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When Periodicity Enforces Aperiodicity

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When Periodicity Enforces Aperiodicity

Nicolas B´edaride & Thomas Fernique

Marseille, january 17th, 2013

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1 Settings

2 Main result

3 Examples

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Settings Main result Examples

Planar rhombus tilings

npairwise non-colinear vectors ofR2 tilings ofR2 by n2

rhombi.

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Planar rhombus tilings

Lift: homeomorphism which maps tiles on 2-faces of unit n-cubes.

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Settings Main result Examples

Planar rhombus tilings

Planar: lift in E+ [0,t]n, whereE is the slopeandt thethickness.

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Local rules

Definition

A slopeE haslocal rules (LR) if there is a finite set ofpatchess. t.

any rhombus tiling without any such patch is planar with slopeE.

LR are said to be

strongif the tilings satisfying them have thickness 1;

natural if the thickness 1 tilings satisfy them;

weak otherwise (the thickness is just bounded).

Mathieu’s talk focused on weak LR. We here focus on natural LR.

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Settings Main result Examples

Shadows and subperiods

Shadow: projection on a space generated by three basis vectors.

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Shadows and subperiods

Shadow: projection on a space generated by three basis vectors.

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Settings Main result Examples

Shadows and subperiods

Shadow: projection on a space generated by three basis vectors.

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Shadows and subperiods

Shadow: projection on a space generated by three basis vectors.

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Settings Main result Examples

Shadows and subperiods

Subperiod: shadow period (tiling); shadow rational vector (slope).

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1 Settings

2 Main result

3 Examples

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Settings Main result Examples

A Characterization

Theorem

A slope has natural LR iff finitely many slopes have its subperiods.

This result is moreover constructive (see examples hereafter).

Slopes with natural LR must be algebraic (Le’95). Here, we refine: Corollary

A slope with natural LR is generated by vectors defined over a number field of degree at mostbn2c. Degreebφ(n)2 c is reached.

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A Characterization

Theorem

A slope has natural LR iff finitely many slopes have its subperiods.

This result is moreover constructive (see examples hereafter).

Slopes with natural LR must be algebraic (Le’95). Here, we refine:

Corollary

A slope with natural LR is generated by vectors defined over a number field of degree at mostbn2c. Degreebφ(n)2 c is reached.

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Settings Main result Examples

Necessity (sketch)

Definition

Singular points of orderk ofE: Singk(E) :=∂(E + [0,1]n) +Znk.

Lemma

Singk(E)cuts up the window into convex connected components corresponding to “size k” patches of slope E thickness1 tilings.

Lemma

Subperiods characterize either finitely many slopes, or a continuum.

Lemma

Subperiod'intersection of boundaries of connected component.

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Necessity (sketch)

Definition

Singular points of orderk ofE: Singk(E) :=∂(E + [0,1]n) +Znk.

Lemma

Singk(E)cuts up the window into convex connected components corresponding to “size k” patches of slope E thickness1 tilings.

Lemma

Subperiods characterize either finitely many slopes, or a continuum.

Lemma

Subperiod'intersection of boundaries of connected component.

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Settings Main result Examples

Sufficiency (sketch)

Lemma

Subperiods can be enforced by forbidding finitely many patches.

Definition

A slope satisfies theP-conditionif it contains three non-collinear vectors which project onto subperiods in three irrational shadows.

Lemma

P-condition⇔ planarity of the tilings with the same subperiods.

Lemma

Subperiods characterize finitely many slopes⇒ P-condition holds.

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Settings Main result Examples

Sufficiency (sketch)

Lemma

Subperiods can be enforced by forbidding finitely many patches.

Definition

A slope satisfies theP-conditionif it contains three non-collinear vectors which project onto subperiods in three irrational shadows.

Lemma

P-condition⇔ planarity of the tilings with the same subperiods.

Subperiods characterize finitely many slopes⇒ P-condition holds.

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Settings Main result Examples

Sufficiency (sketch)

Lemma

Subperiods can be enforced by forbidding finitely many patches.

Definition

A slope satisfies theP-conditionif it contains three non-collinear vectors which project onto subperiods in three irrational shadows.

Lemma

P-condition⇔ planarity of the tilings with the same subperiods.

Lemma

Subperiods characterize finitely many slopes⇒ P-condition holds.

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1 Settings

2 Main result

3 Examples

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Settings Main result Examples

Grassmann-Pl¨ ucker coordinates

Definition

The planeR~u+R~v has GP-coordinates (Gij)i<j = (uivj−ujvi)i<j.

Proposition (Grassmann-Pl¨ucker)

GP-coordinates satisfy all the relations GijGkl =GikGjl −GilGjk.

Proposition

Whenever a planar tiling admits p~ei+q~ej +r~ek as a subperiod, the GP-coordinates of its slope satisfy pGjk−qGik+rGij = 0.

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Generalized Penrose tilings

The slope has GP-coordinates (ϕ,1,−1,−ϕ, ϕ,1,−1, ϕ,1, ϕ).

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Settings Main result Examples

Generalized Penrose tilings

The slope has GP-coordinates (ϕ,1,−1,−ϕ, ϕ,1,−1, ϕ,1, ϕ).

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Generalized Penrose tilings

Subperiods yield

G13=G41=G24=G52=G35= 1 G12=G51=G45=G34=G23=:x

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Settings Main result Examples

Generalized Penrose tilings

Plugged into the five GP-relations, this yieldsx2 =x+ 1.

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Generalized Penrose tilings

Subperiods characterize finitely many slopes: the theorem applies!

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Settings Main result Examples

Generalized Penrose tilings

Subperiods are easily enforced in each shadow by forbidden patches.

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Generalized Penrose tilings

This still holds in the tilings, at least in those of thickness 1.

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Settings Main result Examples

Generalized Penrose tilings

This still holds in the tilings, at least in those of thickness 1.

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Generalized Penrose tilings

This still holds in the tilings, at least in those of thickness 1.

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Settings Main result Examples

Generalized Penrose tilings

This still holds in the tilings, at least in those of thickness 1.

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Generalized Penrose tilings

This still holds in the tilings, at least in those of thickness 1.

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Settings Main result Examples

Generalized Penrose tilings

Considering all the shadows yields (simple) natural LR for the tilings.

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Ammann-Beenker tilings

The slope has GP-coordinates (1,√

2,1,1,√ 2,1).

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Settings Main result Examples

Ammann-Beenker tilings

Subperiods yieldG12=G14=G23=G34; GP-relation G13G24= 2.

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Ammann-Beenker tilings

Subperiods thus characterize all the slopes (1,t,1,1,2/t,1),t ∈R.

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Settings Main result Examples

Ammann-Beenker tilings

Subperiods thus characterize all the slopes (1,t,1,1,2/t,1),t ∈R.

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Ammann-Beenker tilings

Subperiods thus characterize all the slopes (1,t,1,1,2/t,1),t ∈R.

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Settings Main result Examples

Ammann-Beenker tilings

Subperiods thus characterize all the slopes (1,t,1,1,2/t,1),t ∈R.

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Ammann-Beenker tilings

The theorem does not apply, but planarity is nevertheless enforced!

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Settings Main result Examples

Ammann-Beenker tilings

Moreover, AB tilings are those maximizing the rhombus frequencies.

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Thank you for your attention!

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