Aperiodic Tilings: Notions and Properties Michael Baake & Uwe - - PowerPoint PPT Presentation

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Aperiodic Tilings: Notions and Properties Michael Baake & Uwe Grimm Faculty of Mathematics University of Bielefeld, Germany Department of Mathematics and Statistics The Open University, Milton Keynes, UK Fields Institute, Toronto, 20

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Aperiodic Tilings: Notions and Properties

Michael Baake & Uwe Grimm

Faculty of Mathematics University of Bielefeld, Germany Department of Mathematics and Statistics The Open University, Milton Keynes, UK

Fields Institute, Toronto, 20 October 2011 – p.1

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Quasicrystals

Fields Institute, Toronto, 20 October 2011 – p.2

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Quasicrystals

Fields Institute, Toronto, 20 October 2011 – p.2

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Dan Shechtman

Wolf Prize in Physics 1999 Nobel Prize in Chemistry 2011

Fields Institute, Toronto, 20 October 2011 – p.3

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Periodic point sets

Definition: A (discrete) point set Λ ⊂ Rd is called periodic, when t + Λ = Λ holds for some t = 0. It is called crystallographic when the group of periods,

per(Λ) = {t ∈ Rd | t + Λ = Λ}, is a lattice.

Crystallographic restriction: If (t, M) is a Euclidean motion that maps a crystallographic point set Λ ⊂ Rd onto itself, the characteristic polynomial of M has integer coefficients only. In particular, for d ∈ {2, 3}, the possible rotation symmetries have order 1, 2, 3, 4 or 6.

Fields Institute, Toronto, 20 October 2011 – p.4

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Non-periodic point sets

Definition: A discrete point set Λ ⊂ Rd is called non-crystallographic when per(Λ) is not a lattice, and non-periodic when per(Λ) = {0}. Examples:

Z \ {0} (Z \ {0}) × Z

Definition: The hull of a discrete point set Λ is defined as

X(Λ) := {t + Λ | t ∈ Rd},

where the closure is taken in the local (rubber) topology.

Fields Institute, Toronto, 20 October 2011 – p.5

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Non-periodic point sets

Definition: A discrete point set Λ ⊂ Rd is called aperiodic when X(Λ) contains only non-periodic elements. It is called strongly aperiodic when the remaining symmetry group of the hull is a finite group.

Fields Institute, Toronto, 20 October 2011 – p.6

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Aperiodic point sets

Silver mean substitution:

a → aba, b → a

(λPF = 1 + √ 2 )

Silver mean point set:

Λ =

  • x ∈ Z[

√ 2 ] | x′ ∈ [−

√ 2 2 , √ 2 2 ]

  • Fields Institute, Toronto, 20 October 2011

– p.7

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Model sets

CPS:

Rd

π

← − − − Rd × Rm

πint

− − − − → Rm ∪ ∪ ∪ dense π(L)

1−1

← − − − − L − − − − → πint(L)

  • L

⋆

− − − − − − − − − − − − − − − − − − − − → L⋆

Model set:

Λ = {x ∈ L | x⋆ ∈ W }

(assumed regular)

with W ⊂ Rm compact, λ(∂W) = 0

Diffraction:

  • γ =

k∈L⊛|A(k)|2 δk

with L⊛ = π(L∗)

(Fourier module of Λ)

and amplitude A(k) = dens(Λ)

vol(W)

1

W (−k⋆)

