Polygons and polygrams

1. Polygons and polygrams

2. A regular polygram is generalization of a (regular) polygon on sides (i.e., an -gon) obtained by connecting every th vertex around a circle with every th, "picking up" the pencil as needed to repeat the procedure after traversing the circle until none of the vertices remain unconnected.

3. The best-known polygrams are the pentagram and hexagram. The following table summarizes some named polygrams.

4. Patterns in Islamic art Fez, Morocco, 1325

5. Isfahan, Iran, end of 15th century

6. Patterns at Plaza Singapore

7. Fullerton Hotel

9. Shaw House

10. Symmetry at Scotts Road C8 D6

11. Marriott Hotel

12. Bugis Junction

13. More cool stuff in Singapore

14. Where in Ithaca is this?

15. 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

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

17. 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

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

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

20. Reflection A reflection flips across an axis of reflection

21. Rotation A rotation has a centre of rotation and an angle of rotation

22. 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

23. Rotational symmetry

24. Glide reflection A glide reflection is a combination of a reflection and a translation

25. Four types of plane isometries Translation Reflections Rotations Glide reflections

26. Sumerian symmetry

27. 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

28. 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

29. Examples of rosette patterns

30. 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

31. Frieze patterns on cloth

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

33. No sym LLLL Half turnNNN Hor refDDD Ver refVVV Glide ref Hor and ver refHHH Glide ref and ver ref Examples of frieze patterns

34. Chart for the 7 frieze groups

35. Wallpaper floor tilings

36. Wallpaper cloth

37. The 17 types of wall paper groups

38. Chart for the 17 wall paper groups

39. Examples of the 17 groups

40. 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. What does this have to do with art?

41. Distribution in Islamic art

42. We will study Ming ceramics as an example Ming ceramics

43. The p111 pattern (no symmetry) No symmetry

44. The p1m1 pattern (horizontal reflection) Horizontal reflection

45. The pm11 pattern (vertical reflection) Vertical reflection

46. The p112 pattern (half turn) Half turn

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

48. The pma2 pattern (glide reflection and vertical reflection) Glide reflection and vertical reflection

49. The p1a1 pattern (glide reflection) Glide reflection

50. Ming porcelain patterns