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3 D Symmetry (2 weeks)

3 D Symmetry (2 weeks). Next we would move a step further into 3D symmetry. Leonhard Euler :. http://en.wikipedia.org/wiki/Leonhard_Euler. Google search: Euler. Spherical trigonometry. Small circle R <1. For convenience, set R = 1. Great circle (GC), R =1. o.

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3 D Symmetry (2 weeks)

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  1. 3D Symmetry (2 weeks)

  2. Next we would move a step further into 3D symmetry. Leonhard Euler : http://en.wikipedia.org/wiki/Leonhard_Euler Google search: Euler

  3. Spherical trigonometry Small circle R<1 For convenience, set R = 1 Great circle (GC), R=1 o Distance: AOB =  (GC)  A B P Pole90o to arc AB. o A B

  4. c B A arc BC = a arc AC = b arc AB = c. GC b a C Spherical Angles BAC =B’OC’ A GC o B C ? B’OC’

  5. Polar triangle A GC A A, pole of arc BC B, pole of arc AC C, pole of arc AB GC B C B C ABC and ABC are mutually polar! Proof: B: pole of arc AC B is 90o away from point A. C: pole of arc AB C is 90o away from point A.  A:pole of arc BC. Similarly, B: AC, C: pole of arc AB.

  6. Proof: BAC = , arc BC = a,   + a = . A B A  o Q B C A C B C a P Q B : pole of arc AC arc BQ = /2 C : pole of arc AB arc CP = /2  arc BQ + arc CP =  = (arc BP+ arc PQ) + (arc PQ+ arc QC) = (arc BP+ arc PQ+ arc QC) + arc PQ = a + arc PQ 

  7. Law of cosines Plane geometry length b c A B C angle a Spherical geometry

  8. (3) O O 90o (2) (4) a (1) b 1 1 y u w z y z u u (4) (1) (3) b v o (1)= tana (3)= tanb a 1/(2)= cosa  (2) = seca 1/(4)= cosb  (4) = secb (2) Unit circle (From uyz) (From oyz) http://en.wikipedia.org/wiki/Spherical_law_of_cosines

  9. Combination of two rotation operations: A C B A : (1)  (2) B : (2)  (3) 2 R 1 R 3 R (1) and (3) relation? c 3-D: translation, reflection, rotation, and inversion. must be crystallographic

  10. A Locate the position of axis C B C A c B c A B b a A B b a C c c Euler construction: N C’ M’ A symmetry element is the locus of a point that is left unmoved by an operation.   B A A: AMAM’. B: BNBN’. C C (the point unmoved). OC: the axis N’ M

  11. (1) A: leave A unmoved. (2) B: move A to A’. A B /2 /2  A’ ABC = A’BC = /2 AB = A’B ABC = A’BC  ACB = A’CB  /2 C N’ M /2 A c B /2 b a The law of cosine (spherical trigonometry) /2 C

  12. 180o-/2 A c B 180o-b /2 180o-a /2 /2 b a 180o-/2 180o-/2 180o-c C Law of cosine to the polar triangle

  13. Combination to be tested B 1 2 3 4 6 A 111 1 112 113 114 116 212 222 2 213 223 214 224 216 226 313 323 333 3 314 324 334 316 326 336 414 424 434 444 4 416 426 436 446 6 616 626 636 646 666

  14. /2 /2 /2 Axis at A, B, or C , , or  1-fold 2-fold 3-fold 4-fold 6-fold 360o 180o 120o 90o 60o 180o 90o 60o 45o 30o -1 0 1/2 1/21/2 31/2/2 0 1 31/2/2 1/21/2 1/2

  15. /2 Case: 11n A c B A: 1,  = 360o, cos( /2) = -1; sin( /2) = 0 /2 b a B: 1,  = 360o, cos( /2) = -1; sin( /2) = 0 /2 C C: n,  = 360o /n , cos( /2); sin( /2) 180o A c B 180o b a 180o/n C None exist!

  16. /2 Case: 22n A c B A: 2,  = 180o, cos( /2) = 0; sin( /2) = 1 /2 b a B: 2,  = 180o, cos( /2) = 0; sin( /2) = 1 /2 C C: n,  = 360o /n , cos( /2); sin( /2) 90o A c B 90o b a 180o/n C

  17. C Angle between A and B axis b a 222 B A 223 B A 224 B A 226 B A What are a and b?

  18. A: 2,  = 180o, cos( /2) = 0; sin( /2) = 1 B: 2,  = 180o, cos( /2) = 0; sin( /2) = 1 C: n,  = 360o /n , cos( /2); sin( /2)  a = 90o.  b = 90o.

  19. C C C C B B B 45o 30o 60o 90o B A A A A 222 223 224 226

  20. Case: 23n A: 2,  = 180o, cos( /2) = 0; sin( /2) = 1 90o A c B B: 3,  = 120o, cos( /2) = 0.5; sin( /2) = 30.5/2 60o b C: n,  = 360o /n , cos( /2); sin( /2) a 360o/n C 233 234 236 None exist The rest of combination does not exist!

