Short-course on symmetry and crystallography
Part 1: Point symmetry
Michael Engel Ann Arbor, June 2011 Euclidean move
Defini on 1: An Euclidean move = A, b is a linear transforma on 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
Defini on 2: The product of two transforma ons 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!)
Defini on 3: The order of a transforma on T is the smallest integer n such that n(x)= (x)=x T T ◦ T ◦ T ◦ ···◦ T One can also say this transforma on is n-fold.
Observa ons: 1 1 1 1. The inverse is: − = A− , A− b T1 {1 − } (Check: − = − =1) T ◦ T T ◦ T 2. Every transforma on of finite order n (i.e. T n = 1) leaves at least one point invariant. Group
h p://en.wikipedia.org/wiki/Group_(mathema cs) Formal defini on of symmetry group
Defini on 4: • A symmetry of an object in space (cluster, ling, la ce, …) is an Euclidean move that leaves the object indis nguishable. • A symmetry group is a group of symmetries.
Defini on 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: Classifica on: 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: Iden ty = 1, (i) Reflec on, (ii) Rota on, (iii) Transla on
Composite types: (iv) Glide reflec on, (v) Rotoreflec on (Inversion), (iv) Helical symmetry Reflec on/mirror symmetry (S2 = 1)
Kyoto, June 2008
Ambigramm (segerman.org) (n-fold) Rota onal symmetry (Sn = 1)
Mandala, n = 6
Ambigramm (segerman.org), n = 2
Flag, n = 3 Transla onal symmetry (n > 1: Sn ≠ 1)
SEM image of the wing Giant’s causeway, of a Papilio bu erfly Northern Ireland Composite Symmetries
Rota on + Reflec on = Rotoreflec on (Inversion)
Transla on + Reflec on = Transla on + Rota on = Helical symmetry Glide reflec on Group ac on
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 transforma on. • The orbit consists of all points that are equivalent under symmetry. • The stabilizer consists of all symmetries that leave a point invariant. Point symmetries
Defini on 6: A point symmetry is a symmetry which leaves a point x0 invariant: (x0)=x0 T Observa ons: • Transla ons 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
Defini on 7: A point group is a group of point symmetries, which leave a common point x0 invariant.
Observa on: 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)
Classifica on strategy: Determine finite subgroups of SO(3). Then extend them into O(3). Comparing groups
Defini on 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. Classifica on 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 rota ons about a fixed point by mul ples of 360/n.
Dihedral groups: D1, D2, D3, D4,... where Dn (of order 2n) consists of the rota ons in Cn together with reflec ons 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. Classifica on of 3D point groups – Part I
h p://en.wikipedia.org/wiki/Point_groups_in_three_dimensions Exercise 1
Exercise 2 Exercise 3 Exercise 4
Point symmetry? Classifica on of 3D point groups – Part II
The 7 remaining point groups: • T (332) of order 12 - chiral tetrahedral symmetry. Rota on 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. Rota on 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. Rota on 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
Determina on 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 rota on axis: C1, Ci, Cs. 4. Determine the symmetry element with highest order and use the following table:
Group Order
n 2n ver cal mirror 2n horizontal mirror 2n 4n horizontal mirror orthogonal rota ons 4n ver cal mirror n Example: Carolyn’s packings of small spheres on a big sphere
1. Six trimers of spheres arrange on the ver ces of an octahedron into two different orienta ons. 2. What are the point groups? Ignore the numerical inaccuracy (fluctua ons in the orienta on). Exam ques ons, 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 iden fy the point group of an object?