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## The Mathematics of Ceramics

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**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**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**Why are these patterns nice?**• Symmetry • What is symmetry? • Most people think of vertical mirror symmetry (left/right)**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**An isometry of the plane is a mapping that preserves**distance, and therefore shape What is an isometry?**A translation moves a fixed distance in a fixed direction**Translation**Rotation**• A rotation has a centre of rotation and an angle of rotation**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**Glide reflection**• A glide reflection is a combination of a reflection and a translation**Four types of isometries**• Translation • Reflections • Rotations • Glide reflections**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**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**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**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**Wallpaper**• There are 17 types of wall paper patterns**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.**Ming ceramics**• We will study Ming ceramics as an example**No symmetry**• The p111 pattern (no symmetry)**Horizontal reflection**• The p1m1 pattern (horizontal reflection)**Vertical reflection**• The pm11 pattern (vertical reflection)**Half turn**• The p112 pattern (half turn)**Horizontal and vertical reflection**• The pmm2 pattern (horizontal and vertical reflections)**Glide reflection and vertical reflection**• The pma2 pattern (glide reflection and vertical reflection)**Glide reflection**• The p1a1 pattern (glide reflection)**Peranakan Ceramics**• We also looked at the Peranakan ceramics at the Asian Civilisations Museum in Singapore**No symmetry**• The p111 pattern**Vertical reflection**• The pm11 pattern**Half turn**• The p112 pattern**Horizontal and vertical reflection**• The pmm2 pattern**Glide reflection and vertical reflection**• The pma2 pattern pma2 pm11**Glide reflection**• The p1a1 pattern