taipei teaching a course on mathematics in art and architecture
DESCRIPTION
Prof. Helmer Aslaksen's SlidesTRANSCRIPT
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Teaching a course on Mathematics in Art and Architecture
Helmer Aslaksen
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What’s the goal of this talk? • I used to teach two General Education Modules
at the National University of Singapore• Heavenly Mathematics & Cultural Astronomy• Mathematics in Art and Architecture
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Content of art course
• Tilings and polyhedra• Symmetry• Frieze and wallpaper patterns• Perspective in painting
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Polyhedra
• There is a lot of interesting mathematics regarding polyhedra
• It is fun to make polyhedral models
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What is the best way to make models?
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There is no best way!
• All the methods have advantages and disadvantages.
• My goal is to help you make the choice that is right for you.
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Why make polyhedral models?
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They are beautiful!
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They are fun to make!
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They are great for learning!
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They are great for teaching!
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They are great for your department!
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How to make polyhedral models?• Paper• Plastic
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Paper Models
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Platonic and Archimedean solids• Several web pages have nets for the Platonic
and Archimedean solids.• Build your own Polyhedra• Paper Models of Polyhedra• douglas zongker polyhedra models.
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Simple classroom activity
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Plastic Model Kits
• Zome Tool• Polydron• Jovo
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Polydron and Frameworks
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Face or edge?
Polydron/Frameworks and Jovo are face based
Zome is edge based
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Advantages of Polydron and Jovo• Easier to assemble. • Green struts in Zome require some practice.• Makes more sense for non-convex models.• Colored faces.• Models are smaller, especially with Jovo.• Tilings.
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Advantages of Zome
• Unlimited possibilities.• Nested models.
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Zome possibilities
• Zome Geometry: Hands on Learning With Zome Models by George W. Hart and Henri Picciotto.
• Soap bubbles.
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Which Platonic/Archimedean solids can you make?• Zome: All except the snub cube and snub
dodecahedron. The struts can only be pointed in certain directions.
• Polydron: All except the truncated dodecahedron and the great rhombicosidodecahedron. No decagon.
• Jovo: The basic set only contain triangle, square and pentagon. Hexagon in an additional package.
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Jovo models
• Basic Jovo can only make six of the 13 Archimedean solids. With hexagons we can make three more. But truncated cube and great rhombicuboctahedron require octagons.
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What are your needs?
• Do you need to quickly make some models for demonstration purposes or simple student activities?
• Do you or the students want to explore further?
• Do you have a large class or a small group?• What is your budget?
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How much space do you have?
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More
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More!
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More!!
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More!!!
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Symmetry and patterns
• Rosette, frieze and wallpaper patterns occur all around us
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Where in Singapore is this?
Lau Pa Sat
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Mystery pattern
Odd number of kites at 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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More cool stuff in Singapore
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Marriott Hotel
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Ming Porcelain
• One of my students studied frieze patterns on Ming porcelain
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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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Frieze Patterns Found
• The p111 pattern
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Frieze Patterns Found
• The p1m1 pattern
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Frieze Patterns Found
• The pm11 pattern
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Frieze Patterns Found
• The p112 pattern
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Frieze Patterns Found
• The pmm2 pattern
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Frieze Patterns Found
• The pma2 pattern
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Frieze Patterns Found
• The p1a1 pattern
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Analysis-Ming Porcelains
66
2921 20
13 91
0
20
40
60
pm11 p111 p1a1 p112 pma2 pmm2 p1m1
Frieze Patterns Types
Seven Types of Frieze Pattern
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Analysis-Ming Porcelains
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
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Perspective in painting
• Perspective in painting and photographs has many applications to the world around us
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Giotto, The Flight into Egypt, c1313
• Notice how the trees are the same size
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Lorenzetti, The Presentation in the Temple, c1342
• Notice how the tiles get smaller
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Masaccio, Trinity, 1427
• One of the first perspective pictures
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Side Vanishing Points
• One of the basic results in inverse projective geometry is that the distance between the central vanishing point and side vanishing point of a square is equal to the distance between the observer (camera) and the picture plane
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Side Vanishing Points 2
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Where’s the best view point?
