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G4G9

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G4G9. Carlo H. Séquin. The Beauty of Knots. EECS Computer Science Division University of California, Berkeley. Classical Knot Tables. Flat (2.5D), uninspiring, lack of symmetry …. Trefoil Knot. Figure-8 Knot Bronze, Dec. 2007 Carlo S é quin. 2 nd Prize, AMS Exhibit 2009.

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G4G9

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  1. G4G9 Carlo H. Séquin The Beauty of Knots EECS Computer Science Division University of California, Berkeley

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

  3. Trefoil Knot

  4. Figure-8 KnotBronze, Dec. 2007Carlo Séquin 2nd Prize, AMS Exhibit 2009

  5. “The Beauty of Knots” • Make aesthetically pleasing artifacts! Undergraduate research group in 2009 • What is the most symmetrical configuration? • What is the most 3-dimensional configuration?

  6. Some Results Emphasizing Symmetry Knot 77 Knot 74

  7. Knot Transformations Bring out special, desirable qualities: • A graceful “evenly-spaced” curve:Minimize electrostatic repulsion potential on a flexible wire. • Tightest configuration:Pull tight a rope of fixed diameter without self-intersections. • The least “wiggly” curve:Minimize the arc-length integral of curvature squared. • The most 3D-filling configuration:Wrap knot around a sphere or a cylinder;Turn configuration inside out (point inversion);Play with wires, alu-foil, pipe cleaners!

  8. Knot 52

  9. Knot 61

  10. “Signature Knot” for G4G9 • Has to be a 9-crossing knot … -- but which one ?

  11. the same Knot ! It has 3-fold symmetry! “Signature Knot” for G4G9 … a 9-crossing knot: Knot 940

  12. Knot 940: “Chinese Button Knot” • It has interesting 3D properties !

  13. Knot 940: Chinese Button Knot

  14. Knot 940: Chinese Button Knot

  15. ChineseButton Knot(Knot 940)Bronze, Dec. 2007Carlo Séquincast & patina bySteve Reinmuth

  16. Knot 940 in Ribbon Form • Will be the subject of some hands-on “constructivist activities” on Fri./Sat. pm.

  17. From Simple Knots to Complicated Knots • “Hilbert Cube 512” – looks complicated … but it is not; -- just a simple, unknotted loop!

  18. Generating Complicated Knots Is there a procedure to make knots of arbitrary complexity…? • Perhaps by fusing simple knots together… • Perhaps by applying recursive techniques… • Start with: 2.5D - Celtic Knots

  19. 2.5D Celtic Knots – Basic Step

  20. Celtic Knot – Denser Configuration

  21. Celtic Knot – Second Iteration

  22. Another Approach: Knot-Fusion • Combine 3 trefoils into a 12-crossing knot

  23. Sierpinski Trefoil Knot

  24. Close-up of Sierpinski Trefoil Knot

  25. 3rd Generation of Sierpinski Knot

  26. Another Approach: Mesh-Infilling ... • Robert Fathauer, Bridges Conference, 2007 Map “the whole thing” into all meshes of similar shape

  27. 2.5D Recursive (Fractal) Knot • Robert Fathauer: “Recursive Trefoil Knot” Trefoil Recursion 3 views step

  28. Result after 2 more recursion steps Mark crossings over/under, form alternating knot Recursive Figure-8 Knot (4 crossings) Recursion step

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

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

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

  32. Recursive Figure-8 Knot • 9 loop iterations

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

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