Transcript Symmetry Groups in Arts and Architecture
The Mathematics of Ceramics A/P Helmer Aslaksen Dept. of Mathematics National Univ. of Singapore www.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 an n-fold rotation n = 360 o / θ is a whole number, then we call the rotation
Rotational symmetry Order of Rotation 2 Angle of Rotation 180° Figure Symmetry Regions 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.
C n , which has n-fold rotational symmetry and no reflectional symmetry D n , 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 Half turn Hor ref Ver ref Glide ref Hor and ver ref Glide ref and ver ref LLLL NNN DDD VVV HHH
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 Porcelains
66 Seven Types of Frieze Pattern 60 40 29 20 21 20 13 9 0 pm11 p111 p1a1 p112 pma2 Frieze Patterns Types pmm2 p1m1 1
Analysis-Ming Porcelains
Top Rim 33% Distribution of Frieze patterns on Ming Porcelains Top 12% Body 24% Foot Ring 12% Base 19%
25 20 15 10 5 0
Analysis-Ming Porcelains
Distribution of Frieze Patterns on Ming Porcelains Top Rim p111 Top Body Base p112 Area on Ming Porcelains p1a1 pm11 pmm2 pma2 Foot Ring p1m1
Analysis-Ming Porcelains
Distribution of Frieze Patterns on Ming Porcelains in Different Periods 12 10 8 6 4 2 0 Yu an H on gw u Yo ng le Xu an de Zh en gt on g C he ng hu a H on gz hi Zh en gd e Period Top Rim Top Body Base Ji aj in g Foot Ring W an li T& C
12 10 8 6 4 2 0
Analysis-Ming Porcelains
Distribution of Frieze Patterns on Ming Porcelains in Different Time Periods Yuan Yongle Xuande Jiajing Tim e Pe riod Wanli Top Rim Top Body Base Foot Ring T&C
Analysis-Ming Porcelains
Distribution of Frieze Patterns Types in Different Time Periods 16 14 12 10 8 6 4 2 0 Yuan Yongle p111 p112 Xuande Jiajing Time Period p1a1 pm11 pmm2 Wanli pma2 p1m1 T&C
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 50 40 30 20 10 0 pm 11 13 Seven Types of Frieze Pattern 7 1 1 p111 p112 p1a1 pm m 2 Frieze Patterns Types pm a2 1 p1m 1 0
Analysis-Peranakan Porcelains
35 30 25 20 15 10 5 0
Analysis- Peranakan Porcelains
Distribution of Frieze Patterns on Peranakan Porcelains Top Rim p111 Top Body Base Area on Peranakan Porcelains p112 p1a1 pm11 pmm2 pma2 Foot Ring p1m1