Intro Crystiles Graphs Criteria Fractal crystallographic tilings Benoˆ ıt Loridant Leoben University - TU Vienna, Austria April, 2007 Supported by FWF, Projects S9604, S9610, and S9612. Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Introduction Purpose: self-similar tiles T providing a tiling of the plane with respect to a crystallographic group. Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Introduction Purpose: self-similar tiles T providing a tiling of the plane with respect to a crystallographic group. Question: when is T homeomorphic to a closed disk? Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Introduction Purpose: self-similar tiles T providing a tiling of the plane with respect to a crystallographic group. Question: when is T homeomorphic to a closed disk? Results: criteria involving the configuration of the neighbors of T in the tiling. Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Crystallographic tiling If T is a compact set with T = T o , Γ a family of isometries of R 2 such that R 2 = � γ ∈ Γ γ ( T ) and the γ ( T ) do not overlap, we say that T tiles R 2 by Γ . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Crystallographic tiling If T is a compact set with T = T o , Γ a family of isometries of R 2 such that R 2 = � γ ∈ Γ γ ( T ) and the γ ( T ) do not overlap, we say that T tiles R 2 by Γ . Γ ≤ Isom( R 2 ) is a crystallographic group if Γ ≃ Z 2 ⋉ { id, r 2 , . . . , r d } with r 2 , . . . , r d isometries of finite order greater than 2 . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Crystallographic reptile - Γ crystallographic group, Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Crystallographic reptile - Γ crystallographic group, - g expanding affine map such that g Γ g − 1 ≤ Γ , Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Crystallographic reptile - Γ crystallographic group, - g expanding affine map such that g Γ g − 1 ≤ Γ , - D ⊂ Γ digit set (complete set of right coset representatives of Γ /g Γ g − 1 ). Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Crystallographic reptile - Γ crystallographic group, - g expanding affine map such that g Γ g − 1 ≤ Γ , - D ⊂ Γ digit set (complete set of right coset representatives of Γ /g Γ g − 1 ). A crystallographic reptile with respect to (Γ , D , g ) is a set T ⊂ R 2 such that T tiles R 2 by Γ and � g ( T ) = δ ( T ) . δ ∈D Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria An example of crystile We consider the group p 3 = { a i b j r k , i, j ∈ Z , k ∈ { 0 , 1 , 2 } } where a ( x, y ) = ( x + 1 , y ) √ � � b ( x, y ) = x + 1 / 2 , y + 3 / 2 , r = rot[0 , 2 π/ 3] Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria An example of crystile We consider the group p 3 = { a i b j r k , i, j ∈ Z , k ∈ { 0 , 1 , 2 } } where a ( x, y ) = ( x + 1 , y ) √ � � b ( x, y ) = x + 1 / 2 , y + 3 / 2 , r = rot[0 , 2 π/ 3] the digit set { id, ar 2 , br 2 } , Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria An example of crystile We consider the group p 3 = { a i b j r k , i, j ∈ Z , k ∈ { 0 , 1 , 2 } } where a ( x, y ) = ( x + 1 , y ) √ � � b ( x, y ) = x + 1 / 2 , y + 3 / 2 , r = rot[0 , 2 π/ 3] the digit set { id, ar 2 , br 2 } , √ the map g ( x, y ) = 3( y, − x ) . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria An example of crystile Figure: Terdragon T defined by g ( T ) = T ∪ ar 2 ( T ) ∪ br 2 ( T ) . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Digit representation T is the union of its n -th level subpieces: δ 1 ∈D g − 1 δ 1 ( T ) � T = Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Digit representation T is the union of its n -th level subpieces: δ 1 ∈D g − 1 δ 1 ( T ) � T = δ 1 ,δ 2 ∈D g − 1 δ 1 g − 1 δ 2 ( T ) = � Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Digit representation T is the union of its n -th level subpieces: δ 1 ∈D g − 1 δ 1 ( T ) � T = δ 1 ,δ 2 ∈D g − 1 δ 1 g − 1 δ 2 ( T ) = � lim n →∞ g − 1 δ 1 . . . g − 1 δ n ( a ) , δ j ∈ D � � = ( a is any point of R 2 ). Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Digit representation T is the union of its n -th level subpieces: δ 1 ∈D g − 1 δ 1 ( T ) � T = δ 1 ,δ 2 ∈D g − 1 δ 1 g − 1 δ 2 ( T ) = � lim n →∞ g − 1 δ 1 . . . g − 1 δ n ( a ) , δ j ∈ D � � = ( a is any point of R 2 ). Therefore, each x ∈ T has an adress x = ( δ 1 δ 2 . . . ) . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Known results [Gelbrich - 1994] Two crystiles ( T ; Γ , D , g ) and ( T ′ ; Γ ′ , D ′ , g ′ ) are isomorphic if there is an affine bijection φ : T → T ′ preserving the pieces of all levels. There are at most finitely many isomorphy classes of disk-like plane crystiles with k digits ( k ≥ 2 ). Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Known results [Gelbrich - 1994] Two crystiles ( T ; Γ , D , g ) and ( T ′ ; Γ ′ , D ′ , g ′ ) are isomorphic if there is an affine bijection φ : T → T ′ preserving the pieces of all levels. There are at most finitely many isomorphy classes of disk-like plane crystiles with k digits ( k ≥ 2 ). [Luo, Rao, Tan - 2002] T connected self-similar tile with T o � = ∅ is disk-like whenever its interior is connected. Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Known results [Gelbrich - 1994] Two crystiles ( T ; Γ , D , g ) and ( T ′ ; Γ ′ , D ′ , g ′ ) are isomorphic if there is an affine bijection φ : T → T ′ preserving the pieces of all levels. There are at most finitely many isomorphy classes of disk-like plane crystiles with k digits ( k ≥ 2 ). [Luo, Rao, Tan - 2002] T connected self-similar tile with T o � = ∅ is disk-like whenever its interior is connected. [Bandt, Wang - 2001] Criterion of disk-likeness for lattice tiles in terms of the number of neighbors of the central tile. Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Neighbors Set of neighbors: S := { γ ∈ Γ \ { id } , T ∩ γ ( T ) � = ∅} . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Neighbors Set of neighbors: S := { γ ∈ Γ \ { id } , T ∩ γ ( T ) � = ∅} . The boundary of T is: � ∂T = T ∩ γ ( T ) . γ ∈S Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Boundary graph The boundary graph G ( S ) is defined as follows: the vertices are the γ ∈ S , Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Boundary graph The boundary graph G ( S ) is defined as follows: the vertices are the γ ∈ S , δ 1 | δ ′ − − − → γ 1 ∈ G ( S ) iff 1 there is an edge γ γ g − 1 δ ′ 1 = g − 1 δ 1 γ 1 with γ, γ 1 ∈ S and δ 1 , δ ′ 1 ∈ D . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Boundary characterization Theorem Let δ 1 , δ 2 , . . . a sequence of digits and γ ∈ S . Then the following assertions are equivalent. x = ( δ 1 δ 2 . . . ) ∈ T ∩ γ ( T ) . There is an infinite walk in G ( S ) of the shape: δ 1 | δ ′ δ 2 | δ ′ δ 3 | δ ′ 1 2 3 γ − − − → γ 1 − − − → γ 2 − − − → . . . (1) for some γ i ∈ S and δ ′ i ∈ D . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Boundary characterization Theorem Let δ 1 , δ 2 , . . . a sequence of digits and γ ∈ S . Then the following assertions are equivalent. x = ( δ 1 δ 2 . . . ) ∈ T ∩ γ ( T ) . There is an infinite walk in G ( S ) of the shape: δ 1 | δ ′ δ 2 | δ ′ δ 3 | δ ′ 1 2 3 γ − − − → γ 1 − − − → γ 2 − − − → . . . (1) for some γ i ∈ S and δ ′ i ∈ D . Remark. The set of neighbors S and the boundary graph G ( S ) can be obtained algorithmically for given data (Γ , D , g ) . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Neighbor and Adjacent neighbor graphs The neighbor graph of a crystallographic tiling is the graph G N with • vertices γ ∈ Γ • edges γ − γ ′ if γ ( T ) ∩ γ ′ ( T ) � = ∅ , i.e. , γ ′ ∈ γ S . Benoˆ ıt Loridant Crystiles
Intro Crystiles Graphs Criteria Neighbor and Adjacent neighbor graphs The neighbor graph of a crystallographic tiling is the graph G N with • vertices γ ∈ Γ • edges γ − γ ′ if γ ( T ) ∩ γ ′ ( T ) � = ∅ , i.e. , γ ′ ∈ γ S . Adjacent neighbors: γ, γ ′ with γ ( T ) ∩ γ ′ ( T ) contains a point of ( γ ( T ) ∪ γ ′ ( T )) o . A denotes the set of adjacent neighbors of id . It can be obtained with the help of G ( S ) . Benoˆ ıt Loridant Crystiles
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