symmetry in art and architecture
TRANSCRIPT
Symmetry in Art and Architecture
A/P Helmer AslaksenDept. of Mathematics
National Univ. of Singaporewww.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
Symmetry at Scotts Road
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?
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
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 symmetry
Order 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 plane isometries Translation Reflections Rotations Glide reflections
Warning!
Sumerian symmetry
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 on cloth
The 7 frieze groups No sym Glide ref Hor ref Ver ref Half turn Hor and ver ref Glide ref and ver ref
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
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?
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 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)
Ming porcelain patterns
66
2921 20
13 91
0
20
40
60
pm11 p111 p1a1 p112 pma2 pmm2 p1m1
Frieze Patterns Types
Seven Types of Frieze Pattern
Ming porcelain patterns by emperor
Distribution of Frieze Patterns Types in
Diff erent Time Periods
0
2
4
6
8
10
12
14
16
Yuan Yongle Xuande Jiajing Wanli T&C
Time Period
p111 p112 p1a1 pm11 pmm2 pma2 p1m1
Regular tilings
Semiregular tilings
More fun stuff!
False viewpoints
Pozzo’s ceiling (1694) and cupola (1685) in St. Ignatius, Rome
Perspective at SAM