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
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 1 / 54
Introduction
Outline
1 Introduction
2 History and motivations
3 Building aperiodic tilings
4 Symbolic coding of tilings
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 2 / 54
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).
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 3 / 54
Introduction
An example: tiling with squares
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 4 / 54
Introduction
An example: tiling with hexagons
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 5 / 54
Introduction
An example: another periodic tiling
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 6 / 54
Introduction
Tilings with 5-fold symmetry?
Can one produce a regular tiling with 5-fold symmetry?
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 7 / 54
Introduction
An example: the quasiperiodic Penrose tiling
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 8 / 54
Introduction
An example: the quasiperiodic Penrose tiling
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 9 / 54
Introduction
An example: the quasiperiodic Shield tiling
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 10 / 54
Introduction
Aperiodic order?
The last three examples are not periodic, yet highly repetitive.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 11 / 54
History and motivations
Outline
1 Introduction
2 History and motivations
3 Building aperiodic tilings
4 Symbolic coding of tilings
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 12 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 13 / 54
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!
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 14 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 15 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 16 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 16 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 16 / 54
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.”
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 17 / 54
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.”
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 17 / 54
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?
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 18 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 19 / 54
History and motivations
An optimal set of Wang tiles
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 20 / 54
Building aperiodic tilings
Outline
1 Introduction
2 History and motivations
3 Building aperiodic tilings
4 Symbolic coding of tilings
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 21 / 54
Building aperiodic tilings
Different methods
Several methods: substitution, cut-and-project, local matching rules. . .
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 22 / 54
Building aperiodic tilings
The chair substitution
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 23 / 54
Building aperiodic tilings
The chair substitution
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 24 / 54
Building aperiodic tilings
Why is it aperiodic?
The chair substitution has the unique decomposition property.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 25 / 54
Building aperiodic tilings
Why is it aperiodic?
The chair substitution has the unique decomposition property.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 26 / 54
Building aperiodic tilings
Why is it aperiodic?
This substitution rule does not have the unique decomposition property.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 27 / 54
Building aperiodic tilings
Other example: Penrose
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 28 / 54
Building aperiodic tilings
Tilings with local matching rules
Aperiodic tilings can be generated by local adjacency rules.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 29 / 54
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)
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 30 / 54
Building aperiodic tilings
Islamic art
A set of basic “Girih tiles” used to produce Islamic patterns.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 31 / 54
Building aperiodic tilings
Islamic art
A section of the Topkapi scroll (15th–16th century).
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 32 / 54
Building aperiodic tilings
Islamic art
P. Cromwell, The search for quasi-periodicity in Islamic 5-fold ornament, The Mathematical
Intelligencer 31 (2009), 36–56.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 33 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 34 / 54
Building aperiodic tilings
Example: Sturmian sequences
.
α
.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 35 / 54
Building aperiodic tilings
Example: Sturmian sequences
.
.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 36 / 54
Building aperiodic tilings
Example: Sturmian sequences
.
.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 37 / 54
Building aperiodic tilings
Example: Sturmian sequences
.
E
.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 38 / 54
Building aperiodic tilings
Example: Sturmian sequences
.
E
a
a
a
a
a
b
b
b
a
.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 39 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 40 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 41 / 54
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.
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 42 / 54
Symbolic coding of tilings
Outline
1 Introduction
2 History and motivations
3 Building aperiodic tilings
4 Symbolic coding of tilings
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 43 / 54
Symbolic coding of tilings
How many “chair” tilings are there?
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 44 / 54
Symbolic coding of tilings
Building chair tilings: growing a patch
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 45 / 54
Symbolic coding of tilings
Building chair tilings: growing a patch
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 46 / 54
Symbolic coding of tilings
Building chair tilings: growing a patch
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 47 / 54
Symbolic coding of tilings
Building chair tilings: growing a patch
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 48 / 54
Symbolic coding of tilings
Building chair tilings: growing a patch
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 49 / 54
Symbolic coding of tilings
Building chair tilings: growing a patch
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 50 / 54
Symbolic coding of tilings
Building chair tilings: growing further
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 51 / 54
Symbolic coding of tilings
Building chair tilings: growing further
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 52 / 54
Symbolic coding of tilings
Building chair tilings: changing the head of the path
A. Julien (NordU) Aperiodic tilings and quasicrystals October 13th, 2017 53 / 54
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
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