1. Symmetry “the correspondence in size, form and
arrangement of the parts of a recurring array
(for example, patterned wallpaper) with
respect to a point, line or plane”
Zoltai & Stout, 1984

2. is controlled by the symmetry of its atomic structure.
Internal symmetry
not only determines the
morphology of a crystal,
it also determines:
The physical properties and behaviour of a mineral are determined by its atomic structure and bonding, which
is controlled by the symmetry of its atomic structure.
Symmetry also forms a fundamental part of how we classify minerals (the language of mineralogy).

3. Morphology

4. Morphology, the internal arrangement of atoms and the internal symmetry

6. Symmetry
Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern
Operation: some act that reproduces the motif to create the pattern
Element: an operation located at a particular point in space

7. 6 6 2-D + 3D Symmetry Symmetry Elements 1. Rotation Operation
a. Two-fold rotation
= 360o/2 rotation
to reproduce a motif in a symmetrical pattern
= the symbol for a two-fold rotation
Operation 6
Motif
Element 6

8. 6 6 2-D + 3D Symmetry Symmetry Elements 1. Rotation
a. Two-fold rotation
= 360o/2 rotation
to reproduce a motif in a symmetrical pattern
= the symbol for a two-fold rotation 6
first operation step
second operation step 6

9. 6 6 6 2-D + 3D Symmetry Symmetry Elements 1. Rotation
b. Three-fold rotation
= 360o/3 rotation
to reproduce a motif in a symmetrical pattern 6 6 6

10. 6 6 6 2-D + 3D Symmetry Symmetry Elements 1. Rotation
b. Three-fold rotation
= 360o/3 rotation
to reproduce a motif in a symmetrical pattern 6
step 1 6
step 3 6
step 2

11. Symmetry Elements 1. Rotation
2-D + 3D Symmetry
Symmetry Elements
1. Rotation 6 6 6 6 6
1-fold
2-fold
3-fold
4-fold
6-fold
5-fold and greater than 6-fold rotations will not work in combination with translations in crystals

12. 2D + 3D Symmetry
Symmetry Element
2. Inversion
Inversion (i) – symmetry with respect to a point, called an inversion centre 1 1

13. 2-D + 3D Symmetry Symmetry Elements 3. Reflection (m) m m
Reflection across a “mirror plane” reproduces a motif
= symbol for a mirror
plane m m

14. 2-D + 3D Symmetry We now have 6 unique symmetry operations:
Rotations are congruent operations:
reproductions are identical
Inversion and reflection are enantiomorphic operations:
reproductions are mirror images

15. Congruent operations are also called proper operations, or in more elegant terms, operations of the first sort
Incongruent operations are also called improper operations, or in more elegant terms, operations of the second sort

16. 2-D + 3D Symmetry Combinations of symmetry elements are also possible
To create a complete analysis of symmetry about a point in space, we must try all possible combinations of these symmetry elements

17. Try combining a 2-fold rotation axis with a mirror
2-D + 3D Symmetry
Try combining a 2-fold rotation axis with a mirror

18. 2-D + 3D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
(could do either step first)

19. 2-D + 3D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)

20. 2-D + 3D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
Is that all??

21. 2-D + 3D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
No! A second mirror is required

22. 2-D + 3D Symmetry Try combining a 2-fold rotation axis with a mirror
The result is Point Group 2mm
“2mm” indicates 2 mirrors
The mirrors are different
(not equivalent by symmetry)

23. Now try combining a 4-fold rotation axis with a mirror
2-D + 3D Symmetry
Now try combining a 4-fold rotation axis with a mirror

24. Now try combining a 4-fold rotation axis with a mirror Step 1: reflect
2-D + 3D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect

25. 2-D + 3D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 2

26. 2-D + 3D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 3

27. Now try combining a 4-fold rotation axis with a mirror
2-D + 3D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors

28. Now try combining a 4-fold rotation axis with a mirror
2-D + 3D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors
Point group name??

29. Now try combining a 4-fold rotation axis with a mirror
2-D + 3D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors
Point group name??
4mm

30. 3-fold rotation axis with a mirror creates point group 3m
2-D + 3D Symmetry
3-fold rotation axis with a mirror creates point group 3m

31. 6-fold rotation axis with a mirror creates point group 6mm
2-D + 3D Symmetry
6-fold rotation axis with a mirror creates point group 6mm

32. 2-D + 3D Symmetry All other combinations are either: Incompatible
(2 + 2 cannot be done in 2-D)
Redundant with others already tried
m + m  2mm because creates 2-fold
This is the same as 2 + m  2mm

