Aperiodic tilings and quasicrystals
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1 Aperiodic tilings and quasicrystals Perleforedrag, NTNU Antoine Julien Nord universitet Levanger, Norway October 13 th, 2017 A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
2 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 13 th, / 54
3 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 (for us: translation). A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
4 Introduction An example: tiling with squares A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
5 Introduction An example: tiling with hexagons A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
6 Introduction An example: another periodic tiling A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
7 Introduction Tilings with 5-fold symmetry? Can one produce a regular tiling with 5-fold symmetry? A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
8 Introduction An example: the quasiperiodic Penrose tiling A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
9 Introduction An example: the quasiperiodic Penrose tiling A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
10 Introduction An example: the quasiperiodic Shield tiling A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
11 Introduction Aperiodic order? The last three examples are not periodic, yet highly repetitive. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
12 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 13 th, / 54
13 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 classified also. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
14 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 13 th, / 54
15 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 13 th, / 54
16 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 A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
17 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 definition of crystal. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
18 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 definition of crystal. In 2011, Dan Shechtman won the Nobel Prize in chemistry. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
19 History and motivations Sir Roger Penrose and his tilings Penrose tilings were invented before quasicrystals. The reason? They were pretty: R. Penrose, The Rôle of Aesthetics in Pure and Applied Mathematical Research, Institute of Mathematics and its Applications Bulletin (1974). A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
20 History and motivations Sir Roger Penrose and his tilings Penrose tilings were invented before quasicrystals. The reason? They were pretty: R. Penrose, The Rôle 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. Outrage ensued:... when it comes to the population of Great Britain being invited by a multinational [corporation] to wipe their bottoms on what appears to be the work of a Knight of the Realm without his permission, then a last stand must be made. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
21 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 13 th, / 54
22 History and motivations An optimal set of Wang tiles In 2015, a computer search found a set of 11 aperiodic Wang tiles with 4 colours. This is optimal. E. Jandel, M. Rao, An aperiodic set of 11 Wang tiles, prepublication. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
23 History and motivations An optimal set of Wang tiles A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
24 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 13 th, / 54
25 Building aperiodic tilings Different methods Several methods: substitution, cut-and-project, local matching rules... A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
26 Building aperiodic tilings The chair substitution A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
27 Building aperiodic tilings The chair substitution A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
28 Building aperiodic tilings Why is it aperiodic? The chair substitution has the unique decomposition property. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
29 Building aperiodic tilings Why is it aperiodic? The chair substitution has the unique decomposition property. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
30 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 13 th, / 54
31 Building aperiodic tilings Other example: Penrose A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
32 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 13 th, / 54
33 Building aperiodic tilings Islamic art P. Lu and P. Steinhardt, Decagonal and quasi-crystalline tilings in medieval Islamic architecture Science 315 (2007), Gunbad-i Kabud tomb tower in Maragha, Iran (1197) A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
34 Building aperiodic tilings Islamic art A set of basic Girih tiles used to produce Islamic patterns. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
35 Building aperiodic tilings Islamic art A section of the Topkapi scroll (15th 16th century). A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
36 Building aperiodic tilings Islamic art P. Cromwell, The search for quasi-periodicity in Islamic 5-fold ornament, The Mathematical Intelligencer 31 (2009), A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
37 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 13 th, / 54
38 Building aperiodic tilings Example: Sturmian sequences. α. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
39 Building aperiodic tilings Example: Sturmian sequences.. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
40 Building aperiodic tilings Example: Sturmian sequences.. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
41 Building aperiodic tilings Example: Sturmian sequences. E. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
42 Building aperiodic tilings Example: Sturmian sequences. a b a b a a b a a E. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
43 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 13 th, / 54
44 Building aperiodic tilings Meyer sets These cut-and-project sets (or model sets ) are examples of sets studied by Yves Meyer. Original motivations: harmonic analysis. Which subsets Λ R d have characters which can be approximated by characters of R d? Meyer sets have an approximate analogue of a reciprocal lattice. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
45 Building aperiodic tilings Meyer sets These cut-and-project sets (or model sets ) are examples of sets studied by Yves Meyer. Theorem If a point-set Λ is a Meyer set, and if θ > 1 satisfies θλ Λ, then θ is either a Pisot number or a Salem number. A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
46 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 13 th, / 54
47 Symbolic coding of tilings How many chair tilings are there? A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
48 Symbolic coding of tilings Building chair tilings: growing a patch A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
49 Symbolic coding of tilings Building chair tilings: growing a patch A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
50 Symbolic coding of tilings Building chair tilings: growing a patch A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
51 Symbolic coding of tilings Building chair tilings: growing a patch A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
52 Symbolic coding of tilings Building chair tilings: growing a patch A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
53 Symbolic coding of tilings Building chair tilings: growing a patch A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
54 Symbolic coding of tilings Building chair tilings: growing further A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
55 Symbolic coding of tilings Building chair tilings: growing further A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
56 Symbolic coding of tilings Building chair tilings: changing the head of the path A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
57 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; A. Julien (NordU) Aperiodic tilings and quasicrystals October 13 th, / 54
58 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 13 th, / 54
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