# Symmetry Groups in Arts and Architecture

```Symmetry in Art and
Architecture
A/P Helmer Aslaksen
Dept. of Mathematics
National Univ. of Singapore
www.math.nus.edu.sg
aslaksen@math.nus.edu.sg
Where in Singapore is this?
Lau Pa Sat
Polygons and polygrams
Reuleaux triangle
Patterns in Islamic art
Fez, Morocco, 1325
Patterns in Islamic art
Isfahan, Iran, end of 15th century
Patterns at Plaza Singapore
Mystery pattern
Fullerton Hotel
Where in Singapore is this?
Shaw House
C8
D6
Marriott Hotel
Bugis Junction
Suntec
Tampines
More cool stuff in Singapore
Not so cool stuff in Singapore
What does math have to
do with art?
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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?
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Symmetry
What is symmetry?
Most people think of vertical mirror
symmetry (left/right)
What is symmetry in general?
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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?
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An isometry of the plane is a mapping that
preserves distance, and therefore shape
Translation
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A translation moves a fixed distance in
a fixed direction
Reflection
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A reflection flips across an axis of
reflection
Rotation
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A rotation has a centre of rotation and
an angle of rotation
N-fold rotation
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If the angle is θ and n = 360o/θ is a
whole number, then we call the rotation
an n-fold rotation
Rotational symmetry
Order of
Rotation
Angle of
Rotation
2
180&deg;
3
120&deg;
6
60&deg;
Figure
Symmetry Regions
Glide reflection
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A glide reflection is a combination of a
reflection and a translation
Four types of plane isometries
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Translation
Reflections
Rotations
Glide reflections
Warning!
Sumerian symmetry
Symmetric patterns
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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
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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
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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 on cloth
The 7 frieze groups
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No sym
Glide ref
Hor ref
Ver ref
Half turn
Hor and ver ref
Glide ref and ver ref
Examples of frieze patterns
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No sym
Half turn
Hor ref
Ver ref
Glide ref
Hor and ver ref
Glide ref and ver ref
LLLL
NNN
DDD
VVV
HHH
Chart for the 7 frieze groups
Wallpaper floor tilings
Wallpaper cloth
The 17 types of wall paper
groups
Chart for the 17 wall paper
groups
Examples of the 17 groups
What does this have to do
with art?
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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.
Distribution in Islamic art
Ming ceramics
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We will study Ming ceramics as an
example
No symmetry
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The p111 pattern (no symmetry)
Horizontal reflection
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The p1m1 pattern (horizontal
reflection)
Vertical reflection
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The pm11 pattern (vertical reflection)
Half turn
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The p112 pattern (half turn)
Horizontal and vertical
reflection
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The pmm2 pattern (horizontal and vertical
reflections)
Glide reflection and vertical
reflection
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The pma2 pattern (glide reflection and
vertical reflection)
Glide reflection
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The p1a1 pattern (glide reflection)
Ming porcelain patterns
66
Seven Types of Frieze Pattern
60
40
29
21
20
20
0
pm11
p111
p1a1
p112
13
pma2
Frieze Patterns Types
9
pmm2
1
p1m1
Ming porcelain patterns by
emperor
Distribution of Frieze Patterns Types in
Different Time Periods
16
14
12
10
8
6
4
2
0
Yuan
Yongle
Xuande
Jiajing
Wanli
Time Period
p111
p112
p1a1
pm11
pmm2
pma2
p1m1
T&amp;C
Regular tilings
Semiregular tilings
More fun stuff!
False viewpoints
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Pozzo’s ceiling (1694) and cupola (1685) in
St. Ignatius, Rome
Perspective at SAM
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