Symmetry Elements

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

5. Operation 2-D Symmetry 6 Motif 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 Element 6

6. 2-D Symmetry 6 first operation step 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

7. 2-D Symmetry 6 first operation step 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 second operation step 6

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

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

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

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

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

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

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

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

22. 3-D Symmetry 1 New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Completion of the first sequence 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 3 1 New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Complete second step to create face 3 2

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

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

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

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

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 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 We now have 8 unique 3D symmetry operations: 1 2 3 4 6 m3 4