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Short-‐Course on Symmetry and Crystallography Part 1 Short-course on symmetry and crystallography Part 1: Point symmetry Michael Engel Ann Arbor, June 2011 Euclidean move Definion 1: An Euclidean move = A, b is a linear transformaon that leaves space invariant: T { } x (x)=Ax + b → T Here x is a vector, A an 3x3 orthogonal Matrix and b a 3-vector. Ques&on: Euclidean moves form a ?-dimensional space. Product of Euclidean moves Definion 2: The product of two transformaons 1 = A1,b1 T { } and = A ,b is: = A A ,A b + b T2 { 2 2} T2 ◦ T1 { 2 1 2 1 2} (Note: T1 is applied first!) Definion 3: The order of a transformaon T is the smallest integer n such that n(x)= (x)=x T T ◦ T ◦ T ◦ ···◦ T One can also say this transformaon is n-fold. Observaons: 1 1 1 1. The inverse is: − = A− , A− b T1 {1 − } (Check: − = − =1) T ◦ T T ◦ T 2. Every transformaon of finite order n (i.e. T n = 1) leaves at least one point invariant. Group hp://en.wikipedia.org/wiki/Group_(mathema<cs) Formal definiBon of symmetry group Definion 4: • A symmetry of an object in space (cluster, Bling, lace, …) is an Euclidean move that leaves the object indisBnguishable. • A symmetry group is a group of symmetries. Definion 5: The order of a group is equal to the number of elements: G = g G | | |{ ∈ }| Types of symmetries x (x)=Ax + b → T Normal form: Classificaon: 10 0 1) b = 0 or b ≠ 0? ± A = 0 cos(α) sin(α) 2) Angle α. − 0 sin(α) cos(α) 3) Eigenvalues of A. Basic types: IdenBty = 1, (i) ReflecBon, (ii) Rotaon, (iii) Translaon Composite types: (iv) Glide reflecon, (v) RotoreflecBon (Inversion), (iv) Helical symmetry ReflecBon/mirror symmetry (S2 = 1) Kyoto, June 2008 Ambigramm (segerman.org) (n-fold) Rotaonal symmetry (Sn = 1) Mandala, n = 6 Ambigramm (segerman.org), n = 2 Flag, n = 3 Translaonal symmetry (n > 1: Sn ≠ 1) SEM image of the wing Giant’s causeway, of a Papilio buerfly Northern Ireland Composite Symmetries Rotaon + ReflecBon = RotoreflecBon (Inversion) Translaon + ReflecBon = Translaon + Rotaon = Helical symmetry Glide reflecon Group acBon Here: • The group G is a set of Euclidean moves. • The set X is the three-dimensional space. • An Euclidean move acts on 3D space as an affine transformaon. • The orbit consists of all points that are equivalent under symmetry. • The stabilizer consists of all symmetries that leave a point invariant. Point symmetries Definion 6: A point symmetry is a symmetry which leaves a point x0 invariant: (x0)=x0 T Observaons: • Translaons cannot be point symmetries. • Symmetries with finite order are point symmetries. • Symmetries with infinite order cannot be point symmetries. (Note: Some sources consider spherical and cylindrical symmetry point symmetries.) Point group Definion 7: A point group is a group of point symmetries, which leave a common point x0 invariant. Observaon: 1. A point group is a finite subgroup of O(3), the space of three dimensional orthogonal matrices. 3 3 T Note: O(3) = A × : A A =1 { ∈ 3 3 T } SO(3) = A × : A A =1, det(A)=1 { ∈ } 2. If two symmetries have no common invariant point, then they generate a group of infinite order. (Exercise) Classificaon strategy: Determine finite subgroups of SO(3). Then extend them into O(3). Comparing groups Definion 8: Two subgroups H1 and H2 of a group G are conjugated, if there exists a g G , such that: 1 ∈ H2 = g− H1g (Exercise: Show that conjugated subgroups are isomorphic.) Example: G = O(3). Two point groups are conjugated, if there is a change of basis that maps them into each other. Classificaon of 2D point groups (up to conjugacy) Normal form of an orthogonal Matrix in O(2): cos(α) sin(α) A = − ± sin(α) cos(α) Cyclic groups: C1, C2, C3,… where Cn consists of all rotaons about a fixed point by mulBples of 360/n. Dihedral groups: D1, D2, D3, D4,... where Dn (of order 2n) consists of the rotaons in Cn together with reflecBons in n axes that pass through the fixed point. Proper point groups in 3D (subgroups of SO(3)) • Cyclic groups: Cn with order n • Dihedral groups: Dn with order 2n • Tetrahedral group T with order 12. Octahedral group O with order 24. Icosahedral group I with order 60. Role of dimension Platonic solids in 4D: Higher dimensions: Only simplex, hypercube, cross-polytope. Sands, page 25. Classificaon of 3D point groups – Part I hp://en.wikipedia.org/wiki/Point_groups_in_three_dimensions Exercise 1 Exercise 2 Exercise 3 Exercise 4 Point symmetry? Classificaon of 3D point groups – Part II The 7 remaining point groups: • T (332) of order 12 - chiral tetrahedral symmetry. Rotaon group for a regular tetrahedron. • Td (*332) of order 24 – full tetrahedral symmetry. Full symmetry group of a regular tetrahedron. • Th (3*2) of order 24 – pyritohedral symmetry. Symmetry of a volleyball. • O (432) of order 24 – chiral octahedral symmetry. Rotaon group for a regular octahedron/cube. • Oh (*432) of order 48 - full octahedral symmetry. Full symmetry group of a regular octahedron/cube. • I (532) of order 60 – chiral icosahedral symmetry. Rotaon group for a regular dodecahedron/icosahedron. • Ih (*532) of order 120 - full icosahedral symmetry. Full symmetry group of a regular dodecahedron/icosahedron. Archimedean solids – Part 1 Determinaon of the point group of an object in space 1. Object linear: C∞v or D∞h. 2. High symmetry, non-axial: T, Th, Td, O, Oh, I, Ih. 3. No rotaon axis: C1, Ci, Cs. 4. Determine the symmetry element with highest order and use the following table: Group Order n 2n verBcal mirror 2n horizontal mirror 2n 4n horizontal mirror orthogonal rotaons 4n verBcal mirror n Example: Carolyn’s packings of small spheres on a big sphere 1. Six trimers of spheres arrange on the verBces of an octahedron into two different orientaons. 2. What are the point groups? Ignore the numerical inaccuracy (fluctuaons in the orientaon). Exam quesBons, part 1 • What is a symmetry? • How does a symmetry act on Euclidean space? • What types of symmetries are there? • What is a point symmetry and a point symmetry group? • What does it mean to classify point groups? • What points groups are there in 2D and 3D? • How can you idenBfy the point group of an object? .
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