1. DeYoung Museum, June12, 2013 Carlo H. Séquin University of California, Berkeley Tracking Twisted Toroids MATHEMATICAL TREASURE HUNTS

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

3. Early “Free-Form” Art Cave paintings, LascauxVenus von Willendorf

4. Regular, Geometric Art 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 (for CS) 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. Generalization to 6-Link Torus 6 triangles forming a hexagonal cross section

31. Keizo Ushio’s Multi-Loop Cuts 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. +180°(ccw), 0°, –180°, –540°(cw) Apparent twist (compared to a rotation-minimizing frame) Changing Shapes of a Möbius Band Regular Homotopies u Using a “magic” surface material that can pass through itself.

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

40. Triply Twisted Möbius Space 540°

41. Triply Twisted Moebius Space (2005)

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

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

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

46. Splitting a Band with a Twist of 540° by Keizo Ushio (1994) Bondi, 2001

47. Another Way to Split the Möbius Band Metal band available from Valett Design: conrad@valett.de

48. SOME HANDS-ON ACTIVITIES 1.Splitting Möbius Strips 2.Double-layer Möbius Strips 3.Escher’s Split Möbius Band

49. Activity #1: Möbius Strips For people who have not previously played with physical Möbius strips. u Take an 11” long white paper strip; bend it into a loop; u But before joining the end, flip one end an odd number of times: +/– 180°or 540°; u Compare results among students: How many different bands do you find? u Take a marker pen and draw a line ¼” off from one of the edges... Continue the line until it closes (What happens?) u Cut the strip lengthwise down the middle... (What happens? -- Discuss with neighbors!)

50. Activity #2: Double Möbius Strips u Take TWO 11” long, 2-color paper strips; put them on top of each other so touching colors match; bend sandwich into a loop; join after 1 or 3 flips ( tape the two layers individually! ). u Convince yourself that strips are separate by passing a pencil or small paper piece around the whole loop. u Separate (open-up) the two loops. u Put the configuration back together.

51. Activity #3: Escher’s Split Möbius Band u Take TWO 11”-long, 2-color paper strips; tape them together into a 22”-long paper strip (match color). u Try to form this shape inspired by MC Escher’s drawing: u After you have succeeded, can you reconfigure your model into something that looks like picture #3 ?

52. MUSEUM or WEB ACTIVITIES 1.Find pictures or sculptures of twisted toroids. 2.Find earliest depiction of a Möbius Band.

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

54. Helaman Ferguson: Umbilic Torus

55. 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.

56. Symmetrical 3-Way Split Parts are different, but maintain 3-fold symmetry

57. Split into 3 Congruent Parts u Change the twist of the configuration! u Parts no longer have C3 symmetry, but are congruent.

58. 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).

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

60. The Genus of an Object u Number of tunnels through a solid blob. u Number of handles glued onto a sphere. u Number of cuts needed to break all loops, but still keep object hanging together.

61. Objects with Different Genus g=1 g=2 g=3 g=4 g=5

62. ACTIVITIES related to GENUS 1. Find museum artifacts of genus 1, 2, 3 …. If you cannot find physical artifacts, pictures of appropriate objects are OK too. 2. Determine the genus of a select sculpture. 3. Find a highly complex object of genus 0.

63. Twisted “Chains”

64. “Millennium Arch” (Hole-Saddle Toroid)

65. Brent Collins: Hyperbolic Hexagon

66. A Special Kind of Toroidal Structures Collaboration with sculptor Brent Collins:  “Hyperbolic Hexagon” (1994)  “Hyperbolic Hexagon II” (1996)  “Heptoroid” (1998) = = > “Scherk-towers” wound up into a loop.

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. Sculpture Generator #2

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

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

78. Figure-8 Klein Bottle

79. 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

80. Classical “Inverted-Sock” Klein Bottle

81. The Two Klein Bottles Side-by-Side u Both are composed from two Möbius bands !

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

83. Limerick Mathematicians try hard to floor us with a non-orientable torus. The bottle of Klein, they say, is divine. But it is so exceedingly porus! by Cliff Stoll

84. ACTIVITIES with Twisted Toroids 1. Twisted prismatic toroids. 2. Making a figure-8 Klein bottle.

85. Activity #1: Twisted Prismatic Toroids u Mark one face of the square-profile foam-rubber prism with little patches of masking tape; u Bend the foam-rubber prism into a loop; (or combine two sticks and make a trefoil knot); u Twist the prism, join the ends; fix with tape:  Avoid matching the 2 ends of the marked face to obtain a true Möbius prism. u With your finger, continue to trace the marked face until it closes back to itself (perhaps add tape patches) (How many passes through the loop does it make?) (Have all the prism faces been marked?) u Discuss with neighbors!

86. Activity #2: Figure-8 Klein Bottle u Take a 2.8” x 11” 2-colored paper strip; u Length-wise crease down the middle, and ¼ width; u Give the whole strip a zig-zag Z-shaped profile (assume that the ends that touch the middle crease are connected through the middle to form a figure-8 profile) ; u Connect the ends of the figure-8 tube after a 180°flip. u Draw a longitudinal line with a marker pen. u Why is this a Klein bottle? -- Discuss with neighbors! PROFILE:

87. QUESTIONS ? ?