• 174cm above, 770cm away
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False viewpoints
• Pozzo’s ceiling (1694) and cupola (1685) in St. Ignazio, Rome
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Anamorphic art
• Holbein, The Ambassadors, 1533
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Is there perspective in Chinese paintings?
• Multiple viewpoints, Chen Chong Swee, Snowscape, 1993
• Raphael, The School of Athens, 1511
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What does a sphere look like?
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What’s going on here?
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Vermeer (1632—1675)
The Music Lesson (1662-5)Royal Collection, London
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Did Vermeer use Optical Aids?
• This was suggested already in 1891 by the photographer Joseph Pennell
• Some of his paintings “look like photographs”, including sections that seem to be out of focus or use counterintuitive perspective
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Counterintuitive Perspective
Compare The Procuress by van Honthorst and Officer and Laughing Girl by VermeerMany art historians accept that Vermeer used a camera obscura (pinhole camera)
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Girl with a Pearl Earring
• He is seen using a camera obscura in the movie Girl with a Pearl Earring
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Vermeer’s Studio
• Several of his paintings appear to have been painted in the same studio
• We see similar windows on the left wall, wooden joists in the ceiling and tiles on the floor
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Lady Standing at the Virginals (1670-3), National Gallery, London
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Steadman’s Reconstruction
• Philip Steadman did a 1/6 scale model reconstruction of The Music Lesson in a BBC TV program in 1989
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Vermeer’s Camera
• More details are given in his book Vermeer’s Camera (2001)
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What is Special about The Music Lesson?
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Inverse Projective Geometry
• Several people have studied the problem of reconstructing 3D information from 2D images
• Criminisi: Accurate Visual Metrology from Single and Multiple Uncalibrated Images
• Byers, Henle: Where the Camera Was, Mathematics Magazine
• Crannell: Where the Camera Was, Take Two, Mathematics Magazine
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Student Work
• Inverse projective geometry is suitable for student work at many different levels
• From simple measurements and computation to literature surveys and software implementation
• Unfortunately, serious applications require serious applied math/engineering skills
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The Mystery of the Mirror
• The mirror is central to all mathematical analysis of this paper, but instead of solving our problems, it reveals a slew of questions
• Why would anybody hang a mirror there?• Is it for the lady to look at herself, or for us to
look at her?• Is it for the artist to give us a glimpse into his
secrets?
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The Angel and the Shadow
• In Steadman’s reconstruction, almost everything looks perfect, except for the angle of the mirror and its shadow on the wall
• He had to increase the angle of the mirror to make us see the lady in the mirror
• He could not make the lady and her mirror image line up
• What did Vermeer do?
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Future work
• Gothic architecture• Salsa dance
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Gothic vaults
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More vaults
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Mathematics of Salsa dancing
• How to remember dance moves• Leg work is easy, arm work is hard• Construct a language to describe moves
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Some contrarian thoughts
• Can I convince my department chair and dean that this is math?
• Can I convince the director of an art museum that this is art?
• Can I convince your students that this class will enrich their life?
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What is Mathematics and Art?
• I sometimes find it useful to think of the following four categories
• Mathematics in art• Mathematical art• Mathematics as art• Mathematics is art
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Mathematics in Art
• Topics like perspective in painting, symmetry in ornamental art and musical scales.
• Material that even the most anti-scientific art connoisseur will appreciate.
• You can approach any art museum with an offer of a public lecture on such topics.
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Mathematical Art
• Escher and other mathematically inclined artists.
• Worshiped by mathematicians, frowned upon or ignored by the art community.
• Strict “no Escher” policy at the Singapore Art Museum.
• An offer to an art museum of a public lecture about Escher may not necessarily be accepted.
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Mathematics as Art
• Computers allow us to create beautiful visual mathematics.
• How many art museums would be interested in a public lecture about the Mandelbrot set?
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Mathematics is Art
• Many mathematicians believe that mathematics is an art, not a science.
• No art museum would be interested in a public lecture on Euclid’s axioms.
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Have fun designing your own course!
• Good luck and thank you!