Symmetry in Art and Architecture

1. Symmetry in Art and Architecture A/P Helmer Aslaksen Dept. of Mathematics National Univ. of Singapore www.math.nus.edu.sg aslaksen@math.nus.edu.sg

2. Where in Singapore is this? Lau Pa Sat

3. Polygons and polygrams Reuleaux triangle

4. Patterns in Islamic art Fez, Morocco, 1325

5. Patterns in Islamic art Isfahan, Iran, end of 15th century

6. Patterns at Plaza Singapore

7. Mystery pattern Fullerton Hotel

8. Where in Singapore is this?

9. Shaw House

10. Symmetry at Scotts Road C8 D6

11. Marriott Hotel

12. Bugis Junction

13. Suntec

14. Tampines

15. More cool stuff in Singapore

16. Not so cool stuff in Singapore

17. What does math have to do with art? • What is math? • Math is the abstract study of patterns • What is a pattern? • Concrete geometrical patterns or abstract numerical or logical patterns • What is abstract study? • Generalize to get the underlying concept

18. Why are these patterns nice? • Symmetry • What is symmetry? • Most people think of vertical mirror symmetry (left/right)

19. What is symmetry in general? • A pattern is symmetric if it is built up from related parts • A plane pattern has a symmetry if there is an isometry of the plane that preserves the pattern

20. An isometry of the plane is a mapping that preserves distance, and therefore shape What is an isometry?

21. A translation moves a fixed distance in a fixed direction Translation

22. A reflection flips across an axis of reflection Reflection

23. Rotation • A rotation has a centre of rotation and an angle of rotation

24. N-fold rotation • If the angle is θ and n = 360o/θ is a whole number, then we call the rotation an n-fold rotation

25. Rotational symmetry

26. Glide reflection • A glide reflection is a combination of a reflection and a translation

27. Four types of plane isometries • Translation • Reflections • Rotations • Glide reflections

28. Warning!

29. Sumerian symmetry

30. Symmetric patterns • A plane pattern has a symmetry if there is an isometry of the plane that preserves it. There are three types of symmetric patterns. • Rosette patterns (finite designs) • Frieze patterns • Wallpaper patterns

31. Rosette patterns • Leonardo’s Theorem: There are two types of rosette patterns. • Cn, which has n-fold rotational symmetry and no reflectional symmetry • Dn, which has n-fold rotational symmetry and reflectional symmetry

32. Examples of rosette patterns

33. Frieze patterns • Frieze patterns are patterns that have translational symmetry in one direction • We imagine that they go on to infinity in both directions or wrap around

34. Frieze patterns on cloth

35. The 7 frieze groups • No sym • Glide ref • Hor ref • Ver ref • Half turn • Hor and ver ref • Glide ref and ver ref

36. Examples of frieze patterns • No sym LLLL • Half turn NNN • Hor ref DDD • Ver ref VVV • Glide ref • Hor and ver ref HHH • Glide ref and ver ref

37. Chart for the 7 frieze groups

38. Wallpaper floor tilings

39. Wallpaper cloth

40. The 17 types of wall paper groups

41. Chart for the 17 wall paper groups

42. Examples of the 17 groups

43. What does this have to do with art? • Every culture has a preference for certain symmetry type of patterns. • The important thing is not the motif in the patterns, but the symmetry types. • This can be used to date objects and detect connections between different cultures.

44. Distribution in Islamic art

45. Ming ceramics • We will study Ming ceramics as an example

46. No symmetry • The p111 pattern (no symmetry)

47. Horizontal reflection • The p1m1 pattern (horizontal reflection)

48. Vertical reflection • The pm11 pattern (vertical reflection)

49. Half turn • The p112 pattern (half turn)

50. Horizontal and vertical reflection • The pmm2 pattern (horizontal and vertical reflections)