The Mathematics of Ceramics

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# The Mathematics of Ceramics - PowerPoint PPT Presentation

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

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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
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
Glide reflection
• The p1a1 pattern