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 3 1 2 1 2 1 1 2 1 2 3 3 3 2 1 2 1 2 1 2 1 1 3 3 1 2 2 1 1 2 1 2 1 3 3 3 1 2 1 2 1 1 2 1 2 3 3 1 1 2 1 2 1 2 1
Lozenge tilings Substitutions Continued fractions Stepped surfaces e 3 0 0 0 e 1 e 2 Definition A stepped surface is defined as a union of faces such that the orthogonal projection onto 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 orthogonal 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 orthogonal 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 orthogonal 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. z x y
Lozenge tilings Substitutions Continued fractions Stepped surface Definition A functional discrete surface is defined as a union of pointed faces such that the orthogonal 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. n m 1 1 1 3 2 1 2 3 1 1 1 1 1 1 2 2 3 1 3 3 2 2 2 2 1 2 2 2 3 1 3 3 3 3 3 3
Lozenge tilings Substitutions Continued fractions Arithmetic discrete planes Let v ∈ R 3 and µ ∈ R . The standard arithmetic discrete hyperplane P ( v , µ ) is defined as P ( v , µ ) = { x ∈ Z 3 ; 0 ≤ � x , v � + µ < || v || 1 } . The stepped plane P v ,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 σ ∗ ∗ Θ σ Θ σ ∗ Θ 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 σ ∗ ∗ Θ σ Θ σ ∗ Θ σ ∗ Θ σ
Lozenge tilings Substitutions Continued fractions Multidimensional substitution Exemple [Arnoux-Ito] 2 Θ: 1 → 2 → 3 3 → 1 1 How to iterate such a rule? [N. M. Priebe Frank, C. Radin, E. A. Robinson Jr., B. Solomyak, C. Goodman Strauss, K. Gr¨ ochening, 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 2 1 → 2 → 3 3 → 1 1
Lozenge tilings Substitutions Continued fractions Local rules for Θ 2 1 → 2 → 3 3 → 1 1 2 2 2 2 2 1 1 2 1 �→ 3 1 �→ 1 �→ 2 1 2 1 �→ 1 �→ 1 3 1 1 3 1 1 1 3
Lozenge tilings Substitutions Continued fractions Local rules for Θ 2 1 → 2 → 3 3 → 1 1 2 2 2 2 2 1 1 2 1 �→ 3 1 �→ 1 �→ 2 1 2 1 �→ 1 �→ 1 3 1 1 3 1 1 1 3 2 2 2 2 2 1 1 �→ �→ �→ 3 1 �→ 3 1 3 1 1 1 1 2 2 2 2 3 1 2 1 3 1 2 1 3 1 �→ �→ 3 1 2 1 1 3 1 1
Lozenge tilings Substitutions Continued fractions Local rules for Θ 2 1 → 2 → 3 3 → 1 1 2 2 2 2 2 1 1 2 1 �→ 3 1 �→ 1 �→ 2 1 2 1 �→ 1 �→ 1 3 1 1 3 1 1 1 3 2 2 2 2 2 1 1 �→ �→ �→ 3 1 �→ 3 1 3 1 1 1 1 2 2 2 2 3 1 2 1 3 1 2 1 3 1 �→ �→ 3 1 2 1 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 M σ of σ is defined by M σ = ( | σ ( j ) | i ) ( i , j ) ∈A 2 , where | σ ( j ) | i counts the number of occurrences of the letter i in σ ( j ). Unimodular substitution det M σ = ± 1 Abelianisation Let d be the cardinality of A . Let l : A ⋆ → N d 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 e 3 e 3 e 3 x x x e 2 e 2 e 2 e 1 e 1 e 1 Generalized substitution [Arnoux-Ito][Ei] Let σ be a unimodular morphism of the free froup. X X Θ ∗ σ ( x , i ∗ ) = ( M − 1 ( x − l ( P )) , k ∗ ) . σ k ∈A P , σ ( k )= PiS ∗ Θ σ ∗ ∗ Θ σ Θ σ ∗ Θ σ
Lozenge tilings Substitutions Continued fractions Action on a plane Theorem [Arnoux-Ito, Fernique] Let σ be a unimodular substitution. Let v ∈ R d + be a nonzero vector. The generalized substitution Θ ∗ σ maps without overlaps the stepped plane P v ,µ onto P t M σ v ,µ . ∗ Θ σ
Lozenge tilings Substitutions Continued fractions Action on a plane Theorem [Arnoux-Ito, Fernique] Let σ be a unimodular substitution. Let v ∈ R d + be a nonzero vector. The generalized substitution Θ ∗ σ maps without overlaps the stepped plane P v ,µ onto P t M σ v ,µ . ∗ Θ σ Let σ be a unimodular morphism of the free group. Let v ∈ R d + be a nonzero vector such that t M σ v ≥ 0 . Then, Θ ∗ σ maps without overlaps the stepped plane P v ,µ onto P t M σ v ,µ .
Lozenge tilings Substitutions Continued fractions Theorem Let σ be a unimodular subtitution. The generalized substitution Θ ∗ σ acts without overlaps 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 ( ϕ s n ) n ∈ N is said to be locally finite if, for any n 0 ∈ N , the set { s n ∈ Z 3 , π ( s n ) = π ( s n 0 ) } 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 ( ϕ s n ) n ∈ N such that U = lim n →∞ ϕ s n ◦ . . . ◦ ϕ s 1 ( P ) .
Lozenge tilings Substitutions Continued fractions ∗ Θ σ ∗ ∗ Θ σ Θ σ ∗ Θ σ
Lozenge tilings Substitutions Continued fractions Brun’s algorithm Brun’s transformation is defined on [0 , 1] d \{ 0 } by T ( α 1 , · · · , α d ) = ( α 1 , · · · , α i − 1 , { 1 } , α i +1 , · · · , α d ) , α i α i α i α i α i where i = min { j | α j = || α || ∞ } . For a ∈ N and i ∈ { 1 , . . . , d } , we introduce the following ( d + 1) × ( d + 1) matrix: 0 1 1 a I i − 1 B C B a , i = A . B C 1 0 @ I d − i One has (1 , α ) = || α || ∞ B a , i (1 , T ( α )) . Let β a , i be a substitution with incidence matrix B a , i , then P (1 ,α ) ,µ = Θ ∗ β a , 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 ( α 1 , · · · , α d ) = ( α 1 , · · · , α i − 1 , { 1 } , α i +1 , · · · , α d ) , α i α i α i α i α i where i = min { j | α j = || α || ∞ } . • Unimodular algorithm • Weak convergence (convergence of the type | α − p n / q n | ) • 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 | α j = || α || ∞ } and the partial quotient a = [1 /α i ]? We thus need entries comparisons and floor computations. If the above parameters are known, then P (1 ,α ) ,µ = Θ ∗ β a , i ( P || α || ∞ (1 , T ( α )) ,µ ) ) , where the substitution β a , i has incidence matrix B a , i with || α || ∞ B a , i (1 , T ( α )) = (1 , α ) .
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