1. Simons Center, July 30, 2012 Carlo H. Séquin University of California, Berkeley Artistic Geometry -- The Math Behind the Art

2. ART  MATH

3. What came first: Art or Mathematics ? u Question posed Nov. 16, 2006 by Dr. Ivan Sutherland “father” of computer graphics (SKETCHPAD, 1963).

4. My Conjecture... u Early art: Patterns on bones, pots, weavings... u Mathematics (geometry) to help make things fit:

5. Geometry ! u Descriptive Geometry – love since high school

6. Descriptive Geometry

7. 40 Years of Geometry and Design CCD TV Camera Soda Hall RISC 1 Computer Chip Octa-Gear (Cyberbuild)

8. More Recent Creations

9. Homage a Keizo Ushio

10. ISAMA, San Sebastian 1999 Keizo Ushio and his “OUSHI ZOKEI”

11. The Making of “Oushi Zokei”

12. The Making of “Oushi Zokei” (1) Fukusima, March’04 Transport, April’04

13. The Making of “Oushi Zokei” (2) Keizo’s studio, 04-16-04 Work starts, 04-30-04

14. The Making of “Oushi Zokei” (3) Drilling starts, 05-06-04 A cylinder, 05-07-04

15. The Making of “Oushi Zokei” (4) Shaping the torus with a water jet, May 2004

16. The Making of “Oushi Zokei” (5) A smooth torus, June 2004

17. The Making of “Oushi Zokei” (6) Drilling holes on spiral path, August 2004

18. The Making of “Oushi Zokei” (7) Drilling completed, August 30, 2004

19. The Making of “Oushi Zokei” (8) Rearranging the two parts, September 17, 2004

20. The Making of “Oushi Zokei” (9) Installation on foundation rock, October 2004

21. The Making of “Oushi Zokei” (10) Transportation, November 8, 2004

22. The Making of “Oushi Zokei” (11) Installation in Ono City, November 8, 2004

23. The Making of “Oushi Zokei” (12) Intriguing geometry – fine details !

24. Schematic Model of 2-Link Torus u Knife blades rotate through 360 degrees as it sweep once around the torus ring. 360°

25. Slicing a Bagel...

26. ... and Adding Cream Cheese u From George Hart’s web page: http://www.georgehart.com/bagel/bagel.html

27. Schematic Model of 2-Link Torus u 2 knife blades rotate through 360 degrees as they sweep once around the torus ring. 360°

28. Generalize this to 3-Link Torus u Use a 3-blade “knife” 360°

29. Generalization to 4-Link Torus u Use a 4-blade knife, square cross section

30. Generalize to 6-Link Torus 6 triangles forming a hexagonal cross section

31. Keizo Ushio’s Multi-Loops u There is a second parameter: u If we change twist angle of the cutting knife, torus may not get split into separate rings! 180° 360° 540°

32. Cutting with a Multi-Blade Knife u Use a knife with b blades, u Twist knife through t * 360° / b. b = 2, t = 1; b = 3, t = 1; b = 3, t = 2.

33. Cutting with a Multi-Blade Knife... u results in a (t, b)-torus link; u each component is a (t/g, b/g)-torus knot, u where g = GCD (t, b). b = 4, t = 2  two double loops.

34. “Moebius Space” (Séquin, 2000) ART: Focus on the cutting space ! Use “thick knife”.

35. Anish Kapoor’s “Bean” in Chicago

36. Keizo Ushio, 2004

37. It is a Möbius Band ! u A closed ribbon with a 180° flip; u A single-sided surface with a single edge:

38. Twisted Möbius Bands in Art Web Max Bill M.C. Escher M.C. Escher

39. Triply Twisted Möbius Space 540°

40. Triply Twisted Moebius Space (2005)

42. Splitting Other Stuff What if we started with something more intricate than a torus ?... and then split that shape...

43. Splitting Möbius Bands (not just tori) Keizo Ushio 1990

44. Splitting Möbius Bands M.C.Escher FDM-model, thin FDM-model, thick

45. Splits of 1.5-Twist Bands by Keizo Ushio (1994) Bondi, 2001

46. Splitting Knots … u Splitting a Möbius band comprising 3 half-twists results in a trefoil knot.

47. Splitting a Trefoil into 2 Strands u Trefoil with a rectangular cross section u Maintaining 3-fold symmetry makes this a single-sided Möbius band. u Split results in double-length strand.

