symmetry in art and architecture

Symmetry in Art and Architecture

A/P Helmer AslaksenDept. of Mathematics

National Univ. of Singaporewww.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

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