1. Leonardo Meeting, San Francisco, May 12, 2008
Florida 1999
Leonardo Meeting, San Francisco, May 12, 2008
Naughty Knotty Sculptures
I hope that when you heard about this talk – you did not come with the wrong expectations ...
Nothing naughty – but may raise some knotty problems in knot theory.
Carlo H. Séquin
EECS, Computer Science
University of CA, Berkeley

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

3. Since 1994: Aesthetic Designs …
Florida 1999
What is the role of the computer in:
aesthetic optimization,
the creative process ?
Do computers have a role in the design of artistic objects ?
Do they have a role in conceptual, creative, activities ?

4. Collaboration with Brent Collins
Florida 1999
Collaboration with Brent Collins
For whom I designed certain shapes on the computer which he then built in wood.
“Hyperbolic Hexagon II”

5. “Sculpture Generator I ” GUI

6. When does a mathematical model become a piece of art ?
Math-Art Connection
When does a mathematical model become a piece of art ?

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

8. Hamiltonian Cycles on 4D Cross Polytope

9. Sculptures Made from Knots
: 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. Complex linking (two views)
Tetra Trefoil Tangle
Complex linking (two views)

13. Platonic Trefoil Tangles
Take a Platonic polyhedron made from triangles,
Add a trefoil knot on every face,
Link with neighboring knots across shared edges.

14. Icosahedral Trefoil Tangle
Simplest linking (type 1)

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

16. Arabic Icosahedron

17. Dodecahedral Pentafoil Cluster

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

20. Sculptures Made from Knots
More recently, I have been looking for sculptures
where the whole piece is just a single knot.
Generate knots & increase their complexity in a structured, procedural way
Make aesthetically pleasing artifacts

21. Many Different Ways . . . I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting

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

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

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

25. Metal Sculpture at SIGGRAPH 2006
Florida 1999
Here is a solution that eliminates all these flaws and executed with a metal sintering process which builds this sculpture layer by layer on a 3D printer.

26. It is still just the un-knot !
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:

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

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

29. Recursive Cubist Trefoil Knot

30. 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 . . .

31. 2.5D Celtic Knots – Basic Step

32. Celtic Knot – Denser Configuration

33. Celtic Knot – Second Iteration

34. Recursive 9-Crossing Knot
Florida 1999
Recursive 9-Crossing Knot
9 crossings
I only figured out yesterday which one it is.
I eave it to you as a brainteaser.
Is this really a 81-crossing knot ?

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

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

38. 3rd Generation of Trefoil-Sierpinsky

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

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

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

43. 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

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

45. Strands in the Granny-Knot-Lattice

46. Granny-Knot-Lattice (Séquin, 1981)

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

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

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

50. A Nice Symmetrical Starting Knot
Granny Knot with cross-connected ends
4-fold symmetric Knot 819
(3,4) Torus Knot

51. Placement of the 8 substitution knots
Recursion Step
Placement of the 8 substitution knots

52. Establishing Connectivity
Grow knots until they almost touch

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

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)
Mark crossings over/under to form alternating knot
Result after 2 more recursion steps
Recursion step

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

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

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

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

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

61. Recursive Figure-8 Knot
9 loop iterations

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

63. Split Trefoil (side view, closed)

64. Split Trefoil (side view, open)

65. “Knot Divided” by Team Minnesota

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

67. Florida 1999
Is It Math ? Is It Art ?
it is: “KNOT-ART”