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Knotty Problems in Mathematics and Art

Explore the intersection of knot theory, mathematics, and art through sculptures made from knots, tackling complex linking patterns. Discover the beauty in constructing intricate knots with a structured and procedural approach. Delve into innovative techniques like bottom-up assembly, top-down mesh infilling, and longitudinal knot splitting to create aesthetically pleasing artifacts.

margaretpennington
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Knotty Problems in Mathematics and Art

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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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