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The Beauty of Groups and Symmetry

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The Beauty of Groups and Symmetry The Beauty of Groups and Symmetry Peter Webb Some pictures Groups of Isometries The Beauty of Groups and Symmetry What is a Crystal? Distinguishing different crystals Classification of Peter Webb crystals Final remarks University of Minnesota December 13, 2007 1 The Beauty of Outline Groups and Symmetry Peter Webb Some pictures Some pictures Groups of Isometries Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of What is a Crystal? crystals Final remarks Distinguishing different crystals Classification of crystals Final remarks 2 The Beauty of Introduction Groups and Symmetry Peter Webb Some pictures Groups of Isometries The goal in this talk is to share some of the beauty to be What is a Crystal? found in symmetrical objects, especially crystals, and to Distinguishing show how the use of mathematics can help us to different crystals Classification of understand them, use them, and appreciate them better. crystals In the following pictures, which are the work of M.C. Final remarks Escher, you will be impressed by the wonder of the design, but may find it hard to say what important differences there are between the pictures. 3 The Beauty of Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks 4 The Beauty of Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks 5 The Beauty of Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks 6 The Beauty of A pattern which suggests 5-fold symmetry Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks 7 The Beauty of A question to which I do not know the answer Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Why is it that we should find symmetry so attractive? Distinguishing different crystals It is a famous quotation that ‘God gave us the natural Classification of numbers, the rest is man’s invention.’ We might also say crystals that God gave us symmetry. Final remarks Here are some more pictures. 8 The Beauty of Four Platonic solids and a puzzle Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks 9 The Beauty of Paper model of a projective plane and its net Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks Instant Projective Plane 1. Cut out outline. 2. Cut lines on the inside of the squares. 3. Fold up and glue. 4. Decorate. 10 The Beauty of Two cubical crystals Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks 11 The Beauty of The role of mathematics Groups and Symmetry Peter Webb Some pictures Mathematics can help us to understand better the Groups of structure of these patterns and objects, and how they are Isometries made. Without mathematics we think the patterns are What is a Crystal? Distinguishing beautiful, and complicated, but find it hard to say why one different crystals should differ from another. Classification of crystals In the next section of this talk we introduce the following Final remarks mathematical concepts: I isometry of Euclidean space, I group I crystal structure in Euclidean space 12 The Beauty of What is an isometry? Groups and Symmetry Peter Webb Write En for n-dimensional (Euclidean) space. We live in Some pictures 3 Groups of E . Isometries n An isometry (or rigid motion) of E is a transformation What is a Crystal? n n E ! E which preserves distance. Distinguishing different crystals n Each subset of E (an object of some kind) has a group Classification of crystals of isometries. This is the set of all isometries of n which E Final remarks send the subset exactly to itself. Examples A cube has 48 elements in its group of isometries, a tetrahedron has 24, and an icosahedron has 120. A regular polygon with n sides has 2n isometries. 13 The Beauty of Groups and Theorem Symmetry The possible isometries of 2-dimensional space are Peter Webb translations, rotations and glide reflections. Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks 14 The Beauty of In general, what is a group? Groups and Symmetry The definition is a little abstract, but we will need it later. Peter Webb A group is a set whose elements can be multiplied in Some pictures some formal way, satisfying properties listed below. If a Groups of Isometries and b are elements of the group, there is defined an What is a Crystal? element ab which is the product of a and b. The following Distinguishing must hold. different crystals Classification of I Multiplication is associative, that is, (ab)c = a(bc) crystals always. Final remarks I There is an identity element, that is, an element written 1 with the property that 1a = a1 = a always. I Every element is invertible, that is, for every element a there is an element a−1 so that aa−1 = a−1a = 1. Example A group of isometries is a group. 15 The Beauty of What is a crystal (informal idea)? Groups and Symmetry The property that distinguishes a crystal is that there are Peter Webb planes along which it will fall apart cleanly. Some pictures If we take a sodium chloride crystal (a cube), place a Groups of knife on it parallel to one of the faces, and strike the back Isometries of the knife with a hammer, the crystal will divide cleanly What is a Crystal? Distinguishing along the plane of the knife. There are many places in different crystals each of 3 independent directions we could have put the Classification of crystals knife to achieve this. Final remarks How can we model these properties mathematically? 16 The Beauty of What is not a crystal? Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing A cube of glass (which is amorphous) if treated similarly different crystals will shatter into little pieces. Glass does not have the Classification of crystals structure of a crystal, even though it may be in a shape Final remarks which looks like a crystal. 17 The Beauty of What else is not a crystal? Groups and Symmetry Peter Webb Some pictures In two dimensions we have There are types of structure Groups of the Penrose tiles, which tile Isometries called Fullerenes which are the plane in many ways, but What is a Crystal? sometimes viewed as always without periodicity. Distinguishing crystals, but which are not different crystals The wikipedia entry http: Classification of crystals in our sense. It is //en.wikipedia.org/ crystals possible for atoms to Final remarks wiki/Penrose_tiling arrange themselves with the has an account of this. symmetry of an icosahedron around a point. These are not crystals because they favour a distinguished point. 18 The Beauty of Formal mathematical definition of a crystal Groups and Symmetry Peter Webb Some pictures Groups of A crystal structure is a subset of En whose group of Isometries isometries What is a Crystal? Distinguishing I contains n independent translations, and different crystals Classification of I has the property that every isometry moves every crystals point it does not fix at least as far as some Final remarks prescribed minimum distance. The group of isometries of a crystal structure is called a space group, or crystallographic group. 19 The Beauty of Wallpaper patterns Groups and Symmetry Peter Webb Some pictures In 2-dimensional space we call crystal structures Groups of wallpaper patterns. The definition says that a crystal in Isometries dimension 2 must repeat itself in two independent What is a Crystal? Distinguishing directions (one direction along the roll of paper, the other different crystals direction coming as strips of paper are laid next to one Classification of another) and that the pattern may not be continuous in crystals any direction. Final remarks One of our goals is to explain what is meant by the following: Theorem There are precisely 17 different wallpaper patterns. 20 The Beauty of Some questions Groups and Symmetry Peter Webb Some pictures Groups of Isometries The Escher pictures shown at the beginning are What is a Crystal? wallpaper patterns. Distinguishing different crystals What does this mean to say that there are 17 wallpaper Classification of patterns? Clearly there are more than 17 wallpaper crystals patterns! Final remarks How can we recognize them? How can we prove such a result? 21 The Beauty of List of the 17 wallpaper patterns Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks 22 The Beauty of A plate seen in Kyoto Groups and Symmetry Peter Webb Some pictures Groups of Isometries What is a Crystal? Distinguishing different crystals Classification of crystals Final remarks How can we identify each pattern on the plate with one which of the 17 patterns? 23 The Beauty of Exploring wallpaper patterns on the internet Groups and Symmetry Peter Webb The web site of Professor Helmer Aslaksen (National Some pictures University of Singapore) has much information and many Groups of links: Isometries http://www.math.nus.edu.sg/aslaksen/ What is a Crystal? Distinguishing teaching/math-art-arch.html different crystals Classification of Some different pictures of the 17 wallpaper patterns: crystals http://mathmuse.sci.ibaraki.ac.jp/pattrn/ Final remarks PatternE.html http://www2.spsu.edu/math/tile/symm/types/ index.htm The software ‘kali’ allows us to draw wallpaper patterns: http: //www.geom.uiuc.edu/java/Kali/program.html 24 The Beauty of Equivalence of crystals Groups and Symmetry We say that crystals are the equivalent if they have the Peter Webb ‘same’ symmetry group.
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