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DeYoung Museum, June12, 2013

DeYoung Museum, June12, 2013. MATHEMATICAL TREASURE HUNTS. Tracking Twisted Toroids. Carlo H. Séquin University of California, Berkeley. What came first: Art or Mathematics ?. Question posed Nov. 16, 2006 by Dr. Ivan Sutherland “ father ” of computer graphics ( SKETCHPAD , 1963).

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DeYoung Museum, June12, 2013

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

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

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

  4. Regular, Geometric Art • Early art: Patterns on bones, pots, weavings... • Mathematics (geometry) to help make things fit:

  5. Geometry ! • 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 • Knife blades rotate through 360 degreesas it sweep once around the torus ring. 360°

  25. Slicing a Bagel . . .

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

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

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

  29. Generalization to 4-Link Torus • 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 • There is a second parameter: • 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 • Use a knife with b blades, • 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 ... • results in a(t, b)-torus link; • each component is a (t/g, b/g)-torus knot, • where g = GCD (t, b). b = 4, t = 2  two double loops.

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

  35. Anish Kapoor’s “Bean” in Chicago

  36. Keizo Ushio, 2004

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

  38. Changing Shapes of a Möbius Band • Using a “magic” surface material that can pass through itself. +180°(ccw), 0°, –180°, –540°(cw) Apparent twist (compared to a rotation-minimizing frame) Regular Homotopies

  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)

  42. 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 • Splitting Möbius Strips • Double-layer Möbius Strips • Escher’s Split Möbius Band

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

  50. Activity #2: Double Möbius Strips • 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! ). • Convince yourself that strips are separate by passing a pencil or small paper piece around the whole loop. • Separate (open-up) the two loops. • Put the configuration back together.

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