Fields Institute, Toronto, 20 October 2011 – p.8

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Ammann-Beenker tiling

L = Z[ξ] L ∼ Z4 ⊂ R2 × R2 O: octagon ξ = exp(2πi/8) φ(8) = 4 ⋆-map: ξ → ξ3 ΛAB =

  • x ∈ Z1 + Zξ + Zξ2 + Zξ3 | x⋆ ∈ O
  • 1

ξ ξ2 ξ3 1⋆ ξ⋆ ξ2⋆ ξ3⋆

Fields Institute, Toronto, 20 October 2011 – p.9

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Ammann-Beenker tiling

physical space internal space

Fields Institute, Toronto, 20 October 2011 – p.9

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Ammann-Beenker tiling

Fields Institute, Toronto, 20 October 2011 – p.9

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Aperiodic tilings

Fields Institute, Toronto, 20 October 2011 – p.10

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Aperiodic tilings

Many examples with hierarchical structure (see below). Exception: The Kari-Culik prototile set

1 2

1

1 2 1 2 1 2

1 1 1 2

1 2

1

1 2 1 2

1 2 1 1 1 2 1 2 1 1 2 2 2 1 2 1 1 2 1

Fields Institute, Toronto, 20 October 2011 – p.10

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Question

Is there a single shape that tiles space without gaps or

  • verlaps, but does not admit any periodic tiling?

Fields Institute, Toronto, 20 October 2011 – p.11

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Question

Is there a single shape that tiles space without gaps or

  • verlaps, but does not admit any periodic tiling?

3D: Schmitt-Conway-Danzer ‘einstein’

Fields Institute, Toronto, 20 October 2011 – p.11

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Question

Is there a single shape that tiles space without gaps or

  • verlaps, but does not admit any periodic tiling?

3D: Schmitt-Conway-Danzer ‘einstein’ 2D: Penrose tiling (two tiles)

Fields Institute, Toronto, 20 October 2011 – p.11

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Question

Is there a single shape that tiles space without gaps or

  • verlaps, but does not admit any periodic tiling?

3D: Schmitt-Conway-Danzer ‘einstein’ 2D: Penrose tiling (two tiles) No monotile known — but Penrose’s 1 + ε + ε2 tiling

Fields Institute, Toronto, 20 October 2011 – p.11

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The Taylor Tiling: Story

19 Feb 2010: Email from Joshua Socolar announcing An aperiodic hexagonal tile (joint preprint with Joan M. Taylor)

Fields Institute, Toronto, 20 October 2011 – p.12

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The Taylor Tiling: Story

19 Feb 2010: Email from Joshua Socolar announcing An aperiodic hexagonal tile (joint preprint with Joan M. Taylor) 28 Feb 2010: Visit Joan Taylor in Burnie, Tasmania

Fields Institute, Toronto, 20 October 2011 – p.12

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The Taylor Tiling: Story

19 Feb 2010: Email from Joshua Socolar announcing An aperiodic hexagonal tile (joint preprint with Joan M. Taylor) based on Joan’s unpublished manuscript Aperiodicity of a functional monotile which is available (with hand-drawn diagrammes) from http://www.math.uni-bielefeld.de/sfb701/ preprints/view/420 (slight difference in definition of matching rules)

Fields Institute, Toronto, 20 October 2011 – p.12

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Joan Taylor

Fields Institute, Toronto, 20 October 2011 – p.13

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Joan Taylor

Fields Institute, Toronto, 20 October 2011 – p.13

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Joan Taylor

Fields Institute, Toronto, 20 October 2011 – p.13

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Robinson’s tiling

Fields Institute, Toronto, 20 October 2011 – p.14

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Robinson’s tiling

Fields Institute, Toronto, 20 October 2011 – p.14

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Half-hex tiling

Fields Institute, Toronto, 20 October 2011 – p.15

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Half-hex tiling

Fields Institute, Toronto, 20 October 2011 – p.15

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Half-hex tiling

Fields Institute, Toronto, 20 October 2011 – p.15

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Half-hex tiling

Fields Institute, Toronto, 20 October 2011 – p.15

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Half-hex tiling

Fields Institute, Toronto, 20 October 2011 – p.15

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Half-hex tiling

Fields Institute, Toronto, 20 October 2011 – p.15

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Half-hex tiling

hexagonal tile still admits periodic tilings of the plane

Fields Institute, Toronto, 20 October 2011 – p.15

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Half-hex tiling

Fields Institute, Toronto, 20 October 2011 – p.15

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Penrose’s 1 + ε + ε2 tiling

3 tiles: 1 + ε + ε2 ‘key tiles’ encode matching rule information proof of aperiodicity (Penrose) the ε tile transmits information along edge