48. Split Moebius Trefoil (Séquin, 2003)

49. “Infinite Duality” (Séquin 2003)

50. Final Model Thicker beams Wider gaps Less slope

51. “Knot Divided” by Team Minnesota

52. Splitting a Knotted Möbius Band

53. More Ways to Split a Trefoil u This trefoil seems to have no “twist.” u However, the Frenet frame undergoes about 270° of torsional rotation. u When the tube is split 4 ways it stays connected, (forming a single strand that is 4 times longer).

54. Twisted Prisms u An n-sided prismatic ribbon can be end-to-end connected in at least n different ways

55. Helaman Ferguson: Umbilic Torus

56. Splitting a Trefoil into 3 Strands u Trefoil with a triangular cross section (twist adjusted to close smoothly, maintain 3-fold symmetry). u 3-way split results in 3 separate intertwined trefoils. u Add a twist of ± 120° (break symmetry) to yield a single connected strand.

57. Another 3-Way Split Parts are different, but maintain 3-fold symmetry

58. Split into 3 Congruent Parts u Change the twist of the configuration! u Parts no longer have 3-fold symmetry

59. A Split Trefoil u To open: Rotate one half around central axis

60. Split Trefoil (side view, closed)

61. Split Trefoil (side view, open)

62. Triple-Strand Trefoil (closed)

63. Triple-Strand Trefoil (opening up)

64. Triple-Strand Trefoil (fully open)

65. A Special Kind of Toroidal Structures Collaboration with sculptor Brent Collins:  “Hyperbolic Hexagon” 1994  “Hyperbolic Hexagon II”, 1996  “Heptoroid”, 1998

66. Brent Collins: Hyperbolic Hexagon

67. Scherk’s 2nd Minimal Surface 2 planes the central core 4 planes bi-ped saddles 4-way saddles = “Scherk tower”

68. Scherk’s 2nd Minimal Surface Normal “biped” saddles Generalization to higher-order saddles (monkey saddle) “Scherk Tower”

69. V-art (1999) Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

70. Closing the Loop straight or twisted “Scherk Tower”“Scherk-Collins Toroids”

71. Sculpture Generator 1, GUI

72. Shapes from Sculpture Generator 1

73. The Finished Heptoroid u at Fermi Lab Art Gallery (1998).

74. On More Very Special Twisted Toroid u First make a “figure-8 tube” by merging the horizontal edges of the rectangular domain

75. Making a Figure-8 Klein Bottle u Add a 180° flip to the tube before the ends are merged.

76. Figure-8 Klein Bottle

77. What is a Klein Bottle ? u A single-sided surface u with no edges or punctures u with Euler characteristic: V – E + F = 0 u corresponding to: genus = 2 u Always self-intersecting in 3D

78. Classical “Inverted-Sock” Klein Bottle

79. How to Make a Klein Bottle (1) u First make a “tube” by merging the horizontal edges of the rectangular domain

80. How to Make a Klein Bottle (2) u Join tube ends with reversed order:

81. How to Make a Klein Bottle (3) u Close ends smoothly by “inverting one sock”

82. Limerick A mathematician named Klein thought Möbius bands are divine. Said he: "If you glue the edges of two, you'll get a weird bottle like mine."

83. 2 Möbius Bands Make a Klein Bottle KOJ = MR + ML

84. Fancy Klein Bottles of Type KOJ Cliff Stoll Klein bottles by Alan Bennet in the Science Museum in South Kensington, UK

85. Klein Knottles Based on KOJ Always an odd number of “turn-back mouths”!

86. A Gridded Model of Trefoil Knottle

87. Some More Klein Bottles...

88. Topology u Shape does not matter -- only connectivity. u Surfaces can be deformed continuously.

89. Smoothly Deforming Surfaces u Surface may pass through itself. u It cannot be cut or torn; it cannot change connectivity. u It must never form any sharp creases or points of infinitely sharp curvature. OK

90. Regular Homotopy u Two shapes are called regular homotopic, if they can be transformed into one another with a continuous, smooth deformation (with no kinks or singularities). u Such shapes are then said to be: in the same regular homotopy class.

91. Regular Homotopic Torus Eversion

92. Three Structurally Different Klein Bottles u All three are in different regular homotopy classes!

93. Conclusions u Knotted and twisted structures play an important role in many areas of physics and the life sciences. u They also make fascinating art-objects...

94. 2003: “Whirled White Web”

95. Inauguration Sutardja Dai Hall 2/27/09

96. Brent Collins and David Lynn

97. Sculpture Generator #2

98. Is It Math ? Is It Art ? u it is: “KNOT-ART”

99. QUESTIONS ? ?