Aperiodic tilings and quasicrystals - NTNU · Aperiodic order? The last three examples arenot...
Transcript of Aperiodic tilings and quasicrystals - NTNU · Aperiodic order? The last three examples arenot...
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Aperiodic tilings and quasicrystalsPerleforedrag, NTNU
Antoine Julien
Nord universitetLevanger, Norway
October 13th, 2017
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Introduction
Outline
1 Introduction
2 History and motivations
3 Building aperiodic tilings
4 Symbolic coding of tilings
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Introduction
What is a tiling?
A tiling is a covering of the space by geometric shapes (tiles) such that:
they cover the space;
there is no overlap.
Often, we ask for finitely many tiles types up to some kind of motion (forus: translation).
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Introduction
An example: tiling with squares
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Introduction
An example: tiling with hexagons
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Introduction
An example: another periodic tiling
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Introduction
Tilings with 5-fold symmetry?
Can one produce a regular tiling with 5-fold symmetry?
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Introduction
An example: the quasiperiodic Penrose tiling
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Introduction
An example: the quasiperiodic Penrose tiling
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Introduction
An example: the quasiperiodic Shield tiling
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Introduction
Aperiodic order?
The last three examples are not periodic, yet highly repetitive.
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History and motivations
Outline
1 Introduction
2 History and motivations
3 Building aperiodic tilings
4 Symbolic coding of tilings
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History and motivations
Crystals are ordered
The groups of symmetries of periodic tilings of the plane (or of the space)are entirely classified.So perfect crystals in which atoms are arranged periodically are classifiedalso.
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History and motivations
The first quasicrystal
In 1982, Dan Shechtman observed a diffraction pattern of a material with:
sharp peaks (order);
a forbidden 10-fold symmetry (non periodic).
First documented observation of aperiodic order. Need for new models!
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History and motivations
The diffraction pattern: 10 fold???
On the left, the diffraction pattern of a zinc–manganese–holmium alloy.On the right, Dan Shechtman’s logbook.
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History and motivations
Quasicrystals
Shechtman’s observation was first thought to be an experimental error.
“There are no quasicrystals, just quasi-scientists!”
Linus Pauling, Nobel Prize winner
In 1992, the International Union of Crystallography revised the definitionof crystal.
In 2011, Dan Shechtman won the Nobel Prize in chemistry.
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History and motivations
Quasicrystals
Shechtman’s observation was first thought to be an experimental error.
“There are no quasicrystals, just quasi-scientists!”
Linus Pauling, Nobel Prize winner
In 1992, the International Union of Crystallography revised the definitionof crystal.
In 2011, Dan Shechtman won the Nobel Prize in chemistry.
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History and motivations
Quasicrystals
Shechtman’s observation was first thought to be an experimental error.
“There are no quasicrystals, just quasi-scientists!”
Linus Pauling, Nobel Prize winner
In 1992, the International Union of Crystallography revised the definitionof crystal.
In 2011, Dan Shechtman won the Nobel Prize in chemistry.
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History and motivations
Sir Roger Penrose and his tilings
Penrose tilings were invented before quasicrystals. The reason? They werepretty:R. Penrose, The Role of Aesthetics in Pure and Applied Mathematical Research, Institute of
Mathematics and its Applications Bulletin (1974).
In 1997, Kleenex sold toilet paper printed with Penrose tilings. Outrageensued:
“. . . when it comes to the population of Great Britain being invited by amultinational [corporation] to wipe their bottoms on what appears to bethe work of a Knight of the Realm without his permission, then a laststand must be made.”
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History and motivations
Sir Roger Penrose and his tilings
Penrose tilings were invented before quasicrystals. The reason? They werepretty:R. Penrose, The Role of Aesthetics in Pure and Applied Mathematical Research, Institute of
Mathematics and its Applications Bulletin (1974).
In 1997, Kleenex sold toilet paper printed with Penrose tilings. Outrageensued:
“. . . when it comes to the population of Great Britain being invited by amultinational [corporation] to wipe their bottoms on what appears to bethe work of a Knight of the Realm without his permission, then a laststand must be made.”
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History and motivations
Wang tiles and the tiling problem
Even before, aperiodic tilings appeared to answer this question:Given a set of tiles, does it tile the plane? (answer: the general problem is
undecidable.)
The tiles above (with matching rules) tile the plane, but only aperiodically.K. Culik, J. Kari, On aperiodic sets of Wang tiles, Lecture Notes in Computer Science (1997).
Is the number of tiles optimal?
