Tilings and discrete geometry V. Berth e, T. Fernique - - PowerPoint PPT Presentation

Lozenge tilings Substitutions Continued fractions

Tilings and discrete geometry

• V. Berth´

e, T. Fernique

LIRMM-CNRS-Montpellier-France berthe@lirmm.fr http://www.lirmm.fr/˜berthe

Workshop on combinatorial and computational aspects of tilings –London 2008

Lozenge tilings Substitutions Continued fractions

Tilings by lozenges

We work with lozenge tilings of the plane (tilings with 60◦ rhombi, dimer covering of the honeycomb graph).

3 1 2 1 1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 1 2 1 1 3 2 2 1 1 2 2 2 1 1 1 2 2 1

Lozenge tilings Substitutions Continued fractions

Stepped surfaces

e1 e2 e3 Definition

A stepped surface is defined as a union of faces such that the orthogonal projection

• nto the diagonal plane x + y + z = 0 induces an homeomorphism from the stepped

surface onto the diagonal plane.

Lozenge tilings Substitutions Continued fractions

Tilings and stepped surfaces

Lift [Thurston]

Let T be a lozenge tiling of the plane x + y + z = 0. Then there exists a unique stepped surface, up to translation by the vector (1, 1, 1), that projects onto T by the

• rthogonal projection onto the plane x + y + z = 0.

Lozenge tilings Substitutions Continued fractions

Stepped surface

Definition

A functional discrete surface is defined as a union of pointed faces such that the

• rthogonal projection onto the diagonal plane x + y + z = 0 induces an

homeomorphism from the discrete surface onto the diagonal plane. Being a stepped surface is a local property.

Lozenge tilings Substitutions Continued fractions

Stepped surface

Definition

A functional discrete surface is defined as a union of pointed faces such that the

• rthogonal projection onto the diagonal plane x + y + z = 0 induces an

homeomorphism from the discrete surface onto the diagonal plane. Being a stepped surface is a local property.

x y z

Lozenge tilings Substitutions Continued fractions

Stepped surface

Definition

A functional discrete surface is defined as a union of pointed faces such that the

• rthogonal projection onto the diagonal plane x + y + z = 0 induces an

homeomorphism from the discrete surface onto the diagonal plane. Being a stepped surface is a local property.

1 1 1 1 1 3 1 2 1 1 2 3 2 1 1 2 2 2 2 3 1 2 3 3 3 3 2 2 3 3 1 3 3 3 n m 1 2

Lozenge tilings Substitutions Continued fractions

Arithmetic discrete planes

Let v ∈ R3 and µ ∈ R. The standard arithmetic discrete hyperplane P(v, µ) is defined as P(v, µ) = {x ∈ Z3; 0 ≤ x, v + µ < ||v||1}. The stepped plane Pv,µ is defined as the stepped surface whose set of edges is P(v, µ).

Lozenge tilings Substitutions Continued fractions

Some objects...

• discrete lines, planes, surfaces,

...and some transformations/dynamical systems acting on them

• substitutions

Θ

∗

Θ Θσ

∗ σ ∗

Θσ

∗ σ

• flips

Lozenge tilings Substitutions Continued fractions

Generalized substitutions

Generalized substitutions belong to the family of Combinatorial tiling substitutions according to the terminology of [N. Priebe Frank, A primer of substitution tilings of the Euclidean plane].

Motivation

• Define substitution rules acting on stepped surfaces
• Give a geometric version of a multidimensional continued fraction algorithm

Θ

∗

Θ Θσ

∗ σ ∗

Θσ

∗ σ

Θ

∗ σ

Lozenge tilings Substitutions Continued fractions

Multidimensional substitution

Exemple [Arnoux-Ito]

Θ: 1 → 2 1 2 → 3 3 → 1 How to iterate such a rule? [N. M. Priebe Frank, C. Radin, E. A. Robinson Jr., B. Solomyak, C. Goodman Strauss, K. Gr¨

• chening, A. Haas, J. Lagarias, Y. Wang...]

Based on Arnoux-Ito’s formalism:

• With a morphism of the free group (+ Hypothesis) σ is associated a generalized

substitution Θ(σ)∗.

