The Mathematics of Ceramics

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

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

8. A reflection flips across an axis of reflection 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 = 360o/θ is a whole number, then we call the rotation an n-fold rotation

11. Rotational symmetry

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. • Cn, which has n-fold rotational symmetry and no reflectional symmetry • Dn, 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 sym LLLL • Half turn NNN • Hor ref DDD • Ver ref VVV • Glide ref • Hor and ver ref HHH • 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

31. Analysis-Ming Porcelains

32. Analysis-Ming Porcelains

33. Analysis-Ming Porcelains

34. Analysis-Ming Porcelains

35. 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

44. Analysis-Peranakan Porcelains

45. Analysis- Peranakan Porcelains