1. Valencia, November 2006 Artistic Geometry Carlo H. Séquin U.C. Berkeley

2. Homage a Keizo Ushio

3. Performance Art at ISAMA’99 San Sebastian 1999 (also in 2007) 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 u Use a 3-blade “knife”

19. Generalize to 4-Link Torus u 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 u 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 u Use a knife with b blades, u Rotate through t * 360°/b. b = 2, t = 1; b = 3, t = 1; b = 3, t = 2.

23. 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.

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

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

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

27. Cut-Profiles around the Toroid

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

29. A First (Approximate) Model u Individual parts made on the FDM machine. u 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 u Made on a Zcorporation 3D-Printer. u Define the cuts rather than the solid parts.

32. Separating the Three Loops u 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 u Nat Friedman: “The voids in sculptures may be as important as the material.”

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

37. “Moebius Space” (Séquin, 2000)

39. Keizo Ushio, 2004

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

41. Triply Twisted Moebius Space 540°

42. Triply Twisted Moebius Space (2005)

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

45. Splitting Moebius Bands Keizo Ushio 1990

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

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

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

49. Splitting Knots u Splitting a Moebius band comprising 3 half-twists results in a trefoil knot.

50. Splitting 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).

51. Splitting a Trefoil into 3 Strands u Trefoil with a triangular cross section (Twist adjusted to close smoothly and maintain 3-fold symmetry). u Add a twist of ± 120° (break symmetry) to yield a single connected strand.

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

53. Split Moebius Trefoil (Séquin, 2003)

54. “Infinite Duality” (Séquin 2003)

55. Final Model Thicker beams Wider gaps Less slope

56. “Knot Divided” by Team Minnesota

57. What would happen if the original band were double-sided? u ==> True split into two knots ! u Probably tangled result u How tangled is it ? u How much can the 2 parts move ? u Explore these issues, and others...

58. Splitting the Knot into 3 Strands 3-deep stack

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

60. Split into 3 Congruent Parts u Change the twist of the configuration! u Parts no longer have C3 symmetry

61. Split Trefoil (closed)

62. Split Trefoil (open)

63. Triple-Strand Trefoil (closed)

64. Triple-Strand Trefoil (opening up)

65. Triple-Strand Trefoil (fully open)

66. How Much Wiggle Room ? u Take a simple trefoil knot u Split it lengthwise u See what happens...

67. Trefoil Stack

68. An Iterated Trefoil-Path of Trefoils

69. Linking Knots... Use knots as constructive building blocks !

70. Tetrahedral Trefoil Tangle (FDM)

71. Tetra Trefoil Tangles u Simple linking (1) -- Complex linking (2)

72. Tetra Trefoil Tangle (2) Complex linking -- two different views

73. Tetra Trefoil Tangle Complex linking (two views)

74. Octahedral Trefoil Tangle

75. Octahedral Trefoil Tangle (1) Simplest linking

76. Platonic Trefoil Tangles u Take a Platonic polyhedron made from triangles, u Add a trefoil knot on every face, u Link with neighboring knots across shared edges. u Tetrahedron, Octahedron,... done !

77. Arabic Icosahedron

78. Icosahedral Trefoil Tangle Simplest linking (type 1)

79. Icosahedral Trefoil Tangle (Type 3) u Doubly linked with each neighbor

80. Arabic Icosahedron, UniGrafix, 1983

81. Arabic Icosahedron

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

83. Space-filling Sculptures u Can we pack knots so tightly u that they fill all of 3D space ? u First: Review of Space-Filling Curves

84. The 2D Hilbert Curve (1891) A plane-filling Peano curve Fall 1983: CS Graduate Course: “Creative Geometric Modeling” Do This In 3 D !

85. Construction of the 2D Hilbert Curve 1 2 3

86. Construction of 3D Hilbert Curve

87. “Hilbert” Curve in 3D u Start with Hamiltonian Path on Cube Edges

88. “Hilbert_512_3D”

89. ProMetal Division of Ex One Company Headquarters in Irwin, Pennsylvania, USA.

90. Questions ?

91. Spares

92. V. Splitting Graphs u Take a graph with no loose ends u Split all edges of that graph u Reconnect them, so there are no junctions u Ideally, make this a single loop!

93. Splitting a Junction u For every one of N arms of a junction, there will be a passage thru the junction.

94. Flipping Double Links u To avoid breaking up into individual loops.

95. Splitting the Tetrahedron Edge-Graph 4 Loops 3 Loops 1 Loop

96. “Alter-Knot” by Bathsheba Grossman u Has some T-junctions

97. Turn this into a pure ribbon configuration! Some of the links had to be twisted.

98. “Alter-Alterknot” “Alter-Alterknot” Inspired by Bathsheba Grossman QUESTIONS ?