Symmetry in Art and Architecture

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Symmetry in Art and Architecture
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Where in Singapore is this?
Lau Pa Sat
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Polygons and polygrams
Reuleaux triangle
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Patterns in Islamic art
Fez, Morocco, 1325
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Patterns in Islamic art
Isfahan, Iran, end of 15th century
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Patterns at Plaza Singapore
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Mystery pattern
Fullerton Hotel
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Where in Singapore is this?
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Shaw House
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Symmetry at Scotts Road
C8
D6
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Marriott Hotel
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Bugis Junction
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Suntec
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Tampines
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More cool stuff in Singapore
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Not so cool stuff in Singapore
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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

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Why are these patterns nice?

• Symmetry
• What is symmetry?
• Most people think of vertical mirror symmetry
  (left/right)

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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

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What is an isometry?

• An isometry of the plane is a mapping that
  preserves distance, and therefore shape

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Translation

• A translation moves a fixed distance in a fixed
  direction

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Reflection

• A reflection flips across an axis of reflection

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Rotation

• A rotation has a centre of rotation and an angle
  of rotation

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N-fold rotation

• If the angle is ? and n 360o/? is a whole
  number, then we call the rotation an n-fold
  rotation

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Rotational symmetry
Order of Rotation Angle of Rotation Figure Symmetry Regions
2 180
3 120
6 60
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Glide reflection

• A glide reflection is a combination of a
  reflection and a translation

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Four types of plane isometries

• Translation
• Reflections
• Rotations
• Glide reflections

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Warning!
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Sumerian symmetry
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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

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Rosette patterns

• Leonardos 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

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Examples of rosette patterns
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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

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Frieze patterns on cloth
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The 7 frieze groups

• No sym
• Glide ref
• Hor ref
• Ver ref
• Half turn
• Hor and ver ref
• Glide ref and ver ref

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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

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Chart for the 7 frieze groups
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Wallpaper floor tilings
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Wallpaper cloth
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The 17 types of wall paper groups
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Chart for the 17 wall paper groups
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Examples of the 17 groups
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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.

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Distribution in Islamic art
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Ming ceramics

• We will study Ming ceramics as an example

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No symmetry

• The p111 pattern (no symmetry)

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Horizontal reflection

• The p1m1 pattern (horizontal reflection)

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Vertical reflection

• The pm11 pattern (vertical reflection)

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Half turn

• The p112 pattern (half turn)

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Horizontal and vertical reflection

• The pmm2 pattern (horizontal and vertical
  reflections)

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Glide reflection and vertical reflection

• The pma2 pattern (glide reflection and vertical
  reflection)

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Glide reflection

• The p1a1 pattern (glide reflection)

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Ming porcelain patterns
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Ming porcelain patterns by emperor
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Regular tilings
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Semiregular tilings
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More fun stuff!
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False viewpoints

• Pozzos ceiling (1694) and cupola (1685) in St.
  Ignatius, Rome

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Perspective at SAM