is the set of its r-patterns.

Definition (Strong Local Rules (SLR))

AC-tiling has SLR if, for somer >0, itsr-atlas characterizesC~. SLR fix the “slope” of a canonical tiling.

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Forr >0:

Definition (r-atlas)

Ar-pattern of a tilingT is a union of tiles of T appearing in an open ball of radiusr. The r-atlas ofT is the set of its r-patterns.

Definition (Strong Local Rules (SLR))

AC-tiling has SLR if, for somer >0, itsr-atlas characterizesC~. SLR fix the “slope” of a canonical tiling.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules A necessary condition

A necessary condition for the existence of Strong Local Rules:

Theorem (Levitov)

If a2-dim. C -tiling has SLR, thenC~^⊥ is spanned by vectors with entries inQ(√

n), for some n ∈N.

One speaks aboutquadratic-basedslopes.

Corollary

Only3,4,5,6,8,10,12-fold symmetries can be SLR-compatible. Proof: cos(^2π_D)∈Q(√

n) yieldsn = 1|2|3|5,D = 3,4,6|8|12|5,10.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules A necessary condition

A necessary condition for the existence of Strong Local Rules:

Theorem (Levitov)

If a2-dim. C -tiling has SLR, thenC~^⊥ is spanned by vectors with entries inQ(√

n), for some n ∈N.

One speaks aboutquadratic-basedslopes.

Corollary

Only3,4,5,6,8,10,12-fold symmetries can be SLR-compatible.

Proof: cos(^2π_D)∈Q(√

n) yieldsn = 1|2|3|5,D = 3,4,6|8|12|5,10.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules A sufficient condition?

Can we effectively find SLR for all these symmetries?

D ∈ {3,4,6}: yes, with periodic tilings;

D ∈ {5,10}: yes (Thang);

D = 8: no (Beenker);

D = 12: ?

Recall: 8-fold symmetry experimentally observed gap.

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2 Canonical tilings

4 Weak Local Rules

5 Matching Rules

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Definition (Weak Local Rules (WLR))

AC-tilingT has WLR if, for somer >0, the lift of any tiling with the samer-atlas asT stays at bounded distance from C.

WLR fix the “global slope” of a canonical tiling.

Note: ifT hasD-fold symmetry, D∈ {2,/ 3,4,6}, then any tiling whose lift stays at bounded distance fromC is aperiodic.

aperiodic sets of tiles (`a la Kari)

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Definition (Weak Local Rules (WLR))

AC-tilingT has WLR if, for somer >0, the lift of any tiling with the samer-atlas asT stays at bounded distance from C.

WLR fix the “global slope” of a canonical tiling.

Note: ifT hasD-fold symmetry, D∈ {2,/ 3,4,6}, then any tiling whose lift stays at bounded distance fromC is aperiodic.

aperiodic sets of tiles (`a la Kari)

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Two positive results

A sufficient condition for the existence of Weak Matching Rules:

Theorem (Levitov)

If a2-dim. C -tiling is such thatC~^⊥ is spanned by vectors with entries inQ(√

n), for some n ∈N, then it has WLR.

Again, quadratic-base slopes.

Another existence result: Theorem (Socolar)

For D ∈/4N, D ≥3, D-fold symmetries are WLR-compatible. Here, the above sufficient condition does not always hold

(hint: cos(2π/D) is quadratic only forD∈ {3,4,5,6,8,10,12}).

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Two positive results

A sufficient condition for the existence of Weak Matching Rules:

Theorem (Levitov)

If a2-dim. C -tiling is such thatC~^⊥ is spanned by vectors with entries inQ(√

n), for some n ∈N, then it has WLR.

Again, quadratic-base slopes.

Another existence result:

Theorem (Socolar)

For D ∈/4N, D ≥3, D-fold symmetries are WLR-compatible.

Here, the above sufficient condition does not always hold

(hint: cos(2π/D) is quadratic only forD∈ {3,4,5,6,8,10,12}).

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules The Alternation Condition

Proof of Socolar’s result: relies on anAlternation Condition(AC):

AC can be enforced by Weak Local Rules (not trivial).

AC could be easily enforced if tile’s coloring would be allowed. . .

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules The Alternation Condition

Proof of Socolar’s result: relies on anAlternation Condition(AC):

AC can be enforced by Weak Local Rules (not trivial).

AC could be easily enforced if tile’s coloring would be allowed. . .

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2 Canonical tilings

4 Weak Local Rules

5 Matching Rules

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Colored tiling: each tile has a color (finite set of colors).

sort of “colored Local Rules”, called Matching Rules:

Definition (Strong Matching Rules (SMR))

AC-tiling has SMR if, for some r >0, there is a coloredC-tiling whoser-atlas characterizesC~.

Definition (Weak Matching Rules (WMR))

AC-tilingT has WMR if, for some r >0, there is a colored C-tilingT_c such that the lift of any tiling with the samer-atlas as T_c stays at bounded distance from C.

Note: MR'Sofic subshift, whereas LR 'Subshift of Finite Type.

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Strong Matching Rules and substitutions

Strong Matching Rules: maybe the most classic problem.

Widespread approach: for fixed-points of a substitution.

Principle: enforce the hierarchical structure by Matching Rules.

Probably the most general result in this direction: ToBeProven

If aC-tiling is such that ~C is an eigenspace of a non-negative integer matrix, then it has SMR.

Proof would rely ongeneralized substitutions(Arnoux-Ito) to show the pseudo-self-similarityof this tiling, then obtain an equivalentself-similar tiling (Solomyak) and, last, derive matching rules (Goodmann-Strauss).

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Strong Matching Rules and substitutions

Strong Matching Rules: maybe the most classic problem.

Widespread approach: for fixed-points of a substitution.

Principle: enforce the hierarchical structure by Matching Rules.

Probably the most general result in this direction:

ToBeProven

If aC-tiling is such that ~C is an eigenspace of a non-negative integer matrix, then it has SMR.

Proof would rely ongeneralized substitutions(Arnoux-Ito) to show the pseudo-self-similarityof this tiling, then obtain an equivalentself-similar tiling (Solomyak) and, last, derive matching rules (Goodmann-Strauss).

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Some words about crystals Canonical tilings Strong Local Rules Weak Local Rules Matching Rules Other Matching Rules

Other approaches for obtaining Strong Matching Rules?

And for Weak Matching Rules?

At least, there is the ones enforcing the Socolar’s AC.

(in theD= 5 case, they enforce a Penrose tiling SMR).

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Conclusion

SLR SMR

WLR

WMR

Strong Local Rules (SLR):'quadratic-based slopes (i.e.,C~).

Weak Local Rules (WLR): strictly more than SLR. Less than SMR?

Strong Matching Rules (SMR): what?

Weak Matching Rules (WMR): strictly more than SMR?

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Annexe Some questions

1 Do weak rules still ensure diffraction?

2 Directed flip-acc: asynchronous flips on a tiling with matching errors, with probability depending on the number of corrected matching rules. Convergence towards aC-tiling?

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Annexe Remark

A different definition of Matching Rules (Senechal):

Perfect MR : non-periodic and repetitive tilings, same LI-class;

Strong MR : non-periodic and repetitive tilings;

Weak MR : non-periodic tilings (Kari’s rules)