Leonardo Meeting, SETI Institute, Feb. 10, 2010

1. Leonardo Meeting, SETI Institute, Feb. 10, 2010 Carlo H. Séquin U.C. Berkeley • NaughtyKnotty Sculptures

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

9. Rapid Prototyping Model of the 24-Cell • Noticethe 3-foldpermutationof colorsMade on the Z-corp machine.

10. 3 Hamiltonian Cycles on 4D Cross Polytope

11. Hamiltonian Cycles on 4D Cross Polytope

12. PART AKnots as Constructive Building Blocks

13. Tetrahedral Trefoil Tangle (FDM)

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

15. Tetra Trefoil Tangle • Complex linking (two views)

16. Platonic Trefoil Tangles • Take a Platonic polyhedron made from triangles, • Add a trefoil knot on every face, • 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 by ProMetal (Ex One Co.) • Metal sintering and infiltration process

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

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

25. Knot 5.2

26. Knot 6.1

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

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

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

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

31. Metal Sculpture at SIGGRAPH 2006

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

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

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

35. Recursive Cubist Trefoil Knot

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

37. 2.5D Celtic Knots – Basic Step

38. Celtic Knot – Denser Configuration

39. Celtic Knot – Second Iteration

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

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

42. Sierpinski Trefoil Knot

43. Close-up of Sierpinski Trefoil Knot

44. 3rd Generation of Sierpinski Knot

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

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

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