33. 2-D Symmetry
The original 6 elements plus the 4 combinations creates 10 possible 2-D planar point Groups (which also occur in 3-D):
m 2mm 3m 4mm 6mm
Any 2-D pattern of objects surrounding a point must conform to one of these groups

34. New 3-D Symmetry Elements Rotoinversion (a) 1-fold rotoinversion 1

35. 3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion _
a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)

36. 3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion _
a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)
Step 2: invert

37. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2

38. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Step 2: invert

39. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
b. 2-fold rotoinversion ( 2 )
The result:

40. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
b. 2-fold rotoinversion ( 2 )
This is the same as m, so not a new operation

41. New Symmetry Elements 4. Rotoinversion _ c. 3-fold rotoinversion ( 3 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
c. 3-fold rotoinversion ( 3 )

42. 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 _
c. 3-fold rotoinversion ( 3 )
Step 1: rotate 360o/3 1

43. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
c. 3-fold rotoinversion ( 3 )
Step 2: invert through center

44. 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 2 _
c. 3-fold rotoinversion ( 3 )
Completion of the first sequence 1 2

45. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
c. 3-fold rotoinversion ( 3 )
Rotate another 360/3

46. 3-D Symmetry New Symmetry Elements 4. Rotoinversion 3 1 2_
c. 3-fold rotoinversion ( 3 )
Complete second step to create face 3 3 1 2

47. 3-D Symmetry New Symmetry Elements 4. Rotoinversion 3 1 4 2
c. 3-fold rotoinversion ( 3 )
Third step creates face 4
(3  (1)  4) 3 1 4 2

48. 3-D Symmetry New Symmetry Elements 4. Rotoinversion 1 5 2
c. 3-fold rotoinversion ( 3 )
Fourth step creates face 5
(4  (2)  5) 5 1 2

49. 3-D Symmetry New Symmetry Elements 4. Rotoinversion 5 1 6
c. 3-fold rotoinversion ( 3 )
Fifth step creates face 6
(5  (3)  6)
Sixth step returns to face 1 5 1 6

50. 3-D Symmetry New Symmetry Elements 4. Rotoinversion This is unique 3 5
c. 3-fold rotoinversion ( 3 )
This is unique 3 5 1 4 2 6

51. New Symmetry Elements 4. Rotoinversion _ d. 4-fold rotoinversion ( 4 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )

52. New Symmetry Elements 4. Rotoinversion _ d. 4-fold rotoinversion ( 4 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )

53. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4

54. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert

55. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert

56. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4

57. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert

58. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert

59. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4

60. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
This is also a unique operation

61. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
d. 4-fold rotoinversion ( 4 )
A more fundamental representative of the pattern

62. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
e. 6-fold rotoinversion ( 6 )
Begin with this framework:

63. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1

64. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1

65. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1 2

66. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1 3 2

67. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1 3 2

68. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1 3 2 4

69. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1 3 2 4

70. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1 3 5 2 4

71. New Symmetry Elements 4. Rotoinversion _ e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion _
e. 6-fold rotoinversion ( 6 ) 1 3 5 2 4

72. New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1 3 5 2 6 4

73. 3-D Symmetry New Symmetry Elements 4. Rotoinversion _
e. 6-fold rotoinversion ( 6 )
Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane
(combinations of elements follows)
Top View

74. 3-D Symmetry New Symmetry Elements 4. Rotoinversion A simpler pattern_
e. 6-fold rotoinversion ( 6 )
A simpler pattern
Top View

75. 3-D Symmetry We now have 10 unique 3-D symmetry operations:
Combinations of these elements are also possible

76. 3-D Symmetry 3-D symmetry element combinations
a. Rotation axis parallel to a mirror
Same as 2-D
2 || m = 2mm
3 || m = 3m, also 4mm, 6mm
b. Rotation axis  mirror
2  m = 2/m
3  m = 3/m, also 4/m, 6/m
c. Most other rotations + m generates already existing symmetry operations 4/m = 4/m

77. 3-D Symmetry 3-D symmetry element combinations
d. Combinations of rotations
2 + 2 at 90o  222 (third 2 required from combination)
4 + 2 at 90o  422 ( “ “ “ )
6 + 2 at 90o  622 ( “ “ “ )

78. 3-D Symmetry
As in 2-D, the number of possible combinations is limited only by incompatibility and redundancy
There are only 22 possible unique 3-D combinations, when combined with the 10 original 3-D elements yields the 32 3-D Point Groups

79. Note that the 32 point groups, also called crystal classes are derived from only 13 operations:
Congruent operations
1, 2, 3, 4, 6
Incongruent operations
_ _ _ _ _
1 = i, 2 = m, 3, 4, 6 = 3/m, 2/m, 4/m, 6/m