1. Symmetry Elements Lecture 5

3. 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

4. 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation = 360 o /2 rotation to reproduce a motif in a symmetrical pattern 6 6 A Symmetrical Pattern

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

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

7. 6 6 first operation step second operation step 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation = 360 o /2 rotation to reproduce a motif in a symmetrical pattern = the symbol for a two-fold rotation

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

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

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

11. 6 6 6 step 1 step 2 step 3 2-D Symmetry Symmetry Elements 1. Rotation b. Three-fold rotation = 360 o /3 rotation to reproduce a motif in a symmetrical pattern

12. Symmetry Elements 1. Rotation 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 1-fold 2-fold 3-fold 4-fold 6-fold 2-D Symmetry

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

14. 3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 2-fold rotoinversion ( 2 )

15. 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Note: this is a temporary step, the intermediate motif element does not exist in the final pattern

16. 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Step 2: invert

17. 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) The result:

18. 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) This is the same as m, so not a new operation

19. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 )

20. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Step 1: rotate 360 o /3 Again, this is a temporary step, the intermediate motif element does not exist in the final pattern 1

21. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Step 2: invert through center

22. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Completion of the first sequence 1 2

23. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Rotate another 360/3

24. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Invert through center

25. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Complete second step to create face 3 1 2 3

26. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Third step creates face 4 (3  (1)  4) (3  (1)  4) 1 2 3 4

27. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Fourth step creates face 5 (4  (2)  5) 1 2 5

28. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Fifth step creates face 6 (5  (3)  6) (5  (3)  6) Sixth step returns to face 1 1 6 5

29. 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) This is unique 1 6 5 2 3 4

30. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 )

31. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 )

32. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4

33. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert

34. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert

35. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4

36. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert

37. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert

38. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4

39. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 6: Invert

40. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) This is also a unique operation

41. 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) A more fundamental representative of the pattern

42. 3-D Symmetry We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m 3 4 1 2 3 4 6 m 3 4 Combinations of these elements are also possible A complete analysis of symmetry about a point in space requires that we try all possible combinations of these symmetry elements