• We have both local and global information
• Preserves the set of stepped planes and even of stepped surfaces
• We have algebraic properties

Θ(σ ◦ τ)∗ = Θ(τ)∗ ◦ Θ(σ)∗

Lozenge tilings Substitutions Continued fractions

Local rules for Θ

1 → 2 1 2 → 3 3 → 1

Lozenge tilings Substitutions Continued fractions

Local rules for Θ

1 → 2 1 2 → 3 3 → 1 2 1 → 2 3 1 3 1 → 2 1 1 1 1 → 2 2 1 1 2 1 → 2 1 3 1 3 → 2 1 1

Lozenge tilings Substitutions Continued fractions

Local rules for Θ

1 → 2 1 2 → 3 3 → 1 2 1 → 2 3 1 3 1 → 2 1 1 1 1 → 2 2 1 1 2 1 → 2 1 3 1 3 → 2 1 1 1 → 2 1 → 2 3 1 → 2 3 1 1 → 2 2 1 3 1 1 → 2 2 3 1 2 1 3 1 1 → 2 2 3 1 2 1 3 1 2 1 3 1 1

Lozenge tilings Substitutions Continued fractions

Local rules for Θ

1 → 2 1 2 → 3 3 → 1 2 1 → 2 3 1 3 1 → 2 1 1 1 1 → 2 2 1 1 2 1 → 2 1 3 1 3 → 2 1 1 1 → 2 1 → 2 3 1 → 2 3 1 1 → 2 2 1 3 1 1 → 2 2 3 1 2 1 3 1 1 → 2 2 3 1 2 1 3 1 2 1 3 1 1

Lozenge tilings Substitutions Continued fractions

Substitutions

Let σ be a substitution on A. Example: σ(1) = 12, σ(2) = 13, σ(3) = 1. The incidence matrix Mσ of σ is defined by Mσ = (|σ(j)|i)(i,j)∈A2 , where |σ(j)|i counts the number of occurrences of the letter i in σ(j).

Unimodular substitution

det Mσ = ±1

Abelianisation

Let d be the cardinality of A. Let l : A⋆ → Nd be the abelinisation map l(w) = t(|w|1, |w|2, · · · , |w|d).

Lozenge tilings Substitutions Continued fractions

Global rule

Let (x, 1∗), (x, 2∗), (x, 3∗) stand for the following faces

e3 e1 e2 x e3 e1 e2 x e3 e1 e2 x

Generalized substitution [Arnoux-Ito][Ei]

Let σ be a unimodular morphism of the free froup. Θ∗

σ(x, i∗) =

X

k∈A

X

P, σ(k)=PiS

(M−1

σ

(x − l(P)) , k∗).

Θ

∗

Θ Θσ

∗ σ ∗

Θσ

∗ σ

Lozenge tilings Substitutions Continued fractions

Action on a plane

Theorem [Arnoux-Ito, Fernique]

Let σ be a unimodular substitution. Let v ∈ Rd

+ be a nonzero vector. The generalized

substitution Θ∗

σ maps without overlaps the stepped plane Pv,µ onto PtMσv,µ.

Θ

∗ σ

Lozenge tilings Substitutions Continued fractions

Action on a plane

Theorem [Arnoux-Ito, Fernique]

Let σ be a unimodular substitution. Let v ∈ Rd

+ be a nonzero vector. The generalized

substitution Θ∗

σ maps without overlaps the stepped plane Pv,µ onto PtMσv,µ.

Θ

∗ σ

Let σ be a unimodular morphism of the free group. Let v ∈ Rd

+ be a nonzero vector

such that

tMσv ≥ 0.

Then, Θ∗

σ maps without overlaps the stepped plane Pv,µ onto PtMσv,µ.

Lozenge tilings Substitutions Continued fractions

Theorem

Let σ be a unimodular subtitution. The generalized substitution Θ∗

σ acts without

• verlaps on stepped surfaces.

Θ

∗ σ

Lozenge tilings Substitutions Continued fractions

A characterization of stepped surfaces by flips

Projection

Let π the orthogonal projection on the diagonal plane x + y + z = 0.

