1. The Mathematics of Ceramics A/P Helmer Aslaksen Dept. of Mathematics National Univ. of Singapore www.math.nus.edu.sg aslaksen@math.nus.edu.sg

2. What does math have to do with ceramics? 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

3. Where in Singapore is this?

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

5. 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

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

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

8. Reflection A reflection flips across an axis of reflection

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

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

11. Rotational symmetry Order of Rotation Angle of Rotation FigureSymmetry Regions 2180° 3120° 660°

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

13. Four types of isometries Translation Reflections Rotations Glide reflections

14. 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

15. Rosette patterns Leonardo’s Theorem: There are two types of rosette patterns. C n, which has n-fold rotational symmetry and no reflectional symmetry D n, which has n-fold rotational symmetry and reflectional symmetry

16. Examples of rosette patterns

17. 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

18. Frieze Patterns

19. Examples of frieze patterns No symLLLL Half turnNNN Hor refDDD Ver refVVV Glide ref Hor and ver refHHH Glide ref and ver ref

20. Wallpaper There are 17 types of wall paper patterns

21. What does this have to do with arts? 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.

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

23. No symmetry The p111 pattern (no symmetry)

24. Horizontal reflection The p1m1 pattern (horizontal reflection)

25. Vertical reflection The pm11 pattern (vertical reflection)

26. Half turn The p112 pattern (half turn)

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

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

29. Glide reflection The p1a1 pattern (glide reflection)

30. Analysis-Ming Porcelains

36. Peranakan Ceramics We also looked at the Peranakan ceramics at the Asian Civilisations Museum in Singapore

37. No symmetry The p111 pattern

38. Vertical reflection The pm11 pattern

39. Half turn The p112 pattern

40. Horizontal and vertical reflection The pmm2 pattern

41. Glide reflection and vertical reflection The pma2 pattern pma2 pm11

42. Glide reflection The p1a1 pattern

43. Analysis-Peranakan Porcelains