1. How to read
and understand…
Title

2. Page

3. crystal
system
Left system

4. point
group
symbol
Left point group

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

6. space
group
symbol
Schönflies
notation
Left space group2

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

8. Left positions diagram
diagram of
equivalent
positions
Left positions diagram

9. origin position vs.
symmetry elements
Left origin

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

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

12. equivalent
positions
Right positions

13. Right special positions

14. subgroups
Right subgroups

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

16. group generators
Right generators

17. Interpretation of
individual items
Individual items

18. crystal
system
Left system

19. 7 (6) Crystal systems Triclinic a ¹ b ¹ c a, b, g ¹ 90º
Monoclinic a ¹ b ¹ c a = g = 90º, b ¹ 90º
Orthorhombic a ¹ b ¹ c a = b = g = 90º
Tetragonal a = b ¹ c a = b = g = 90º
Rhombohedral a = b = c a = b = g
Hexagonal a = b ¹ c a = b = 90º , g = 120º
Cubic a = b = c a = b = g = 90º
Systems

20. point
group
symbol
Left point group

21. describe symmetry of finite objects (at least one point invariant)
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)
Point groups

22. Schönflies International Examples Cn N 1, 2, 4, 6 Cnv Nmm mm2, 4mm
Point groups
describe symmetry of finite objects
(at least one point invariant)
_ _ _ _ _
_ _ _ _ _ _ __
Schönflies International Examples
Cn N , 2, 4, 6
Cnv Nmm mm2, 4mm
Cnh N/m m, 2/m, 6/m
Cni , S2n N , 3, 4, 6
Dn N , 622
Dnh N/mmm mmm, 4/mmm
Dnd N2m, Nm m, 42m, 62m
T , Th , Td , m3, 43m
O , Oh , m3m
Y , Yh , 53m
Point groups general

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

24. Trp RNA-binding protein 1QAW
11-fold
NCS axis
(C11)
Trp

25. Xylose isomerase 1BXB
Xyl

26. Xylose isomerase 1BXB
Tetramer
222 NCS
symmetry
(D2)
Xyl 222

27. space
group
symbols
Left space group

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

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

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

31. Rhombohedral cell 1

32. Rhombohedral cell 2

33. Rhombohedral reciprocal lattice 1

34. Rhombohedral reciprocal lattice 2

35. Rhombohedral reciprocal lattice 3

36. Space group symbols

37. 321 vs. 312

38. diagram of symmetry operations positions of symmetry operations
Left symmetry diagram

39. Symmetry operators symbols

40. origin position vs.
symmetry elements
Left origin

41. Origin P212121

42. Origin P212121b

43. Origin C2

44. Origin C2b

45. Va.u. = Vcell/N rotation axes cannot pass through the asymm. unit
definition of
asymmetric unit
(not unique)
Va.u. = Vcell/N rotation axes cannot pass
through the asymm. unit
Left asymmetric unit

46. Asymmetric unit P21

47. Left positions diagram
diagram of
equivalent
positions
Left positions diagram

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

49. 2-fold axes

50. 3-fold axis 1

51. 3-fold axis 2

52. Various positions 1

53. Various positions 2

54. Various positions 3

55. Various positions 4

56. P43212 symmetry

57. P43212 symmetry 1

58. P43212 symmetry 2

59. P43212 symmetry 2b

60. Multiple symmetry axes
Higher symmetry axes
include lower symmetry ones
includes 2
“ and 2
41 and “ “
“ and 21
“ and 21
“ and 2
“ and 2
“ and 21
Multiple symmetry axes

61. P43212 symmetry 3

62. P43212 symmetry 4

63. P43212 symmetry 4b

64. P43212 symmetry 5

65. P43212 symmetry 6

66. P43212 symmetry 7

67. P43212 symmetry 8

68. P43212 symmetry 8b

69. Right special positions

70. Special positions 0

71. Special positions 1

72. Special positions 2

73. Special positions 3

74. Special positions 3b

75. on non-translational symmetry elements
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)
Special positions

76. subgroups
Right subgroups

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

78. Subgroups 0

79. Subgroups 1a

80. Subgroups 1b

81. Subgroups 3a

82. Subgroups 3b

83. Subgroups 2a

84. Subgroups 2b

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

86. PSCP orthorhombic diffraction 1

87. PSCP orthorhombic diffraction 2

88. PSCP hexagonal diffraction

89. group generators
Right generators

90. Generators 1

91. Generators 2

92. Generators 3

93. Generators 4

94. Generators 5

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

96. Absences 1

97. Absences 2

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

99. My personal remark: P212121, not 19
I hate when people quote space groups
by numbers instead of name.
For me the orthorhombic space group
without any special positions is
P212121, not 19
Personal remark