Substitution cut-and-project tilings with n-fold rotational symmetry

Transcription

1 Substitution cut-and-project tilings with n-fold rotational symmetry Victor Hubert Lutfalla 2017 Lutfalla (LIPN & UTU ) Tilings / 35

2 Introduction Tilings Lutfalla (LIPN & UTU ) Tilings / 35

3 Introduction Lutfalla (LIPN & UTU ) Quasicrystals Tilings / 35

4 Introduction Goal In their 2016 article, Jarkko Kari and Markus Rissanen define a explicit method to construct a substitution tiling with n-fold rotational symmetry for any n. In this talk I will only present the case of 2k + 1-fold symmetry and first the 7-fold symmetry. So our first goal was to find a tiling which is: defined by a substitution cut-and-project invariant by rotation of angle 2π 7 Lutfalla (LIPN & UTU ) Tilings / 35

5 Introduction 1 Substitution tilings 2 Cut-and-project 3 Dilatation matrix 4 7-fold Details of a tiling Other tilings 5 Methodology for odd rotational symmetry Lutfalla (LIPN & UTU ) Tilings / 35

6 Substitution tilings 1 Substitution tilings 2 Cut-and-project 3 Dilatation matrix 4 7-fold Details of a tiling Other tilings 5 Methodology for odd rotational symmetry Lutfalla (LIPN & UTU ) Tilings / 35

7 Substitution tilings Detail of a toy substitution dilatation subdivision Lutfalla (LIPN & UTU ) Tilings / 35

8 Substitution tilings Substitution on a pattern Lutfalla (LIPN & UTU ) Tilings / 35

9 Substitution tilings Tilings We have two ways of defining tilings generated by a substitution: T = lim σ n (T ) n The infinitly desubstitutable tilings Lutfalla (LIPN & UTU ) Tilings / 35

10 Substitution tilings Quasiperiodicity A tiling is quasiperiodic when: it is not periodic it is uniformly recurrent Definition (Uniform recurrence) A tiling is uniformly recurrent when for every finite patern m that appears in the tiling, there exists a radius r m so that for every vertex s of the tiling, the pattern m appears at distance r m of s. Lutfalla (LIPN & UTU ) Tilings / 35

11 Cut-and-project 1 Substitution tilings 2 Cut-and-project 3 Dilatation matrix 4 7-fold Details of a tiling Other tilings 5 Methodology for odd rotational symmetry Lutfalla (LIPN & UTU ) Tilings / 35

12 Cut-and-project 2 1 Cut D + H in grey. Discrete line D d : projection T on D projection Ω on D. Lutfalla (LIPN & UTU ) Tilings / 35

13 Cut-and-project n 2 Setting: R n L = Z n E irrational plane W and R orthogonal complementary spaces Cut E + H where H is a compact set with non empty interior. Discrete plane E d = L (E + H) Projection T = Π E (E d ) and window Ω = Π W R (E d ). Lutfalla (LIPN & UTU ) Tilings / 35

14 Cut-and-project The 7-fold ( 1 ) cos ( 2π 7 ) ( 0 ) sin 4π cos 7 E is generated by (cos( 2kπ 7 )) k=0..6 = ( ) ( 2π 7 ) sin 6π cos 7 ( ) and (sin( 2kπ 7 8π cos 7 ( )) k=0..6 = ( 4π 7 ) sin ) ( 6π 7 ) 8π sin 7 10π cos ( ( ) 10π sin 7 ) ( 7 ) 12π 12π cos sin 7 7 Figure: Projection of the canonical basis R 7 Lutfalla (LIPN & UTU ) Tilings / 35

15 Cut-and-project The 7-fold Irrational subspace W : W = E E where E is generated by (cos( 4kπ 7 )) k=0..6 and (sin( 4kπ 7 )) k=0..6 E is generated by (cos( 6kπ 7 )) k=0..6 and (sin( 6kπ 7 )) k=0..6 Rational subspace R: R = line generated by the vector (1) k=0..6 R 7 = E E E Lutfalla (LIPN & UTU ) Tilings / 35

