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The Ammann-Beenker Tilings Revisited

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The Ammann-Beenker Tilings Revisited

Nicolas B´edaride (LATP, Marseille) Thomas Fernique (LIPN, Paris)

Cairns, September 6th, 2012

(2)

Arrowed tiles (Beenker, 1982)

Square and rhombus tiles form so-called octagonal tilings.

(3)

Arrowed tiles (Beenker, 1982)

Square and rhombus tiles form so-called octagonal tilings.

(4)

Arrowed tiles (Beenker, 1982)

Which tilings do form arrowedsquare and rhombus tiles?

(5)

Arrowed tilings

Theorem

The arrowed tilings digitize the planes(1,t,1,1,2/t,1), t ∈R.e

Corollary

The Ammann-Beenker tilings maximize the ratio rhombi/squares.

Underlying idea

I rhombi = aluminium and squares = manganese (for example);

I Ammann-Beenker tiling = quasicrystal Al√2Mn1;

I Al7Mn5,Al41Mn29,Al239Mn169 = quasicrystal approximants.

(6)

Arrowed tilings

Theorem

The arrowed tilings digitize the planes(1,t,1,1,2/t,1), t ∈R.e

Corollary

The Ammann-Beenker tilings maximize the ratio rhombi/squares.

Underlying idea

I rhombi = aluminium and squares = manganese (for example);

I Ammann-Beenker tiling = quasicrystal Al√2Mn1;

I Al7Mn5,Al41Mn29,Al239Mn169 = quasicrystal approximants.

(7)

Alternating rhombi

Consider an octagonal tiling. Assume it can be arrowed.

(8)

Alternating rhombi

Consider a “stripe” of tiles (also calledConway worms).

(9)

Alternating rhombi

If rhombi do not alternate orientation, then tiles cannot be arrowed.

(10)

Alternating rhombi

Conversely, consider an octagonal tiling where rhombi alternate.

(11)

Alternating rhombi

Endow rhombi with arrows pointing towards the acute angles.

(12)

Alternating rhombi

Endow squares with parallel arrows being equally oriented.

(13)

Alternating rhombi

Gluing each arrow with the tile on its left yields arrowed tiles.

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Planar octagonal tilings

Lift: homeomorphism which maps rhombi on 2-faces of unit 4-cubes.

(15)

Planar octagonal tilings

Planar: lift inE+ [0,t]4, whereE is the slope and t the thickness.

(16)

Shadows and subperiods

Shadow: orthogonal projection of the lift along a basis vector.

(17)

Shadows and subperiods

Shadow: orthogonal projection of the lift along a basis vector.

(18)

Shadows and subperiods

Shadow: orthogonal projection of the lift along a basis vector.

(19)

Shadows and subperiods

Shadow: orthogonal projection of the lift along a basis vector.

(20)

Shadows and subperiods

Shadow: orthogonal projection of the lift along a basis vector.

(21)

Shadows and subperiods

Shadow: orthogonal projection of the lift along a basis vector.

(22)

Shadows and subperiods

Subperiod: shadow period. Rhombus alternation forces simple ones.

(23)

Pl¨ ucker coordinates

Definition (Pl¨ucker, 1865)

E =R~u+R~v ⊂R4 has coord. (Gij)ij = (uivj −ujvi)ij ∈P5(R).

Proposition

The tile proportions of planar tilings are given by the Pl¨ucker coord.

Example

The Ammann-Beenker tilings are the planar tilings of thickness 1 and slope (1,√

2,1,1,√

2,1); they have√

2 rhombi for 1 square.

(24)

Linear and quadratic relations

Proposition

Subperiods of planar tilings yield linear relations on Pl¨ucker coord.

Example

Subperiods forced by arrowed tiles yield: G12=G14=G23=G34.

Proposition (Pl¨ucker, 1865)

(Gij)ij ∈P5(R) describes a plane iff G12G34=G13G24−G14G23.

Lemma

The planar arrowed tilings have slope(1,t,1,1,2/t,1), t ∈Re.

(25)

Linear and quadratic relations

Proposition

Subperiods of planar tilings yield linear relations on Pl¨ucker coord.

