Quasiperiodic Tilings and Quasicrystals

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Thomas Fernique Universit´e Paris 13

Isfahan University of Technology April 22, 2014

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1 Quasiperiodic tilings

2 Local rules

3 Quasicrystals

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2 Local rules

3 Quasicrystals

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Tiling

Overlap-free covering of the plane by compact sets calledtiles.

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Quasiperiodic tilings Local rules Quasicrystals

Periodic tiling

Invariant by two independant translations.

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Periodic tiling

Invariant by two independant translations.

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Periodic tiling

Invariant by two independant translations. Simple description!

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Periodic tiling

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Periodic tiling

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Quasiperiodic tiling

Any patternP reoccurs at distance at most d(P) from any point.

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Quasiperiodic tiling

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Quasiperiodic tiling

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Substitutions

Inflate the tiles and subdivide each one into original tiles. Iterate.

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Substitutions

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Substitutions

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Substitutions

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Cut and projection (DeBruijn 1981)

Select the 2D unit faces ofZⁿ lying intoE + [0,1]ⁿ. Project onE.

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Cut and projection (DeBruijn 1981)

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Cut and projection (DeBruijn 1981)

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Cut and projection (DeBruijn 1981)

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Cut and projection (DeBruijn 1981)

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Cut and projection (DeBruijn 1981)

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2 Local rules

3 Quasicrystals

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Periodic tilings

Alternative description by finitely many allowed/forbidden patterns.

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Periodic tilings

Alternative description by finitely many notched/decorated tiles.

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Non-periodic tiling

Theorem (Berger 1964)

Decorated tile sets that form only non-periodic tilings do exist.

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Substitutive tiling

Theorem (Mozes 1990, Goodmann-Strauss 1998, F.-Ollinger 2010) Substitutive tilings can (generally) be enforced by decorated tiles.

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Cut and project tiling (1/2)

Cuts enforced by allowed patterns are algebraic.

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Cut and project tiling (1/2)

Cuts enforced by allowed patterns are algebraic.

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Cut and project tiling (1/2)

Cuts enforced by allowed patterns are algebraic. More precisely?

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Cut and project tiling (2/2)

q

s

q’ q’

s s q s

q s q s

q s q

s’ s’ s

q s q’

t

q’ t s’

u r

r u

Theorem (F.-Sablik 2012)

A cut is enforced by decorated tiles if and only if it is computable.

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2 Local rules

3 Quasicrystals

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Crystals (18th century)

Periodic atom arrangement sharp peaks diffraction. Conversely?

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The crystallographic restriction

Theorem

If a lattice is invariant under a rotation by ^2π_n, then n∈ {2,3,4,6}.

Proof:

the rotation has an integer matrix in any lattice base;

the trace of a rotation by θ is 2 cos(θ), in any base;

2 cos(θ)∈Z⇒ θ∈ {0,^2π₂ ,^2π₃ ,^2π₄ ,^2π₆ }.

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Shechtmanite (1982)

Sharp peaks diffraction but non-periodic rotational symmetry!!?

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Quasicrystals (1992)

As quasiperiodic tilings!

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Quasicrystals (1992)

As quasiperiodic tilings! With local rules aiming to explain stability.

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Open questions

The 7-fold tilings admit local rules.

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Open questions

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Open questions

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Open questions

The 7-fold tilings admit local rules. Where are 7-fold quasicrystals?