16 Cut-and-project The 7-fold Projections Projection over E, E and E of a small set Lutfalla (LIPN & UTU ) Tilings / 35

17 Dilatation matrix 1 Substitution tilings 2 Cut-and-project 3 Dilatation matrix 4 7-fold Details of a tiling Other tilings 5 Methodology for odd rotational symmetry Lutfalla (LIPN & UTU ) Tilings / 35

18 Dilatation matrix Dilatation matrix b -c -a a c -b a substitution a a -c -b a a With a = ( ) 1 0, b = 0 ( ) 0 1 and c = 0 M = ( ) We have ϕ(a) = M a... so ϕ is described by M. The study of M leads to a good understanding of the dilatation and substitution in R 3. Lutfalla (LIPN & UTU ) Tilings / 35

19 Dilatation matrix Admissible dilatation Lutfalla (LIPN & UTU ) Tilings / 35

20 Dilatation matrix Conditions on the matrix With the setting R n = E W R we need: E, W, R are eigenspaces λ E > 1 λ W, λ R 1 Lutfalla (LIPN & UTU ) Tilings / 35

21 7-fold 1 Substitution tilings 2 Cut-and-project 3 Dilatation matrix 4 7-fold Details of a tiling Other tilings 5 Methodology for odd rotational symmetry Lutfalla (LIPN & UTU ) Tilings / 35

22 7-fold Details of a tiling We found C = Which has eigenspaces E, E, E and with eigensvalues λ , λ , λ and λ = 0 This matrix only defines the edge of the substitution Lutfalla (LIPN & UTU ) Tilings / 35

23 7-fold Details of a tiling Tiling the metarhombi Criterion and tiling algorithm in Kenyon93. Lutfalla (LIPN & UTU ) Tilings / 35

24 7-fold Details of a tiling Figure: Complete substitution σ over E Lutfalla (LIPN & UTU ) Tilings / 35

25 7-fold Details of a tiling Projections over E, E Lutfalla (LIPN & UTU ) Tilings / 35

26 7-fold Details of a tiling Main result Theorem (Main result) The substitution σ defines a set E d = lim σ n (R2 1 ) that satisfies: n T = Π E (E d ) is a rhombus tiling with invariance by rotation of angle 2π 7 the closure of Ω = Π W R (E d ) is compact and has a non-empty interior E d is a cut-and-project set. Lutfalla (LIPN & UTU ) Tilings / 35

27 7-fold Other tilings Lutfalla (LIPN & UTU ) Tilings / 35

28 7-fold Other tilings Lutfalla (LIPN & UTU ) Tilings / 35

29 Methodology for odd rotational symmetry 1 Substitution tilings 2 Cut-and-project 3 Dilatation matrix 4 7-fold Details of a tiling Other tilings 5 Methodology for odd rotational symmetry Lutfalla (LIPN & UTU ) Tilings / 35

30 Methodology for odd rotational symmetry Cut-and-project setting The ambiant space is R n with n = 2k + 1. We have R n = E i where i=0..k 1 E i is the space generated by the vectors (cos( 2(i+1)jπ )) n j=0..n 1, (sin( 2(i+1)jπ )) n j=0..n 1. is the line generated by (1) j=0..n 1 E 0 is the tiling plane. Lutfalla (LIPN & UTU ) Tilings / 35

31 Methodology for odd rotational symmetry Tiles We have k rhombus tiles r 0... r k 1 The rhombi r i has angles (2i+1)π n So r 0 has narrow angle π n and 2(k i)π n. 2kπ and wide angle = (n 1)π. n n The edges of a substitution will be a sequence of such rhombi w 1 r 1 + w 2 r w k r k Example : Lutfalla (LIPN & UTU ) Tilings / 35

