1 / 34

G4G9

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.

hao
Download Presentation

G4G9

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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”

More Related