
e0d5a4cff3dddf751ff74bbab24a1821.ppt
- Количество слайдов: 59
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