Tribonacci tiling is sofic. - IRISA · Tribonacci tiling is so c ... Speci cities of Tribonacci...

IntroductionMetaresultTribonacci

Tribonacci tiling is sofic.

Xavier Bressaud

Universite Paul SabatierInstitut de Mathematiques de Toulouse

Substitutive Tiling and Fractal Geometry, Guangzhou

July 2010

Xavier Bressaud Tribonacci / Sofic


1 Introduction

2 Metaresult

3 Tribonacci



A discrete plane

Let Π be the contracting plane of the matrix A =0@ 1 1 0

1 0 11 0 0

1AApproximation by unit cubes : discrete plane.

Projection (parallel to the PF direction) of the faces of thecubes on a transversal plane.

Tiling by a family of three rhombi T (Tribonacci).

Observe that this tiling has some self-similar stucture.

Space Ω generated by translations (R2) and closure.



Tribonacci Tiling

Fig.: A piece of the discrete plane



Sofic

Tiling by a family of rhombi T , labelled, C.

Local rules L : some glueing are forbiden.

Underlying tiling : forget the labels.

Space ΩT ,C,L of tilings by labelled tiles (T , C) satisfying thelocal rules L (SFT)

Space U(ΩT ,C,L) of the underlying (non labelled) tilings(Sofic).



Tribonacci tiling is sofic

Theorem

There exist a set of colors C and a set of local rules L such that

U(ΩT ,C,L) = Ω.

Last computation : 9776 labelled tiles



Contrast with dimension 1

Discrete line of slope α : Sturmian sequences and tilings.

For α rational : the tiling is periodic (an SFT).

For α quadratic : the tiling is self-similar.

Such tiling is (certainly) not of finite type (no perodic orbits).



Motivations

The result is not surprising ONCE you know the metaresult.

Existing proofs did not yield the result : it had to be checked.

Unerstand existing proofs.

Following a general proof yields a huge set of tiles.

Specificities of Tribonacci tiling (symetries, geometry of tiles)



Tilings

Tilings and spaces of tilings. Topology. Action of R2.

Patch, Language, Finite local complexity.

Substitutive tilings and space of tilings.

SFT and sofic spaces of tilings.



Substitutive tilings are sofic

Preliminaries : Does a set of tiles tile ? Can we decide ?

Berger, Robinson and aperiodic set of tiles.

Moses for squared substitutive tilings.

Goodman-Strauss for polygonal substitutive tilings.

Solomyak to see pseudo-self-similar tilings as self-similar ones.



Hierarchical structure (informal)

A tiling T by a set of tiles T .

A sequence of tilings T (n) by sets of tiles T (n).

Tiles in T (n) are unions of tiles in T (n−1).

Decomposition of all tiles of T (n) yields T (n−1).

The tiling is self-similar if the families T (n) are similar.

Pseudo self-similar if the decomposition is the ”same” at all n.



Hierarchical structure

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Metatile

Consider a self similar tiling and its hierarchy.

Skeleton and boundary of a metatile.

Skeleton intersects the boundary (dim > 1 !)

In the tiling : union of skeletons = union of boundaries

Partitions of the boundaries (scale changes).



Betaproof

A priori the (initial) set of tiles may have a huge universe.

Put more informations on tiles to specify the roles they canplay in the hierarchy.

Put local constraints that force them play the specified role.

So that the tilings obtained following these rules respect thehierarchical structure.

It remains to forget about the additional information.



First idea

Specify the position of the tile in the upper level of thehierarchy (for instance say which part of its boundary it mustglue to which tile to produce a metatile).

Consider the SFT where the rules allow only these glueings.

Automatically, a given tile belongs to a patch forming ametatile.

Hence we arrive at level 2. But how to iterate ?

New tiles may not have the good colors on their boundaries.And we need to put only a finite amount of information oneach tile.



Second idea

Send information from one level to the next. Through theskeletons.

Recall the skeletons are connected to the boundary of theirmetatile.

Let the skeleton ”know” the global configuration of itsmetatile.

Check at the connection that the information is coherent.



Conclusion

On the one hand, this is enough to force the SFT to respectthe hierarchical structure.

On the other hand, it must be checked that the self-similartiling is itself in the SFT.

In practice, we produce a decoration of the self-similar tilingto determine the alphabet that is to be used.



Case of Tribonacci

Understand the combinatorics of the tiles as the scale grows.

Then, use the geometry of tiles to recover information.

Lot of symmetries : the three tiles play similar similar roles.

Main goal : reduce the number of tiles needed.



Tribonacci

Discrete plane.

Generalized substitution (dual). and on Z2.

IFS.

”Combinatorial” substitution.



Tribonacci tiling : substitution rule

B

Fig.: τ (12,1)Xavier Bressaud Tribonacci / Sofic


Tribonacci tiling : inflation

T (n+1) = T (n) ∪(T (n−1) + ω

(n)1

)∪(T (n−2) + ω

(n)2

).

Fig.: Tiles T (3) ⊂ · · · ⊂ T (11).



Tribonacci tiling

τ (n,n−3)

τ (n,k).

τ (∞,k). Combinatorics appears in the geometry from k = 5.

