Prof. Helmer Aslaksen's Slides

1. Teaching a course on Mathematics
in Art and Architecture
Helmer Aslaksen

2. 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

3. Content of art course
• Tilings and polyhedra
• Symmetry
• Frieze and wallpaper patterns
• Perspective in painting

4. Polyhedra
• There is a lot of interesting mathematics
regarding polyhedra
• It is fun to make polyhedral models

5. What is the best way to make
models?

6. 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.

7. Why make polyhedral models?

8. They are beautiful!

9. They are fun to make!

10. They are great for learning!

11. They are great for teaching!

12. They are great for your
department!

13. How to make polyhedral
models?
• Paper
• Plastic

14. Paper Models

15. 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.

16. Simple classroom activity

17. Plastic Model Kits
• Zome Tool
• Polydron
• Jovo

18. Zome Tool

19. Polydron

20. Polydron and Frameworks

21. Jovo

22. Face or edge?
Polydron/Frameworks
and Jovo are face based
Zome is edge based

23. 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.

24. Advantages of Zome
• Unlimited possibilities.
• Nested models.

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

26. 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.

27. 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.

28. 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?

29. How much space do you
have?

30. More

31. More!

32. More!!

33. More!!!

34. Symmetry and patterns
• Rosette, frieze and wallpaper patterns occur
all around us

35. Where in Singapore is this?
Lau Pa Sat

36. Mystery pattern
Odd number of kites at Fullerton Hotel

37. Where in Singapore is this?

38. Shaw House

39. Symmetry at Scotts Road
C8 D6

40. More cool stuff in Singapore

41. Marriott Hotel

42. Ming Porcelain
• One of my students studied frieze patterns
on Ming porcelain

43. The 7 frieze groups
• No sym
• Glide ref
• Hor ref
• Ver ref
• Half turn
• Hor and ver ref
• Glide ref and ver ref

44. 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

45. Frieze Patterns Found
• The p111 pattern

46. Frieze Patterns Found
• The p1m1 pattern

47. Frieze Patterns Found
• The pm11 pattern

48. Frieze Patterns Found
• The p112 pattern

49. Frieze Patterns Found
• The pmm2 pattern

50. Frieze Patterns Found
• The pma2 pattern

51. Frieze Patterns Found
• The p1a1 pattern

52. Analysis-Ming Porcelains
66
29
21 20
13 9
1
0
20
40
60
pm11 p111 p1a1 p112 pma2 pmm2 p1m1
Frieze Patterns Types
Seven Types of Frieze Pattern

53. Analysis-Ming Porcelains
Distribution of Frieze Patterns Types in
Different 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

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

55. Giotto, The Flight into Egypt,
c1313
• Notice how the trees are the same size

56. Lorenzetti, The Presentation in the
Temple, c1342
• Notice how the tiles
get smaller

57. Masaccio, Trinity, 1427
• One of the first perspective pictures

58. 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

59. Side Vanishing Points 2

60. Where’s the best view point?
• 174cm above, 770cm away

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

62. Anamorphic art
• Holbein, The Ambassadors, 1533

63. Is there perspective in Chinese
paintings?
• Multiple viewpoints, Chen Chong Swee,
Snowscape, 1993
• Raphael, The School of Athens, 1511

64. What does a sphere look like?

65. What’s going on here?

66. Vermeer (1632—1675)
The Music Lesson (1662-5)
Royal Collection, London

67. 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

68. Counterintuitive Perspective
Compare The Procuress by van
Honthorst and Officer and Laughing
Girl by Vermeer
Many art historians accept that Vermeer
used a camera obscura (pinhole
camera)

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

70. 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

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

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

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

74. What is Special about
The Music Lesson?

75. 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

76. 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

77. 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?

78. 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?

79. Future work
• Gothic architecture
• Salsa dance

80. Gothic vaults

81. More vaults

82. Mathematics of Salsa dancing
• How to remember dance moves
• Leg work is easy, arm work is hard
• Construct a language to describe moves

83. 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?

84. 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

85. 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.

86. 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.

87. 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?

88. 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.

89. Have fun designing your own course!
• Good luck and thank you!