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Knotting Mathematics and Art University of Southern Florida, Nov.3, 2007

Knotting Mathematics and Art University of Southern Florida, Nov.3, 2007. Carlo H. Séquin U.C. Berkeley.  Knotty problems in knot theory. Naughty Knotty Sculptures. Sculptures Made from Knots (1). 2004 - 2007: Knots as constructive building blocks. Tetrahedral Trefoil Tangle (FDM).

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Knotting Mathematics and Art University of Southern Florida, Nov.3, 2007

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

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

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

  15. Sculptures Made from Knots (2) 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. • 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

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

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

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

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

  20. Metal Sculpture at SIGGRAPH 2006

  21. 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 !

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

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

  24. Recursive Cubist Trefoil Knot

  25. A Knot Theorist’s View Thus our assembly step should cause a more serious entanglement: Perhaps knotting together crossing strands . . . • This is just a compound-knot ! • It does not really lead to a complex knot !

  26. 2.5D Celtic Knots – Basic Step

  27. Celtic Knot – Denser Configuration

  28. Celtic Knot – Second Iteration

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

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

  31. INTERMEZZO:Homage toFrank Smullin (1943 – 1983)

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

  33. 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... ”

  34. 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” 

  35. Strands in the Granny-Knot-Lattice

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

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

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

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

  40. A Nice Symmetrical Starting Knot • Granny Knot with cross-connected ends 4-fold symmetric Knot 819

  41. Recursion Step • Placement of the 8 substitution knots

  42. Establishing Connectivity • Grow knots until they almost touch

  43. Work in Progress ... • Connectors added to close the knot

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

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

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

  47. 2.5D Recursive (Fractal) Knot Trefoil Recursion • Robert Fathauer: “Recursive Trefoil Knot”

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

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

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