Symmetry in Art and Architecture A P Helmer Aslaksen

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 15 th century

Patterns at Plaza Singapore

Mystery pattern Fullerton Hotel

Where in Singapore is this?

Shaw House

Symmetry at Scotts Road C 8 D 6

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

What is symmetry in general? n n 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? n An isometry of the plane is a mapping that preserves distance, and therefore shape

Translation n A translation moves a fixed distance in a fixed direction

Reflection n A reflection flips across an axis of reflection

Rotation n A rotation has a centre of rotation and an angle of rotation

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

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

Glide reflection n A glide reflection is a combination of a reflection and a translation

Four types of plane isometries n n Translation Reflections Rotations Glide reflections

Warning!

Sumerian symmetry

Symmetric patterns n n 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 n n n 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 n n 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 n n n n No sym Glide ref Hor ref Ver ref Half turn Hor and ver ref Glide ref and ver ref

Examples of frieze patterns n n n n 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? n n n 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 n We will study Ming ceramics as an example

No symmetry n The p 111 pattern (no symmetry)

Horizontal reflection n The p 1 m 1 pattern (horizontal reflection)

Vertical reflection n The pm 11 pattern (vertical reflection)

Half turn n The p 112 pattern (half turn)

Horizontal and vertical reflection n The pmm 2 pattern (horizontal and vertical reflections)

Glide reflection and vertical reflection n The pma 2 pattern (glide reflection and vertical reflection)

Glide reflection n The p 1 a 1 pattern (glide reflection)

Ming porcelain patterns

Ming porcelain patterns by emperor

Regular tilings

Semiregular tilings

More fun stuff!

False viewpoints n Pozzo’s ceiling (1694) and cupola (1685) in St. Ignatius, Rome

Perspective at SAM