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Where in Singapore is this?
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The Mathematics of Ceramics
A/P Helmer AslaksenDept. of Mathematics
National Univ. of Singaporewww.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
Where in Singapore is this?
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
What is an isometry? An isometry of the plane is a mapping that
preserves distance, and therefore shape
Translation A translation moves a fixed
distance in a fixed direction
Reflection A reflection flips across an axis of
reflection
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
Rotational symmetryOrder of Rotation
Angle of Rotation
Figure Symmetry Regions
2 180°
3 120°
6 60°
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
Examples of rosette 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
Frieze Patterns
Examples of frieze patterns No sym LLLL Half turnNNN 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)
Analysis-Ming Porcelains66
2921 20
13 91
0
20
40
60
pm11 p111 p1a1 p112 pma2 pmm2 p1m1
Frieze Patterns Types
Seven Types of Frieze Pattern
Analysis-Ming PorcelainsDistribution of Frieze patterns on
Ming Porcelains
Top Rim33%
Top 12%
Body 24%
Base19%
Foot Ring12%
Analysis-Ming PorcelainsDistribution of Frieze Patterns on
Ming Porcelains
0
5
10
15
20
25
Top Rim Top Body Base Foot Ring
Area on Ming Porcelainsp111 p112 p1a1 pm11 pmm2 pma2 p1m1
Analysis-Ming PorcelainsDistribution of Frieze Patterns on
Ming Porcelains in Diff erent Periods
0
2
4
6
8
10
12
Period
Top Rim Top Body Base Foot Ring
Analysis-Ming PorcelainsDistribution of Frieze Patterns on
Ming Porcelains in Diff erent Time Periods
0
2
4
6
8
10
12
Yuan Yongle Xuande Jiajing Wanli T&CTime Period
Top Rim Top Body Base Foot Ring
Analysis-Ming PorcelainsDistribution of Frieze Patterns Types in
Diff erent Time Periods
02468
10121416
Yuan Yongle Xuande Jiajing Wanli T&C
Time Period
p111 p112 p1a1 pm11 pmm2 pma2 p1m1
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
Analysis-Peranakan Porcelains
50
137
1 1 1 00
1020304050
pm11 p111 p112 p1a1 pmm2 pma2 p1m1
Frieze Patterns Types
Seven Types of Frieze Pattern
Analysis-Peranakan Porcelains
Distribution of Frieze patterns on Peranakan Porcelains
Top Rim21%
Top 16%
Body 10%
Base50%
Foot Ring3%
Analysis- Peranakan Porcelains
Distribution of Frieze Patterns on Peranakan Porcelains
0
5
10
1520
25
30
35
Top Rim Top Body Base Foot Ring
Area on Peranakan Porcelainsp111 p112 p1a1 pm11 pmm2 pma2 p1m1