32 Methodology for odd rotational symmetry Decomposition For every rhombus r i we define a dilatation ϕ i associated to r i The diagonal vector of r 0 is e 0 e k so ϕ 0 is defined by the matrix ( ) (2i+1)(2j+1)π ϕ i has, E j for eigenspaces with eigenvalues 0 and λ (i,j) = 2 cos 2n. ϕ = i=0..k 1 w i ϕ i Lutfalla (LIPN & UTU ) Tilings / 35

33 Methodology for odd rotational symmetry Eigenvalues of ϕ w 0 The edges of the substitution are defined by the vector. w k 1 The dilatation is ϕ = w i ϕ i i=0..k 1 ϕ has, E j for eigenspaces with eigenvalues 0 and λ j = w i λ (i,j) i=0..k 1 λ 0 λ (0,0)... λ (k 1,0) w 0 So we have. = λ k 1 λ (0,k 1)... λ (k 1,k 1) w k 1 ( ) (2i + 1)(2j + 1)π With λ (i,j) = 2 cos 2n Lutfalla (LIPN & UTU ) Tilings / 35

34 Methodology for odd rotational symmetry w 0 So now we want an edge such that. such that: w k λ 0 > 1 and the other λ j < 1 This makes the dilatation admissible for cut-and-project the meta-tiles are tilable over E 0 This makes it an actual substitution of the plane Lutfalla (LIPN & UTU ) Tilings / 35

35 Conclusion 1 The 7-fold case is quite well known now We have 2 explicit substitution 7-fold cut-and-project tilings We have a caracterisation of these tilings 2 We designed the methodology for arbitrary dimension n For any dimension we can easily have the existence of admissble substitution matrices The only thing missing is tilability of such dilated-tiles w 0 So now we need to find a sequence. w k 1 substitution (2k + 1)-fold cut-and-project tiling. k N such that for all k it defines a Lutfalla (LIPN & UTU ) Tilings / 35

36 Lutfalla (LIPN & UTU ) Tilings / 6

37 Annexes Kari and Rissanen λ 1 = S λ λ λ 4 = 0 Lutfalla (LIPN & UTU ) Tilings / 6

38 Annexes Technical lemma Let n N. Let R n with the canonical basis (e 1,..., e n), the set of canonical vectors S = {±e 1,..., ±e n} and a subspace W. We define Π W as the orthogonal projection on W. Definition We define the property linked over the sets by x 0 = x linked(x) x, y X, k, x0... x k, x k = y 0 i < 1, ε S = {±e 1,..., ±e n}, x i+1 = x i + ε. We call unit square a set X such that x R n, e, e S with e e, X = {x, x + e, x + e, x + e + e }. Given a set Y, a family (X i ) i I of unit square is called a total covering by unit squares of Y when { x Y, i I, x X i unit square X Y, i I, X = X i, such a covering is called exact when i I, x X i, x Y. Lutfalla (LIPN & UTU ) Tilings / 6

39 Annexes Technical lemma Lemma Let ϕ a function Z n Zn, σ a function P (Z n ) P (Z n ) and R 0 a finite linked set such that 1 σ is the substitution associated to dilatation ϕ: x, σ({x}) = {ϕ(x)}, X Y, σ(x) σ(y ) X, linked(x) linked(σ(x)) D R +, for any unit square X, σ(x) has a diametre D ie: x, y σ(x), d(x, y) D Y, for any total covering by unit squares X 1,..., X p of Y, the image σ(y ) is the union of the images of the X i : σ(y ) σ(x i ) and furthermore if the covering is exact σ(y ) = σ(x i ) i=1...p 2 ϕ is contracting on W : λ < 1, x Z n, Π W (ϕ(x)) λ Π W (x) 3 R 0 σ(r 0 ) We have the following: 1 R W = lim i Π W (σ i (R 0 )) exists 2 M R +, x, y R W, x y M 3 0 R W i=1...p Lutfalla (LIPN & UTU ) Tilings / 6

40 Annexes Rotated substitution Lutfalla (LIPN & UTU ) Tilings / 6

41 Annexes Rosa Lutfalla (LIPN & UTU ) Tilings / 6