Fields Institute, Toronto, 20 October 2011 – p.16

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The monotile

(figures from Socolar & Taylor An aperiodic hexagonal monotile)

Fields Institute, Toronto, 20 October 2011 – p.17

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The monotile

(figures from Socolar & Taylor An aperiodic hexagonal monotile)

Fields Institute, Toronto, 20 October 2011 – p.17

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Forced patterns

(figures from Socolar & Taylor An aperiodic hexagonal monotile)

Fields Institute, Toronto, 20 October 2011 – p.18

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Filling the gaps

(figures from Socolar & Taylor An aperiodic hexagonal monotile)

Fields Institute, Toronto, 20 October 2011 – p.19

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Filling the gaps

(figures from Socolar & Taylor An aperiodic hexagonal monotile)

Fields Institute, Toronto, 20 October 2011 – p.19

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Filling the gaps

(figures from Socolar & Taylor An aperiodic hexagonal monotile)

Fields Institute, Toronto, 20 October 2011 – p.19

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Composition-decomposition method

(Franz Gähler 1993)

method to show that matching rules (local rules) enforce non-periodicity based on inflation (self-similarity) requirements: Inflation rule has to respect matching rules: Tiles that match must have decompositions that match In any admitted tiling, each tile can be composed, together with part of its neighbours, to a unique supertile The supertiles inherit markings that enforce equivalent matching rules

Fields Institute, Toronto, 20 October 2011 – p.20

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Taylor’s substitution

(figures from Taylor’s manuscript Aperiodicity of a functional monotile)

Fields Institute, Toronto, 20 October 2011 – p.21

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Taylor’s substitution

(figures from Taylor’s manuscript Aperiodicity of a functional monotile)

Fields Institute, Toronto, 20 October 2011 – p.21

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Inflation tiling

Fields Institute, Toronto, 20 October 2011 – p.22

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Inflation tiling

Fields Institute, Toronto, 20 October 2011 – p.22

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Inflation tiling

Relation to Penrose’s 1 + ε + ε2 tiling:

Fields Institute, Toronto, 20 October 2011 – p.22

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References

  • M. Baake and U. Grimm, Theory of Aperiodic Order: A Mathematical Invitation

(CUP , Cambridge), in preparation.

  • F. Gähler, Matching rules for quasicrystals: The composition-decomposition method,
  • J. Non-Cryst. Solids 153& 154 (1993) 160–164.
  • B. Grünbaum and G.C. Shephard, Tilings and Patterns (Freeman, New York, 1987).
  • R. Penrose, Remarks on tiling: Details of a (1 + ε + ε2)-aperiodic set,

in: The Mathematics of Long-Range Aperiodic Order, ed. R.V. Moody (Kluwer, Dordrecht, 1997) pp. 467–497.

  • D. Shechtman, I. Blech, D. Gratias and J.W. Cahn, Metallic phase with long-range
  • rientational order and no translational symmetry,
  • Phys. Rev. Lett. 53 (1984) 1951–1953.

J.E.S. Socolar and J.M. Taylor, An aperiodic hexagonal tile,

  • J. Combin. Th. A 118 (2011) 2207–2231.

J.E.S. Socolar and J.M. Taylor, Forcing nonperiodicity with a single tile, preprint arXiv:1009.1419 (2010). J.M. Taylor, Aperiodicity of a functional monotile, preprint Bielefeld CRC 701: 0-015 (2010), available via http://www.math.uni-bielefeld.de/sfb701/preprints/view/420

Fields Institute, Toronto, 20 October 2011 – p.23