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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?

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the mathematics of ceramics

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 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
Why are these patterns nice?
  • Symmetry
  • What is symmetry?
  • Most people think of vertical mirror symmetry (left/right)
what is symmetry in general
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
rotation
Rotation
  • A rotation has a centre of rotation and an angle of rotation
n fold 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
Glide reflection
  • A glide reflection is a combination of a reflection and a translation
four types of isometries
Four types of isometries
  • Translation
  • Reflections
  • Rotations
  • Glide reflections
symmetric patterns
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
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
  • 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
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
Wallpaper
  • There are 17 types of wall paper patterns
what does this have to do with arts
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
Ming ceramics
  • We will study Ming ceramics as an example
no symmetry
No symmetry
  • The p111 pattern (no symmetry)
horizontal reflection
Horizontal reflection
  • The p1m1 pattern (horizontal reflection)
vertical reflection
Vertical reflection
  • The pm11 pattern (vertical reflection)
half turn
Half turn
  • The p112 pattern (half turn)
horizontal and vertical reflection
Horizontal and vertical reflection
  • The pmm2 pattern (horizontal and vertical reflections)
glide reflection and vertical reflection
Glide reflection and vertical reflection
  • The pma2 pattern (glide reflection and vertical reflection)
glide reflection29
Glide reflection
  • The p1a1 pattern (glide reflection)
peranakan ceramics
Peranakan Ceramics
  • We also looked at the Peranakan ceramics at the Asian Civilisations Museum in Singapore
no symmetry37
No symmetry
  • The p111 pattern
vertical reflection38
Vertical reflection
  • The pm11 pattern
half turn39
Half turn
  • The p112 pattern
glide reflection42
Glide reflection
  • The p1a1 pattern