the mathematics of ceramics a/p helmer aslaksen dept. of mathematics national univ. of singapore

The Mathematics of Ceramics

A/P Helmer AslaksenDept. of Mathematics

National Univ. of Singaporewww.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

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