µ−kτ (∞,k) → τ



Tribonacci tiling

Fig.: τ (12,1)



Tribonacci tiling

Fig.: τ (15,1)



Tribonacci tiling

Fig.: τ (∞,∞)



Tribonacci tiling : scale 5

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Fig.: T (5)



Tribonacci tiling

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Fig.: Skeletons



Tribonacci tiling

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On veut comprendre ce qu'il faut transmettre sur les aretes pour controler ce qui se passe sur les bords "exterieurs"1. Le long de chaque frontiere on dit le nom de la cellule (de gauche) et celui à laquelle elle veut se coller. La ou c'est clair. 2. La ou c'est indeterminé, on laisse de a liberté. Mais bien sur ce "sera" fixé. 3. Il faut qd meme transmettre des contraintes aux bords exterieurs. Il faut pouvoir demander une uniformité de la couleur.

Fig.: SkeletonsXavier Bressaud Tribonacci / Sofic


Tribonacci tiling

Fig.: Skeletons



Tribonacci tiling : geometry of the boundary

(W

(1)1 , . . . ,W

(1)7 ) = (∅, c , ∅, a, c−1, ∅, a−1),

(W(2)1 , . . . ,W

(2)7 ) = (∅, a, ∅, b, a−1, ∅, b−1),

(W(3)1 , . . . ,W

(3)7 ) = (b, ∅, ∅, c , b−1, ∅, c−1).

W(n+1)1 = W

(n−2)1 W

(n−2)2 W

(n−1)6

W(n+1)2 = W

(n−1)7

W(n+1)3 = W

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(n)7

W(n+1)6 = W

(n)1

W(n+1)7 = W

(n)2 W

(n)3 W

(n−2)7 .



Topological metatile

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Fig.: Meta-tile T (8) with informations on marked vertices.



v -configuration

State of a vertex : 0, ±1 or 2.

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Seven states of the vertices of a tile T : v -configuration V (T ).

V = VP ∪ VM ∪ VG set of possible v -configurations.

Describes how the boundary is cut in pieces of skeletons ofdifferent scales.



Automaton

Let T be a metatile : T = TP ∪ TM ∪ TG .

If V = V (T ), V P = V (TP),V M = V (TM),V G = V (TG ),

then,V = V P

1 V M7 V M

1 V M2 V G

7 V G1 V G

2 .

and, V P = V1 0 1 0 2 0 −1V M = V3 V4 1 0 2 −1 V2

V G = V6 V7 0 1 2 −1 V5



V

VP 0 0 1 0 2 0 −1−1 0 1 0 2 0 −1

1 0 1 0 2 0 −1

VM 1 0 1 0 2 −1 11 0 1 0 2 −1 00 1 1 0 2 −1 20 1 1 0 2 −1 −10 1 1 0 2 −1 10 1 1 0 2 −1 0

VG −1 2 0 1 2 −1 20 −1 0 1 2 −1 2−1 1 0 1 2 −1 2−1 0 0 1 2 −1 2−1 −1 0 1 2 −1 2



Labelled tiles

Tn : T (n),T (n+1),T (n+2) × VP (3× 3 = 9)

T (n+1),T (n+2) × VM (2× 6 = 12)

T (n+2) × VG (1× 5 = 5)

]Tn = 26.

Fig.: Labelled tiles



Labelled tiles

Fig.: The labelled tiles (partial)



First step

Set T5 of labelled tiles.

Local rule : at all triple point : (0, 0), (2, 2, 2) ou (0,−1, 1).

Thanks to the geometry/combinatorics this constraint isenough to ”climb one scale”, i.e. all tile belong to a metatile.

To control the configuration of the metatile, need to transmit”information”



Coloring edges

Each edge E belong to the skeleton of a metatile T (of scaleK ) ; T = TP ∪ TM ∪ TG .

Type : S(E) = P,M ou G according to whether the edgebelongs to TP ∩ TM , TM ∩ TG ou TG ∩ TP .

The edge E is colored by :

c(E) = (S(E),V (T ))



Topological metatile

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Fig.: Meta-tile T (8) with informations on marked vertices.



Automaton (2)

Again we use the automatic rule :

C = CP1 CM

7 CM1 CM

2 CG7 CG

1 CG2 .

and, CP = C1 C1 (P,V ) (P,V ) (G ,V ) (G ,V ) C7

CM = C3 C4 (M,V ) (M,V ) (P,V ) C2 C2

CG = C6 C7 C7 (G ,V ) (G ,V ) C6 C5

to check the colorings that really appear in Tribonacci tiling.

Denote V ⊂ (P,M,G × V)7 these colorings. ]V = 4893.

Let Dn be the set of decorated tiles (label + color) at scale n.



Rules

It remains to fix a set rules L.

Constraint on the vertices as before.

Tiles have same colors on common boundaries.

At vertices with states (−1, 1, 0), with ”incoming” edge oftype M : check that the v -configuration on the incoming edgeis coherent with that on the passing edge.



SFT

Consider the SFT space of tilings with tiles in Dn satisfying localrules L. We claim that the space of underlying tilings is exactly theTribonacci tiling space.



Labelled tiles

]D5 = 9776.

Fig.: Labelled tiles. Put colors



Further

Reduce the number of tiles (to something that can be drawn).

Tilings arising from other substitutions. What are theimportant combinatorial/topological properties of the tiles ?

Non stationary hierarchical structure (among a finite set andin an effective order).


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