  21. Case: 233 A: 2,  = 180o, cos( /2) = 0; sin( /2) = 1 B: 3,  = 120o, cos( /2) = 0.5; sin( /2) = 30.5/2 C: 3,  = 120o, cos( /2) = 0.5; sin( /2) =30.5/2 a = 70o32. b = 54o44.

  22. B 70o32’ z C 000 54o44’ y A x 54o44’ 233 Angle between A and B is Angle between A and C is Angle between B and C is

  23. Case: 234 A: 2,  = 180o, cos( /2) = 0; sin( /2) = 1 B: 3,  = 120o, cos( /2) = 0.5; sin( /2) = 30.5/2 C: 4,  = 90o, cos( /2) = 1/20.5; sin( /2) = 1/20.5 a = 54o44. b = 45o.

  24. A 35o16’ B z 45o 000 y 54o44’ x C 234 Angle between A and B is Angle between A and C is Angle between B and C is

  25.  Geometry of the permissible nontrivial combination of rotations: Combination 2A =  c 2B =  a 2C =  b 222 223 224 226 233 234 180o 180o 180o 180o 180o 180o 90o 60o 45o 30o 54o44 35o16 180o 180o 180o 180o 120o 120o 90o 90o 90o 90o 70o32 54o44 180o 120o 90o 60o 120o 90o 90o 90o 90o 90o 54o44 45o

  26. International symbol 222 322 422 622 (1) (1) (2) (2) (3) (3) 32(2) 22 operation is basically on the plane! Just like 3m(m) Only one kind of 2 fold rotation symmetry

  27. n22 222 32(2) 422 622 DnD2D3D4D6 Schonllies notation dihedral different dihedral angle 233 23 is enough to specify the symmetry! = 23 Schonllies notation: T Tetrahedral

  28. International symbol http://en.wikipedia.org/wiki/Tetrahedron

  29. International symbol 35o16’ 45o 54o44’ Schonllies notation: O Octahedron 234 or 432 http://en.wikipedia.org/wiki/Octahedron

  30. 11 axial combinations 1 2 3 4 6 222 322 422 622 233 432 11 axial combinations + Extender 222 +  45o, 2-fold rotation 4-fold rotation  extender 

  31. Ways to add m: horizontal  422 n n diagonal  for Dn, T, O vertical  horizontal m vertical m Not forCn Extender: v, h, d, ! + extender  create new rotation axis!

  32. 1 2 3 4 6 222 32 422 622 23 432 Cnv, Dnv Tv, Ov v h d Cnh, Dnh Th, Oh See reading crystal4.pdf Dnd Td, Od http://ocw.mit.edu/courses/materials-science-and-engineering/3-60-symmetry-structure-and-tensor-properties-of-materials-fall-2005/readings/crystal4.pdf Cni, Dni Ti, Oi

  33. 1 2 3 4 6 222 32 Cnv, Dnv Tv, Ov v h d m 2mm 3m 4mm 6mm Cnh, Dnh Th, Oh m Dnd Td, Od - - - - - Cni, Dni Ti, Oi

  34. ?: (1)  (3) (1) R (2) R at the point of intersection 1 1 1 (3) L 1 1 1 up R R down L L Extender:

  35. (1)  No inversion (2) (3) Use inversion as an extender! (1)   (1)  (1)   (2)  (1)   (3)  (1)   (1)  (1)   (2)  (1)   (3)  New two step operations Roto-inversion

  36. (1)  (1) (1)  (2) (1)  (3) (1)  (4) (1)  (5) (1)  (6) (1) (6) (5) (2) (3) (4)

  37. Sphenoid (Greek word for axe) (1) (2) (1) To (2) (2) To (3) (4) (3) To (4) (3) (4) To (1) Equal length The rest four: equal length. Not tetrahedron

  38. (2) R (1) R (2) R   (1) R Roto-reflection? (3) L (3) L Roto-reflection

  39. Roto-inversion

  40. R(D) L(U) R(U) L(D) R (D) R(U) + h R (D) R(D) L(U) R (U) R(U) L(D) 222 222

  41. R (2) R (1) R R (4) L L + d (3) L L R R R R 222 (1)  (2): A (1)  (3):  or (1)  (4): R R R R

  42. R R + d R R R R 32 R R L L L L R R R R L L

  43. 3L 1R 2R Add   Add v  S4v R(U) L(D) R(D) L(U) R(U) R(D) R(D) L(U) R(U) L(D) R(D) R(U) + h R(U) R(D) R(U) L(D) R(D) L(U) R(D) R(U) R(D) L(U) R(U) L(D)

  44. all up. all down. up T 23 down Add a horizontal mirror plane T 23 Th Create an inversion center inversion

  45. R R L L Inner circle + h Td L L R R

  46. 3D crystallographic point group (Buerger’s book) Euler’s construction: pg. 35-43 Some combination theorems: chapter 6 Points group: pg: 59-68 2D lattices: chapter 7 (pg. 69-83)

  47. 3D lattice: Reading crystal7.pdf Oblique (symmetry 1) + General Triclinic Primitive

  48. Oblique (symmetry 2) + projection 4 choices:

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