Local finiteness

A sequence of flips (ϕsn)n∈N is said to be locally finite if, for any n0 ∈ N, the set {sn ∈ Z3 , π(sn) = π(sn0)} is bounded.

Theorem [Arnoux-B.-Fernique-Jamet]

A union of faces U is a stepped surface if and only if there exist a stepped plane P and a locally finite sequence of flips (ϕsn)n∈N such that U = lim

n→∞ ϕsn ◦ . . . ◦ ϕs1(P).

Lozenge tilings Substitutions Continued fractions Θ

∗

Θ Θσ

∗ σ ∗

Θσ

∗ σ

Lozenge tilings Substitutions Continued fractions

Brun’s algorithm

Brun’s transformation is defined on [0, 1]d\{0} by T(α1, · · · , αd) = ( α1 αi , · · · , αi−1 αi , { 1 αi }, αi+1 αi , · · · , αd αi ), where i = min{j| αj = ||α||∞}. For a ∈ N and i ∈ {1, . . . , d}, we introduce the following (d + 1) × (d + 1) matrix: Ba,i = B B @ a 1 Ii−1 1 Id−i 1 C C A . One has (1, α) = ||α||∞Ba,i(1, T(α)). Let βa,i be a substitution with incidence matrix Ba,i, then P(1,α),µ = Θ∗

βa,i (P||α||∞(1,T(α)),µ)).

Lozenge tilings Substitutions Continued fractions

Brun’s algorithm

Brun’s transformation is defined on [0, 1]d\{0} by T(α1, · · · , αd) = ( α1 αi , · · · , αi−1 αi , { 1 αi }, αi+1 αi , · · · , αd αi ), where i = min{j| αj = ||α||∞}.

• Unimodular algorithm
• Weak convergence (convergence of the type |α − pn/qn|)
• Metric results (natural extension)

Lozenge tilings Substitutions Continued fractions

Brun expansion of a stepped plane

How to read on the stepped plane i = min{j| αj = ||α||∞} and the partial quotient a = [1/αi]? We thus need entries comparisons and floor computations. If the above parameters are known, then P(1,α),µ = Θ∗

βa,i (P||α||∞(1,T(α)),µ)),

where the substitution βa,i has incidence matrix Ba,i with ||α||∞Ba,i(1, T(α)) = (1, α).

Lozenge tilings Substitutions Continued fractions

Brun expansion of a stepped plane P(1,α)

We consider the stepped plane with normal vector (1, (α1, α2)), with α = (α1, α2) ∈ [0, 1]2\{0}. One has α1 > α2

Lozenge tilings Substitutions Continued fractions

Brun expansion of a stepped plane

One has [ 1

α1 ] = 2.

Lozenge tilings Substitutions Continued fractions

Brun expansion of a stepped plane

Finally, one has P(1,α) = Θ∗

β2,1(P(1,T(α))).

We thus can define geometrically Brun expansions of a stepped surfaces.

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

(1, α0, β0) = (1, 11

14 , 19 21 )

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

(1, 11

14 , 19 21 ) ∝ B1,2(1, 33 38 , 2 19 )

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

(1, 33

38 , 2 19 ) ∝ B1,1(1, 5 33 , 4 33 )

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

(1, 5

33 , 4 33 ) ∝ B6,1(1, 3 4 , 4 5 )

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

(1, 3

4 , 4 5 ) ∝ B1,2(1, 3 4 , 1 4 )

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

(1, 3

4 , 1 4 ) ∝ B1,1(1, 1 3 , 1 3 )

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

(1, 1

3 , 1 3 ) ∝ B3,1(1, 0, 1)

Lozenge tilings Substitutions Continued fractions

Arithmetics Geometry d-uple α ∈ [0, 1]d stepped plane P(1,α) (1, αn) ∝ Bn(1, αn+1) P(1,αn) = Θ∗

σn(P(1,αn+1))

with t Bn incidence matrice of σn

(1, 0, 1) ∝ B1,2(1, 0, 0)

Lozenge tilings Substitutions Continued fractions

Applications

• Generation of discrete planes
• Recognition problem: Given a set of points in Zd, does there exist an arithmetic

discrete plane that contains them?

• Generalized Rauzy fractals associated with non-algebraic parameters