1. Knotting Mathematics and Art University of Southern Florida, Nov.3, 2007 Naughty Knotty Sculptures Carlo H. Séquin U.C. Berkeley  Knotty problems in knot theory

2. Sculptures Made from Knots (1) 2004 - 2007: Knots as constructive building blocks.

3. Tetrahedral Trefoil Tangle (FDM)

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

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

6. Tetra Trefoil Tangle Complex linking (two views)

7. Octahedral Trefoil Tangle

8. Octahedral Trefoil Tangle (1) Simplest linking

9. 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. u Tetrahedron, Octahedron,... done !

10. Icosahedral Trefoil Tangle Simplest linking (type 1)

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

12. Arabic Icosahedron

13. Dodecahedral Pentafoil Cluster

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

16. Sculptures Made from Knots (2) Generate knots & increase their complexity in a structured, procedural way: I. Bottom-up assembly of knots II. Top-down mesh infilling III. Longitudinal knot splitting Make aesthetically pleasing artifacts For this conference I have been looking for sculptures where the whole piece is just a single knot and which also involve some “interesting” knots.

17. Outline I. Bottom-up assembly of knots II. Top-down mesh infilling III. Longitudinal knot splitting

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

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

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

21. Metal Sculpture at SIGGRAPH 2006

22. 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:

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

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

25. Recursive Cubist Trefoil Knot

26. 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: Perhaps knotting together crossing strands...

27. 2.5D Celtic Knots – Basic Step

28. Celtic Knot – Denser Configuration

29. Celtic Knot – Second Iteration

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

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

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

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

34. 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

35. 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”

36. Strands in the Granny-Knot-Lattice

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

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

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

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

41. A Nice Symmetrical Starting Knot Granny Knot with cross-connected ends 4-fold symmetric Knot 8 19

42. Recursion Step Placement of the 8 substitution knots

43. Establishing Connectivity Grow knots until they almost touch

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

45. Outline I. Bottom-up assembly of knots II. Top-down mesh infilling III. Longitudinal knot splitting

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

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

48. 2.5D Recursive (Fractal) Knot Robert Fathauer: “Recursive Trefoil Knot” Trefoil Recursion

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

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

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

52. Recursive Figure-8 Knot 9 loop iterations

53. Outline I. Bottom-up assembly of knots II. Top-down mesh infilling III. Longitudinal knot splitting

54. A Split Trefoil To open: Rotate around z-axis

55. Split Trefoil (side view, closed)

56. Split Trefoil (side view, open)

57. Another Split Trefoil How much “wiggle room” is there ?

58. Trefoil “Harmonica”

59. An Iterated Trefoil-Path of Trefoils

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

61. Split Moebius Trefoil (Séquin, 2003)

62. “Knot Divided” by Team Minnesota

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

64. 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 ?

65. Conclusions u Knots are mathematically intriguing and they are inspiring artistic elements. u They can be used as building blocks for sophisticated constellations. u They can be extended recursively to form much more complicated knots. u They can be split lengthwise to make interesting knots and tangles.

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