Graphics Lunch, Feb. 5, 2009

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

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

3. NOT This:

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

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

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

7. 3 Hamiltonian Cycles on 4D Cross Polytope

8. Hamiltonian Cycles on 4D Cross Polytope

9. PART AKnots 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 • 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: Extrude Hone - ProMetal • Metal sintering and infiltration process

23. 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;explore several different methods… --> • Make aesthetically pleasing artifacts!

24. PART BWays 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 Replaces an “elbow” • Start with Hamiltonian path on cube edges and recurse ...

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 Thus our construction element should use a “more knotted thing”: e.g. an overhand knot: It is still just the un-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 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 !

34. 2.5D Celtic Knots – Basic Step

35. Celtic Knot – Denser Configuration

36. Celtic Knot – Second Iteration

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

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. 3rd 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 toFrank Smullin (1943 – 1983)

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

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

48. 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” 

49. Strands in the Granny-Knot-Lattice

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