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Tiling Spaces: Quasicrystals & Geometry

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Tiling Spaces: Quasicrystals & Geometry Tiling Spaces: Quasicrystals & Geometry MA4054 Mathematics Project: Project Dissertation Thesis submitted at the University of Leicester in partial fullment of the requirements for the degree of Master of Mathematics by Daniel Bragg Department of Mathematics University of Leicester May 2014 Contents Declaration iii Abstract iv Introduction 1 1 Lattices & Voronoi Cells 3 1.1 Lattices . 3 1.1.1 Lattices in one-dimension . 4 1.1.2 Lattices in two-dimensions . 4 1.1.3 Lattices in three-dimensions . 5 1.2 Crystallographic Restriction Theorem . 7 1.2.1 Denition of a Crystal . 9 1.3 Projection Lattices . 10 1.3.1 Voronoi Cells . 10 1.3.2 The Projection Method . 13 1.4 Dual Lattices . 17 1.4.1 Properties of the Dual Lattice . 19 1.4.2 Examples of Dual Lattices . 19 2 Penrose Tilings 21 2.1 Tiles and Tilings . 21 2.2 Penrose Tilings of the Plane . 24 2.2.1 Matching Rules . 24 i 2.3 Construction Techniques . 31 2.3.1 Voronoi Decompositions . 31 2.3.2 Substitution . 31 2.3.3 Multigrids . 33 2.3.4 Pentagrids . 36 3 Projection Method for Constructing Tilings of the Plane 41 3.1 The Idea . 41 3.2 Penrose Tilings as Projections . 43 3.2.1 I5 and the Unit Hypercube Q5 ..................... 45 3.2.2 Projecting I5 ............................... 47 3.2.3 Projected Plane as Complex Plane . 50 3.2.4 Shift Vector −!γ .............................. 55 4 Diraction Geometry 58 4.1 Far Field Diraction Geometry . 59 4.1.1 One Point Diraction . 60 4.1.2 Two Point Diraction . 61 4.1.3 N-Point Diraction . 64 4.2 Fourier Transforms and the Diraction Condition . 65 4.2.1 The Fourier Transform . 66 4.2.2 Dirac Deltas and the Diraction Condition . 69 4.3 Topological Bragg Peaks - An Alternate Approach . 77 5 One Dimensional Crystals 80 5.1 One-Dimensional Delone Sets . 80 5.2 Fibonacci Sequences . 82 5.3 Projected One-Dimensional Crystals . 85 Concluding Remarks 91 ii Declaration All sentences or passages quoted in this project dissertation from other people's work have been specically acknowledged by clear cross referencing to author, work and page(s). I understand that failure to do this amounts to plagiarism and will be considered grounds for failure in this module and the degree examination as a whole. Name: Daniel Bragg Signed: Date: iii Abstract Throughout the years a clear link has been established between the structure of crystals and the mathematical concepts of tiling spaces. Until recently crystals were thought to be periodic and ordered structures which can be related to a set of points in the n - dimensional Euclidean space En known as a lattice, where these points act as vertices to the building blocks of crystals. It was known that crystals could only be periodic and ordered if their corresponding lattice had either one, two, three, four or six fold rotational symmetry. However in the mid-20th century quasicrystals were discovered which were ordered and aperiodic but possessed rotational symmetries not listed above. This sparked into life the new area of study known as quasi-crystallography. This report delves into the mathematics behind these interesting structures. First considered are the lattice structures related to the structure of a crystal. I consider the dierent lattice types in various dimensions and then look at the properties such as the Crystallographic Restriction Theorem described above. I then introduce the idea of a dual lattice, Voronoi cell construction and the canonical projection method - all of which are important tools to be used later. Tiling spaces are then considered. In particular I will be exploring the properties of the rhomb Penrose tiling which are derived from the famous kite and dart Penrose tiling. These tilings are aperiodic so therefore provide a good model for quasicrystal structures. With all of this knowledge in place, I will then explain the projection method for constructing tilings of the plane from ve dimensional space. I then digress somewhat and explore the mathematics behind the diraction patterns created by crystals and quasicrystals. This is so that the diraction condition can be introduced, which denes the diraction patterns that represent a crystalline structure. A new method of iv constructing these diraction patterns known as topological Bragg peaks is also discussed. Finally I discuss the possibility of having one dimensional crystals. In particular we look at the Fibonacci sequence and show how they produce crystalline structures. v Introduction There is a rich history of experimentation, misjudgments and discoveries in the eld of crystallography which presents it as an interesting area of research