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Splitting Tori, Knots, and Moebius Bands: The Intriguing Geometry of Keizo Ushio

Explore the fascinating world of Keizo Ushio and his sculptures that involve splitting tori, knots, and Moebius bands. Discover the process of creating these intricate artworks and the unique geometric properties they exhibit.

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Splitting Tori, Knots, and Moebius Bands: The Intriguing Geometry of Keizo Ushio

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  1. BRIDGES, Banff, August 2005 Splitting Tori, Knots, and Moebius Bands Carlo H. Séquin U.C. Berkeley

  2. Homage a Keizo Ushio

  3. Performance Art at ISAMA’99 Keizo Ushio and his “OUSHI ZOKEI”

  4. The Making of “Oushi Zokei”

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

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

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

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

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

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

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

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

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

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

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

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

  17. Schematic of 2-Link Torus Small FDM (fused deposition model) 360°

  18. Generalize to 3-Link Torus • Use a 3-blade “knife”

  19. Generalize to 4-Link Torus • Use a 4-blade knife, square cross section

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

  21. Keizo Ushio’s Multi-Loops • If we change twist angle of the cutting knife, torus may not get split into separate rings. 180° 360°540°

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

  23. 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.

  24. II. Borromean Torus ? Another Challenge: • Can a torus be split in such a way that a Borromean link results ? • Can the geometry be chosen so that the three links can be moved to mutually orthogonal positions ?

  25. “Reverse Engineering” • Make a Borromean Link from Play-Dough • Smash the Link into a toroidal shape.

  26. Result: A Toroidal Braid • Three strands forming a circular braid

  27. Cut-Profiles around the Toroid

  28. Splitting a Torus into Borromean Rings • Make sure the loops can be moved apart.

  29. A First (Approximate) Model • Individual parts made on the FDM machine. • Remove support; try to assemble 2 parts.

  30. Assembled Borromean Torus With some fine-tuning, the parts can be made to fit.

  31. A Better Model • Made on a Zcorporation 3D-Printer. • Define the cuts rather than the solid parts.

  32. Separating the Three Loops • A little widening of the gaps was needed ...

  33. The Open Borromean Torus

  34. III. Focus on SPACE ! Splitting a Torus for the sake of the resulting SPACE !

  35. “Trefoil-Torso” by Nat Friedman • Nat Friedman:“The voids in sculptures may be as important as the material.”

  36. Detail of “Trefoil-Torso” • Nat Friedman:“The voids in sculptures may be as important as the material.”

  37. “Moebius Space” (Séquin, 2000)

  38. Keizo Ushio, 2004

  39. Keizo’s “Fake” Split (2005) One solid piece ! -- Color can fool the eye !

  40. Triply Twisted Moebius Space 540°

  41. Triply Twisted Moebius Space (2005)

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

  43. Splitting Moebius Bands Keizo Ushio 1990

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

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

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

  47. Splitting Knots • Splitting a Moebius band comprising 3 half-twists results in a trefoil knot.

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

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