Example

Subperiods forced by arrowed tiles yield: G12=G14=G23=G34.

Proposition (Pl¨ucker, 1865)

(Gij)ij ∈P5(R) describes a plane iff G12G34=G13G24−G14G23.

Lemma

The planar arrowed tilings have slope(1,t,1,1,2/t,1), t ∈Re.

(26)

Linear and quadratic relations

Proposition

Subperiods of planar tilings yield linear relations on Pl¨ucker coord.

Example

Subperiods forced by arrowed tiles yield: G12=G14=G23=G34.

Proposition (Pl¨ucker, 1865)

(Gij)ij ∈P5(R) describes a plane iff G12G34=G13G24−G14G23.

Lemma

The planar arrowed tilings have slope(1,t,1,1,2/t,1), t ∈Re.

(27)

Planarity

Lemma

Arrowed tilings are planar with a uniformly bounded thickness.

Theorem

The arrowed tilings digitize the planes(1,t,1,1,2/t,1), t ∈R.e

Corollary

The Ammann-Beenker tilings maximize the ratio rhombi/squares.

(28)

Planarity

Lemma

Arrowed tilings are planar with a uniformly bounded thickness.

Theorem

The arrowed tilings digitize the planes(1,t,1,1,2/t,1), t ∈R.e

Corollary

The Ammann-Beenker tilings maximize the ratio rhombi/squares.

(29)

Remark

Arrowing tiles amouts to forbidding arbitrarily big patterns.

Forbidding finite patterns forces closed rational intervals of slopes. Thus not{√

2}, i.e., Ammann-Benker tilings (cf. Burkov, 1988). Thus notG13=G24, i.e., equiprobable orientations of squares.

(30)

Remark

Arrowing tiles amouts to forbidding arbitrarily big patterns.

Forbidding finite patterns forces closed rational intervals of slopes.

Thus not{√

2}, i.e., Ammann-Benker tilings (cf. Burkov, 1988). Thus notG13=G24, i.e., equiprobable orientations of squares.

(31)

Remark

Arrowing tiles amouts to forbidding arbitrarily big patterns.

Forbidding finite patterns forces closed rational intervals of slopes.

Thus not{√

2}, i.e., Ammann-Benker tilings (cf. Burkov, 1988).

Thus notG13=G24, i.e., equiprobable orientations of squares.

(32)

Remark

Arrowing tiles amouts to forbidding arbitrarily big patterns.

Forbidding finite patterns forces closed rational intervals of slopes.

Thus not{√

2}, i.e., Ammann-Benker tilings (cf. Burkov, 1988).

Thus notG13=G24, i.e., equiprobable orientations of squares.

(33)

Beyond Ammann-Beenker tilings

Here, subperiods characterize a family of slopes and the planarity.

When they characterize finitely many slopes, the planarity follows.

All this extends to general canonical projection tilings!

(34)

Beyond Ammann-Beenker tilings

Here, subperiods characterize a family of slopes and the planarity.

When they characterize finitely many slopes, the planarity follows.

All this extends to general canonical projection tilings!

(35)

Beyond Ammann-Beenker tilings

Here, subperiods characterize a family of slopes and the planarity.

When they characterize finitely many slopes, the planarity follows.

All this extends to general canonical projection tilings!

(36)

Beyond Ammann-Beenker tilings

Here, subperiods characterize a family of slopes and the planarity.

When they characterize finitely many slopes, the planarity follows.

All this extends to general canonical projection tilings!

(37)

Beyond Ammann-Beenker tilings

Here, subperiods characterize a family of slopes and the planarity.

When they characterize finitely many slopes, the planarity follows.

All this extends to general canonical projection tilings!

(38)

Beyond Ammann-Beenker tilings

Here, subperiods characterize a family of slopes and the planarity.

When they characterize finitely many slopes, the planarity follows.

All this extends to general canonical projection tilings!

(39)

Beyond Ammann-Beenker tilings

Here, subperiods characterize a family of slopes and the planarity.

When they characterize finitely many slopes, the planarity follows.

All this extends to general canonical projection tilings!

(40)

Thank you for your attention

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