1. IEEE Chapter, Berkeley, April 22, 2010 Naughty Knotty Sculptures Carlo H. Séquin CS Division, U.C. Berkeley

2. NOT This:

3. But This: Sculptures Made from Knots Knots as constructive sculptural building blocks.

4. Technical Designs … CCD Camera, Bell Labs, 1973 Soda Hall, Berkeley, 1994 RISC chip, Berkeley, 1981 “Octa-Gear”, Berkeley, 2000

5. Since 1994: Aesthetic Designs … What is the role of the computer in: aesthetic optimization, the creative process ?

6. Collaboration with Brent Collins “Hyperbolic Hexagon II”

7. “Sculpture Generator I ” GUI

8. Wesleyan College, Math Lounge

9. Hyperbolic Hexagon II in a Movie

10. Dr. Manhattan’s Apartment 15 seconds of fame in The Watchmen

11. Hyperbolic Hexagon II in CITRIS

12. Math  Art Connection When does a mathematical model become a piece of art ?

13. Rapid Prototyping Model of the 24-Cell Notice the 3-fold permutation of colors Made on the Z-corp machine.

14. 3 Hamiltonian Cycles on 4D Cross Polytope

15. Hamiltonian Cycles on 4D Cross Polytope

16. PART A Knots as Constructive Building Blocks

17. Tetrahedral Trefoil Tangle (FDM)

18. Tetra Trefoil Tangles Simple linking (1) -- Complex linking (2) {over-over-under-under} {over-under-over-under}

19. Tetra Trefoil Tangle Complex linking (two views)

20. 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.

21. Icosahedral Trefoil Tangle Simplest linking (type 1)

22. Icosahedral Trefoil Tangle (type 3) Doubly linked with each neighbor

23. Arabic Icosahedron

24. Dodecahedral Pentafoil Cluster

26. Realization by ProMetal (Ex One Co.) Metal sintering and infiltration process

27. “The Beauty of Knots” Undergraduate research group in 2009 More recently, I have been looking for sculptures where the whole piece is just a single knot. Make aesthetically pleasing artifacts!

28. Classical Knot Tables Flat (2.5D), uninspiring, lack of symmetry …

29. Knot 5 2

30. Knot 6 1

31. PART B Computer-Generated Knots I.Bottom-up knot construction II.Fusing simple knots together III. Top-down mesh infilling IV. Longitudinal knot splitting Generate knots & increase their complexity in a structured, procedural way. Explore several different methods…

32. The 2D Hilbert Curve (1891) A plane-filling Peano curve Do This In 3 D !

33. “Hilbert” Curve in 3D (1999) Start with Hamiltonian path on cube edges and recurse... Replaces an “elbow”

34. Jane Yen: “Hilbert Radiator Pipe” (2000) Flaws ( from a sculptor’s. point of view ): 4 coplanar segments Not a closed loop Broken symmetry

35. Metal Sculpture at SIGGRAPH 2006

36. A Knot Theorist’s View It is still just the un-knot ! Thus our construction element should use a “more knotted thing”: e.g. an overhand knot:

37. Recursion Step Replace every 90° turn with a knotted elbow.

38. Also: Start from a True Knot e.g., a “cubist” trefoil knot.

39. Recursive Cubist Trefoil Knot

40. A Knot Theorist’s View This is just a compound-knot ! It does not really lead to a complex knot ! Thus our assembly step should cause a more serious entanglement: adjacent knots should entangle one another, or crossing strands should be knotted together...

41. 2.5D Celtic Knots – Basic Step

42. Celtic Knot – Denser Configuration

43. Celtic Knot – Second Iteration

44. Outline I.Bottom-up knot construction II.Fusing simple knots together III. Top-down mesh infilling IV. Longitudinal knot splitting

45. Knot-Fusion Combine 3 trefoils into a 9-crossing knot

46. Sierpinski Trefoil Knot

47. Close-up of Sierpinski Trefoil Knot

48. 3 rd Generation of Sierpinski Knot

50. From Paintings to Sculptures Do something like this in 3D ! Perhaps using two knotted strands (like your shoe laces).

51. A “Knotty” “3D” Recursion Step Use the Granny knot as a replacement element where two strands cross...

52. Next Recursion Step Substitute the 8 crossings with 8 Granny-knots

53. One More Recursion Step Now use eight of these composite elements; connect; beautify. Too much complexity !

54. Outline I.Bottom-up knot construction II.Fusing simple knots together III. Top-down mesh infilling IV. Longitudinal knot splitting

55. Recursive Figure-8 Knot (4 crossings) Recursion step Mark crossings over/under to form alternating knot Result after 2 more recursion steps

56. Recursive Figure-8 Knot Scale the stroke-width proportional to recursive reduction

57. 2.5D Recursive (Fractal) Knot Robert Fathauer: “Recursive Trefoil Knot” Trefoil Recursion 3 views step

58. From 2D Drawings to 3D Sculpture Too flat ! Switch plane orientations

59. Recursive Figure-8 Knot 3D Maquette emerging from FDM machine

60. Recursive Figure-8 Knot 9 loop iterations

61. Outline I.Bottom-up knot construction II.Fusing simple knots together III. Top-down mesh infilling IV. Longitudinal knot splitting

62. Splitting Moebius Bands Litho by FDM-model FDM-model M.C.Escher thin, colored thick

63. A Split Trefoil To open: Rotate one half around z-axis

64. Split Trefoil (side view, closed)

65. Split Trefoil (side view, open)

66. Split Moebius Trefoil (Séquin, 2003)

67. “Knot Divided” by Team Minnesota

68. Day 1

69. Day 2

73. Day 3

75. Day 4

76. Day 5

78. Knotty Problem How many crossings does this Not-Divided Knot have ?

79. Gathering for Gardner #9 Atlanta, March 24-28, 2010

80. Pagoda by Akio Hizume (2008) John Conway

81. The Triamond Lattice (2008) Chaim Goodman Strauss, John Conway

82. Knot 9 40 in Ribbon Form This was one of my “Knot-Flowers” designs

83. The 3 Pieces

84. A First Faulty Assembly: Knot 3 1

85. Tighter Coiling: Proper Topology

86. Knot 9 40 Sculpture

87. Knot 9 40 = Chinese Button KnotKnot 9 40 = Chinese Button Knot

88. Parts of a Different Sculpture

89. Assembly in Progress

90. The Crucial Center Piece

91. The Finished Trefoil Flower

92. Trefoil Flower Hung In Place

93. Hilbert Cube 512 (Séquin, 2006) Inspiration for a sculpture by Chaim Goodman-Strauss

94. 2-inch Stainless Steel Pipes

95. 32 Prefabricated Segments

96. The Whole Collection of Pieces

97. Assembly of Hilbert Cube 512

98. Artist and Engineer

99. Sponsor Tom Rodgers and Artists

100. Is It Math ? Is It Art ? it is: “KNOT-ART”

101. Figure-8 Knot Bronze, Dec. 2007 Carlo Séquin

102. Chinese Button Knot (Knot 9 40 ) Bronze, Dec. 2007 Carlo Séquin cast & patina by Steve Reinmuth