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Simons Center, July 30, 2012

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

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

  2. ART  MATH

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

  4. My Conjecture ... • 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 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. Generalize to 6-Link Torus 6 triangles forming a hexagonal cross section

  31. Keizo Ushio’s Multi-Loops • 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. 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)

  41. 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 Bandsby Keizo Ushio (1994) Bondi, 2001

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

  47. Splitting a Trefoil into 2 Strands • Trefoil with a rectangular cross section • Maintaining 3-fold symmetry makes this a single-sided Möbius band. • 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

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