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History and motivations
An optimal set of Wang tiles
In 2015, a computer search found a set of 11 aperiodic Wang tiles with 4colours. This is optimal.
E. Jandel, M. Rao, An aperiodic set of 11 Wang tiles, prepublication.
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History and motivations
An optimal set of Wang tiles
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Building aperiodic tilings
Outline
1 Introduction
2 History and motivations
3 Building aperiodic tilings
4 Symbolic coding of tilings
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Building aperiodic tilings
Different methods
Several methods: substitution, cut-and-project, local matching rules. . .
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Building aperiodic tilings
The chair substitution
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Building aperiodic tilings
The chair substitution
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Building aperiodic tilings
Why is it aperiodic?
The chair substitution has the unique decomposition property.
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Building aperiodic tilings
Why is it aperiodic?
The chair substitution has the unique decomposition property.
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Building aperiodic tilings
Why is it aperiodic?
This substitution rule does not have the unique decomposition property.
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Building aperiodic tilings
Other example: Penrose
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Building aperiodic tilings
Tilings with local matching rules
Aperiodic tilings can be generated by local adjacency rules.
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Building aperiodic tilings
Islamic art
P. Lu and P. Steinhardt, Decagonal and quasi-crystalline tilings in medieval Islamic architecture
Science 315 (2007), 1106–1110.
Gunbad-i Kabud tomb tower in Maragha, Iran (1197)
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Building aperiodic tilings
Islamic art
A set of basic “Girih tiles” used to produce Islamic patterns.
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Building aperiodic tilings
Islamic art
A section of the Topkapi scroll (15th–16th century).
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Building aperiodic tilings
Islamic art
P. Cromwell, The search for quasi-periodicity in Islamic 5-fold ornament, The Mathematical
Intelligencer 31 (2009), 36–56.
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Building aperiodic tilings
The cut-and-project method
Idea: cut an “irrational” slice of a regular lattice; project it on a plane.
The resulting pattern will inherit some regularity from the lattice.
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Building aperiodic tilings
Example: Sturmian sequences
.
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Building aperiodic tilings
Example: Sturmian sequences
.
.
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Building aperiodic tilings
Example: Sturmian sequences
.
.
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Building aperiodic tilings
Example: Sturmian sequences
.
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Building aperiodic tilings
Example: Sturmian sequences
.
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Building aperiodic tilings
Example: Sturmian sequences
In this case: one gets Sturmian sequences.
Aperiodic if and only if the slope is irrational.
Properties of this tiling related to arithmetic properties of the slope.
For ex. the tiling is (the rewriting of) a substitution if and only if the slope is a quadratic
irrational.
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Building aperiodic tilings
Meyer sets
These cut-and-project sets (or “model sets”) are examples of sets studiedby Yves Meyer.
Original motivations: harmonic analysis. Which subsets Λ ⊂ Rd havecharacters which can be approximated by characters of Rd?
Meyer sets have an approximate analogue of a reciprocal lattice.
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Building aperiodic tilings
Meyer sets
These cut-and-project sets (or “model sets”) are examples of sets studiedby Yves Meyer.
Theorem
If a point-set Λ is a Meyer set, and if θ > 1 satisfies θΛ ⊂ Λ, then θ iseither a Pisot number or a Salem number.
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Symbolic coding of tilings
Outline
1 Introduction
2 History and motivations
3 Building aperiodic tilings
4 Symbolic coding of tilings
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Symbolic coding of tilings
How many “chair” tilings are there?
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Symbolic coding of tilings
Building chair tilings: growing a patch
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Symbolic coding of tilings
Building chair tilings: growing a patch
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Symbolic coding of tilings
Building chair tilings: growing a patch
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Symbolic coding of tilings
Building chair tilings: growing a patch
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Symbolic coding of tilings
Building chair tilings: growing a patch
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Symbolic coding of tilings
Building chair tilings: growing a patch
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Symbolic coding of tilings
Building chair tilings: growing further
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Symbolic coding of tilings
Building chair tilings: growing further
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Symbolic coding of tilings
Building chair tilings: changing the head of the path
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Symbolic coding of tilings
Diagram encoding: summary
For a good substitution:
Infinite paths on the diagram encode tilings of the plane;
Cofinal paths correspond to tilings which are translate of each other;
The converse does not hold.
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Symbolic coding of tilings
Diagram encoding: summary
For a good substitution:
Infinite paths on the diagram encode tilings of the plane;
Cofinal paths correspond to tilings which are translate of each other;
The converse does not hold.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 54 / 54