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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 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 2 180° 3 120° 6 60° Figure Symmetry Regions 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 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? 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 Seven Types of Frieze Pattern 60 40 29 21 20 20 0 pm11 p111 p1a1 p112 13 pma2 Frieze Patterns Types 9 pmm2 1 p1m1 Ming porcelain patterns by emperor Distribution of Frieze Patterns Types in Different Time Periods 16 14 12 10 8 6 4 2 0 Yuan Yongle Xuande Jiajing Wanli Time Period p111 p112 p1a1 pm11 pmm2 pma2 p1m1 T&C Regular tilings Semiregular tilings More fun stuff! False viewpoints Pozzo’s ceiling (1694) and cupola (1685) in St. Ignatius, Rome Perspective at SAM