1. Graphics Lunch, Feb. 5, 2009 Carlo H. Séquin

2. Graphics Lunch, Feb. 5, 2009 Naughty Knotty Sculptures Carlo H. Séquin U.C. Berkeley  Knotty problems in knot theory

3. NOT This:

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

5. Math-Art Connection When does a mathematical model become a piece of art ? Previous explorations: Minimal surfaces Hyperbolic tessellations 4-dimensional polytopes

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

7. 3 Hamiltonian Cycles on 4D Cross Polytope

8. Hamiltonian Cycles on 4D Cross Polytope

9. PART A Knots as Constructive Building Blocks

10. Tetrahedral Trefoil Tangle (FDM)

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

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

13. Tetra Trefoil Tangle Complex linking (two views)

14. Octahedral Trefoil Tangle

15. Octahedral Trefoil Tangle (1) Simplest linking

16. 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.

17. Icosahedral Trefoil Tangle Simplest linking (type 1)

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

19. Arabic Icosahedron

20. Dodecahedral Pentafoil Cluster

22. Realization: Extrude Hone - ProMetal Metal sintering and infiltration process

23. Sculptures Made from Knots Generate knots & increase their complexity in a structured, procedural way; explore several different methods… --> Make aesthetically pleasing artifacts! More recently, I have been looking for sculptures where the whole piece is just a single knot.

24. PART B Ways to Make Complicated Knots I.Bottom-up knot construction II.Fusing simple knots together III. Top-down mesh infilling IV. Longitudinal knot splitting

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

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

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

28. Metal Sculpture at SIGGRAPH 2006

29. 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:

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

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

32. Recursive Cubist Trefoil Knot

33. 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...

34. 2.5D Celtic Knots – Basic Step

35. Celtic Knot – Denser Configuration

36. Celtic Knot – Second Iteration

37. Recursive 9-Crossing Knot Is this really a 81-crossing knot ? 9 crossings

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

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

40. Sierpinski Trefoil Knot

41. Close-up of Sierpinski Trefoil Knot

42. 3 rd Generation of Sierpinski Knot

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

45. INTERMEZZO: Homage to Frank Smullin (1943 – 1983)

46. Frank Smullin (1943 – 1983) Tubular sculptures; Apple II program for calculating intersections.

47. Frank Smullin (Nashville, 1981): “ The Granny-knot has more artistic merits than the square knot because it is more 3D; its ends stick out in tetrahedral fashion... ” Square Knot Granny Knot

48. Granny Knot as a Building Block Four tetrahedral links, like a carbon atom... can be assembled into diamond-lattice...... leads to the “Granny-Knot-Lattice”  Smullin: “TetraGranny”

49. Strands in the Granny-Knot-Lattice

50. Granny-Knot-Lattice (Squin, 1981) Granny-Knot-Lattice (Séquin, 1981)

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. A Nice Symmetrical Starting Knot Granny Knot with cross-connected ends 4-fold symmetric Knot 8 19 (3,4) Torus Knot

55. Recursion Step Placement of the 8 substitution knots

56. Establishing Connectivity Grow knots until they almost touch

57. Work in Progress... Connectors added to close the knot

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

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

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

61. 2.5D Recursive (Fractal) Knot Robert Fathauer: “Recursive Trefoil Knot” Trefoil Recursion 3 views step

62. Recursion on a 7-crossing Knot Robert Fathauer, Bridges Conference, 2007... Map “the whole thing” into all meshes of similar shape

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

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

65. Recursive Figure-8 Knot 9 loop iterations

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

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

68. Split Trefoil (side view, closed)

69. Split Trefoil (side view, open)

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

71. Split Moebius Trefoil (Séquin, 2003)

72. “Knot Divided” by Team Minnesota

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

74. A More General Question u Take any knot made from an n-sided prismatic cord. u Split that cord lengthwise into n strands. u Cut the bundle of strands at one point and reconnect, after giving the bundle of n strands a twist equivalent of t strand-spacings (where n, t are mutually prime). u How complex is the resulting knot ?

75. PART C Space-filling Knots Can we pack knots so tightly that they fill all of 3D space ? Ian Stewart, Mathematical Recreations, Scientific American, Nov. 1995

76. 4 (convoluted) Trefoils Make a Cube Cubes stack up to fill space

77. A Simpler, More Elegant Solution Three congruent interlocking trefoils make a hexagonal prismatic block.

78. Isohedral Trefoil-Knot Tile of 3D Space

79. What We Would Really Like... u Stacking cubes or prisms in 3D space … is a “cheap” way to fill space with knots! u Neighboring knots should mutually link, so that the “fabric of space holds together.”

80. Toroidal Tile (Linking Unknots) Assembly of 5 Tiles The Basic Tile

81. Extensible Linkage of Toroidal Tiles

82. The Use of Figure-8 Knots Figure-8 knot can also have 4 lobes sticking out in tetrahedral directions.

83. Figure-8 Knots in Diamond Lattice Cell

84. A Denser Lattice of Figure-8 Knots

85. Dense Figure-8 Knot Lattice Model made with selective laser sintering.

86. My Conceptual 3D-CAD Tools

87. PART D Ribbed Sculptures Suspended in Generating Knots Example: Charles O. Perry’s Solstice In Tampa Florida Work with James Hamlin

89. Emulation of Perry’s Solstice Solstice: A ribbed surface in a (3,2) torus knot

90. Generating Principle Line swept along a (2,1) torus knot generates a Moebius band. Hyperbolic triangle swept along a (3,2) torus knot yields Solstice

91. A Variation of the Generating Knot Original (3,2) torus knot Modified (4,3) knot

92. A Variation of the Generating Knot Original (3,2) torus knot Modified (2,3) knot … and of the angle offset of the ribs around major circle

93. Conclusions u Knots are mathematically intriguing and they are also inspiring artistic elements. u They can be used as building blocks for sophisticated aesthetic assemblies. u They can be extended recursively to form much more complicated knots. u They can be split lengthwise to make interesting knots and tangles. u They can be used to tile 3D space. u They can be used as the generating framework for large-scale sculpture.

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

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

96. Figure-8 Knot Bronze, Dec. 2007 Carlo Séquin 2 nd Prize, AMS Exhibit 2009