Teaching a course on Mathematics in Art and Architecture

Helmer Aslaksen

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

Content of art course

• Tilings and polyhedra• Symmetry• Frieze and wallpaper patterns• Perspective in painting

Polyhedra

• There is a lot of interesting mathematics regarding polyhedra

• It is fun to make polyhedral models

What is the best way to make models?

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.

Why make polyhedral models?

They are beautiful!

They are fun to make!

They are great for learning!

They are great for teaching!

They are great for your department!

How to make polyhedral models?• Paper• Plastic

Paper Models

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.

Simple classroom activity

Plastic Model Kits

• Zome Tool• Polydron• Jovo

Zome Tool

Polydron

Polydron and Frameworks

Jovo

Face or edge?

Polydron/Frameworks and Jovo are face based

Zome is edge based

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.

Advantages of Zome

• Unlimited possibilities.• Nested models.

Zome possibilities

• Zome Geometry: Hands on Learning With Zome Models by George W. Hart and Henri Picciotto.

• Soap bubbles.

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.

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.

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?

How much space do you have?

More

More!

More!!

More!!!

Symmetry and patterns

• Rosette, frieze and wallpaper patterns occur all around us

Where in Singapore is this?

Lau Pa Sat

Mystery pattern

Odd number of kites at Fullerton Hotel

Where in Singapore is this?

Shaw House

Symmetry at Scotts Road

C8 D6

More cool stuff in Singapore

Marriott Hotel

Ming Porcelain

• One of my students studied frieze patterns on Ming porcelain

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

Frieze Patterns Found

• The p111 pattern

Frieze Patterns Found

• The p1m1 pattern

Frieze Patterns Found

• The pm11 pattern

Frieze Patterns Found

• The p112 pattern

Frieze Patterns Found

• The pmm2 pattern

Frieze Patterns Found

• The pma2 pattern

Frieze Patterns Found

• The p1a1 pattern

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

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

Perspective in painting

• Perspective in painting and photographs has many applications to the world around us

Giotto, The Flight into Egypt, c1313

• Notice how the trees are the same size

Lorenzetti, The Presentation in the Temple, c1342

• Notice how the tiles get smaller

Masaccio, Trinity, 1427

• One of the first perspective pictures

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

Side Vanishing Points 2

Where’s the best view point?

• 174cm above, 770cm away

False viewpoints

• Pozzo’s ceiling (1694) and cupola (1685) in St. Ignazio, Rome

Anamorphic art

• Holbein, The Ambassadors, 1533

Is there perspective in Chinese paintings?

• Multiple viewpoints, Chen Chong Swee, Snowscape, 1993

• Raphael, The School of Athens, 1511

What does a sphere look like?

What’s going on here?

Vermeer (1632—1675)

The Music Lesson (1662-5)Royal Collection, London

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

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)

Girl with a Pearl Earring

• He is seen using a camera obscura in the movie Girl with a Pearl Earring

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

Lady Standing at the Virginals (1670-3), National Gallery, London

Steadman’s Reconstruction

• Philip Steadman did a 1/6 scale model reconstruction of The Music Lesson in a BBC TV program in 1989

Vermeer’s Camera

• More details are given in his book Vermeer’s Camera (2001)

What is Special about The Music Lesson?

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

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

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?

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?

Future work

• Gothic architecture• Salsa dance

Gothic vaults

More vaults

Mathematics of Salsa dancing

• How to remember dance moves• Leg work is easy, arm work is hard• Construct a language to describe moves

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?

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

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.

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.

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?

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.

Have fun designing your own course!

• Good luck and thank you!