1. Title How to read and understand…

2. Page

3. Left system crystal system

4. Left point group point group symbol

5. Left space group1 space group symbol international (Hermann-Mauguin) notation

6. Left space group2 space group symbol Schönflies notation

7. Left symmetry diagram diagram of symmetry operations positions of symmetry operations

8. Left positions diagram diagram of equivalent positions

9. Left origin origin position vs. symmetry elements

10. Left asymmetric unit definition of asymmetric unit (not unique)

11. Left Patterson Patterson symmetry Patterson symmetry group is always primitive centrosymmetric without translational symmetry operations

12. Right positions equivalent positions

13. Right special positions special positions

14. Right subgroups subgroups

15. Right absences systematic absences systematic absences result from translational symmetry elements

16. Right generators group generators

17. Individual items Interpretation of individual items

18. Left system crystal system

19. Systems 7 (6) Crystal systems Triclinic a  b  c , ,   90 º Monoclinic a  b  c    90 º,   90 º Orthorhombic a  b  c       90 º Tetragonal a  b  c      90 º Rhombohedral a  b  c      Hexagonal a  b  c     90 º,  120 º Cubic a  b  c      90 º

20. Left point group point group symbol

21. Point groups describe symmetry of finite objects (at least one point invariant) Set of symmetry operations: rotations and rotoinversions (or proper and improper rotations) mirror = 2-fold rotation + inversion Combination of two symmetry operations gives another operation of the point group (principle of group theory)

22. Point groups general Point groups describe symmetry of finite objects (at least one point invariant) Schönflies International Examples C n N 1, 2, 4, 6 C nv Nmm mm2, 4mm C nh N/m m, 2/m, 6/m C ni, S 2n N 1, 3, 4, 6 D n N22 222, 622 D nh N/mmm mmm, 4/mmm D nd N2m, Nm 3m, 42m, 62m T, T h, T d 23, m3, 43m O, O h 432, m3m Y, Y h 532, 53m _ _ _ _ _ _ _ _ _ _ _ __

23. Point groups crystallographic 32 crystallographic point groups (crystal classes) 11 noncentrosymmetric Triclinic 1 1 Monoclinic 2 m, 2/m Orthorhombic 222 mm2, mmm Tetragonal 4, 422 4, 4/m, 4mm, 42m, 4/mmm Trigonal 3, 32 3, 3m, 3m Hexagonal 6, 622 6, 6/m, 6mm, 62m, 6/mmm Cubic 23, 432 m3, 43m, m3m _ _ _ _ _ _

24. Trp Trp RNA-binding protein 1QAW 11-fold NCS axis (C 11 )

25. Xyl Xylose isomerase 1BXB

26. Xyl 222 Xylose isomerase 1BXB Tetramer 222 NCS symmetry (D 2 )

27. Left space group space group symbols

28. Space groups Combination of point group symmetry with translations - Bravais lattices - translational symmetry elements Space groups describe symmetry of infinite objects (3-D lattices, crystals)

29. Bravais lattices but the symmetry of the crystal is defined by its content, not by the lattice metric

30. Choice of cell Selection of unit cell - smallest - simplest - highest symmetry

31. Space group symbols

32. 321 vs. 312

33. Left symmetry diagram diagram of symmetry operations positions of symmetry operations

34. Symmetry operators symbols

35. Left origin origin position vs. symmetry elements

36. Origin P212121

37. Origin P212121b

38. Origin C2

39. Origin C2b

40. Left asymmetric unit definition of asymmetric unit (not unique) V a.u. = V cell /N rotation axes cannot pass through the asymm. unit

41. Asymmetric unit P21

42. Left positions diagram diagram of equivalent positions

43. Right positions equivalent positions these are fractional positions (fractions of unit cell dimensions)

44. 2-fold axes

45. P43212 symmetry

46. P43212 symmetry 1

47. P43212 symmetry 2

48. P43212 symmetry 2b

49. Multiple symmetry axes Higher symmetry axes include lower symmetry ones 4 includes 2 6 “ 3 and 2 4 1 and 4 3 “ 2 1 4 2 “ 2 6 1 “ 3 1 and 2 1 6 5 “ 3 2 and 2 1 6 2 “ 3 2 and 2 6 4 “ 3 1 and 2 6 3 “ 3 and 2 1

50. P43212 symmetry 3

51. P43212 symmetry 4

52. P43212 symmetry 4b

53. P43212 symmetry 5

54. P43212 symmetry 6

55. P43212 symmetry 7

56. P43212 symmetry 8

57. P43212 symmetry 8b

58. Right special positions special positions

59. Special positions 0

60. Special positions 1

61. Special positions 2

62. Special positions 3

63. Special positions 3b

64. Special positions on non-translational symmetry elements (axes, mirrors or inversion centers) degenerate positions (reduced number of sites) sites have their own symmetry (same as the symmetry element)

65. Right subgroups subgroups

66. Subgroups reduced number of symmetry elements cell dimensions may be special cell may change

67. Subgroups 0

68. Subgroups 1a

69. Subgroups 1b

70. Subgroups PSCP Dauter, Z., Li M. & Wlodawer, A. (2001). Acta Cryst. D57, 239-249. After soaking in NaBr cell changed, half of reflections disappeared

71. Right generators group generators

72. Right absences systematic presences (not absences) systematic absences result from translational symmetry elements

73. Absences 1

74. Absences 2

75. Personal remark My personal remark: I hate when people quote space groups by numbers instead of name. For me the orthorhombic space group without any special positions is P2 1 2 1 2 1, not 19