for both mathematicians and solid state scientists. Although there was no science directly linked to crystals before the seventeenth century, there is evidence of ideas which are related to crystallography as far back as the ancient Greeks, where Plato assigned each of the `four elements' to a regular polyhedra. In 1984 it was announced that there existed crystals with icosahedral symmetry, made from an alloy of Aluminium and Manganese. This was something which was thought to have been impossible for over 200 years prior to this discovery. This breakthrough opened up the door to new areas of research within geometry and tiling spaces as it was now suggested that crystals need no longer have a periodic atomic pattern. Therefore the denition of what a crystal actually is has been constantly changing throughout all of history. Throughout my research one book has been of particular interest to me - Marjorie Senechal's Quasicrystals and Geometry. The main goal of this project was to try and mathematically relate to crystallography, a subject which is predominantly associated with chemists and physicists, and Senechal's book does exactly that. Therefore the main part of my research below is based on this book and is cited with the notation [1]. A lot of the work carried out on the projection method for constructing Penrose tilings of the plane was by the late N.G De Bruijn (1918-2012). One of his papers, Algebraic Theory of Penrose's non-periodic tiling of the plane, was of great use to me in Chapter 3 and is sited with notation [16]. Johannes Kellendonk's work on topological Bragg peaks has completely changed the way we look at diraction patterns. His paper, entitled Topological Bragg 1 Peaks And How They Characterize Point Sets, views diraction patterns as topological in nature rather than statistical. This is sited with notation [12]. I feel that I have deviated away slightly from the goals I set myself in the project description document. I feel that my naivety of the subject led me to believe that some of my chosen areas of research would take up more time than they actually did. However by pursuing these initial objectives I found that the subject branched o so that I could easily study other areas that I didn't even know existed. This has been one of the main factors as to why I've enjoyed researching crystallography so much - the research possibilities are endless! The content of this project dissertation has successfully covered all the areas which I set as goals to explore in my interim report and therefore feel that this gives a well rounded view into the mathematics of crystalline structures and I feel like I've achieved a great deal over the past eight months. Although following the material set out by the papers above, original material in this dissertation comes in the form of proofs left for the reader to prove, such as the proof of Proposition 5.3.1 in Chapter 5. 2 Chapter 1 Lattices & Voronoi Cells In this chapter we are going to describe some of the basic denitions and notions which I will use throughout this paper. To begin with we will consider lattices in Euclidean space and will then go on to applying the ideas behind such tiling spaces to crystallography. 1.1 Lattices n Consider the n-dimensional Euclidean space E with orthonormal basis ~e1; :::; ~en. Points in our Euclidean space are denoted by lower-case letters (x; y; z etc.) and vectors will always be vectors from the origin 0 2 En to the corresponding lower-case letter in space. n Denition. 1.2.1 [1] Any set of vectors ~u1; :::; ~uk 2 E generates a countably innite group under addition and is called a Z- module; it's elements are vectors of the form ~x = m1 ~u1 + ::: + mk ~uk where mi are integers. A Z- module is the building block for any type of lattice. Before we proceed any further, let's refresh our memory on a particular area of group theory. Let G be a transformation group acting on an object M, where M could be anything from the whole space En to G itself. Let x 2 M. Denition. 1.2.2 [1] The orbit of x under G is the set: OG(x) = fgx 2 M : g 2 Gg 3 Figure 1.1: A lattice in one-dimension is generated by one vector −!u Using this notion of orbits we can generate lattices. n Denition. 1.2.3 [1] A Z- module in E is a lattice (of dimension n ) if it is generated by n linearly independent vectors. We denote a lattice by L, and any orbit of L by Lp. We sometimes refer to lattices of this kind as Bravais lattices, after Auguste Bravais (1850), who studied lattices in great detail. As we will see very shortly, there are classi- cations for Bravais lattices in n-dimensional space. Denition. 1.2.4 [1] Any set of n linearly independent lattice vectors that generates L is called a basis for L.
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