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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
IntroductionMetaresultTribonacci
1 Introduction
2 Metaresult
3 Tribonacci
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci Tiling
Fig.: A piece of the discrete plane
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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).
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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).
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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)
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Hierarchical structure
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Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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).
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci
Discrete plane.
Generalized substitution (dual). and on Z2.
IFS.
”Combinatorial” substitution.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci tiling : substitution rule
B
Fig.: τ (12,1)Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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).
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci tiling
τ (n,n−3)
τ (n,k).
τ (∞,k). Combinatorics appears in the geometry from k = 5.
µ−kτ (∞,k) → τ
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci tiling
Fig.: τ (12,1)
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci tiling
Fig.: τ (15,1)
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci tiling : scale 5
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Fig.: T (5)
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci tiling
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Fig.: Skeletons
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Tribonacci tiling
GG
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MAGA
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MA
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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
IntroductionMetaresultTribonacci
Tribonacci tiling
Fig.: Skeletons
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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
(n−1)1
W(n+1)4 = W
(n−1)2 W
(n)6
W(n+1)5 = W
(n)7
W(n+1)6 = W
(n)1
W(n+1)7 = W
(n)2 W
(n)3 W
(n−2)7 .
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Topological metatile
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Fig.: Meta-tile T (8) with informations on marked vertices.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
v -configuration
State of a vertex : 0, ±1 or 2.
2
-10
12
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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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Labelled tiles
Fig.: The labelled tiles (partial)
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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”
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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 ))
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Topological metatile
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Fig.: Meta-tile T (8) with informations on marked vertices.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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.
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
Labelled tiles
]D5 = 9776.
Fig.: Labelled tiles. Put colors
Xavier Bressaud Tribonacci / Sofic
IntroductionMetaresultTribonacci
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).
Xavier Bressaud Tribonacci / Sofic