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SIAM Conf. on Math for Industry, Oct. 10, 2009

SIAM Conf. on Math for Industry, Oct. 10, 2009. Carlo H. Séquin U.C. Berkeley. Modeling Knots for Aesthetics and Simulations. Modeling, Analysis, Design …. Knots in Clothing . Knotted Appliances . Garden hose Power cable. Intricate Knots in the Realm of . . .

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SIAM Conf. on Math for Industry, Oct. 10, 2009

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  1. SIAM Conf. on Math for Industry, Oct. 10, 2009 Carlo H. Séquin U.C. Berkeley • Modeling Knots for Aesthetics and Simulations Modeling, Analysis, Design …

  2. Knots in Clothing

  3. Knotted Appliances • Garden hose Power cable

  4. Intricate Knots in the Realm of . . . • Boats Horses

  5. Knots in Art • Macrame Sculpture

  6. Knotted Plants • Kelp Lianas

  7. Knotted Building Blocks of Life • Knotted DNA Model of the most complex knotted protein (MIT 2006)

  8. Mathematicians’ Knots unknot • Closed, non-self-intersecting curves in 3D space 0 3 4 6 Tabulated by their crossing-number : = The minimal number of crossings visible after any deformation and projection

  9. Various Unknots

  10. 3D Hilbert Curve (Séquin 2006)

  11. Pax Mundi II (2007) • Brent Collins, Steve Reinmuth, Carlo Séquin

  12. The Simplest Real Knot: The Trefoil • José de Rivera, Construction #35 M. C. Escher, Knots (1965)

  13. Complex, Symmetrical Knots

  14. Tight “Braided” Knots

  15. Composite Knots • Knots can be “opened” at their periphery and then connected to each other.

  16. Links and Linked Knots • A link: comprises a set of loops • – possibly knotted and tangled together.

  17. Two Linked Tori: Link 221 John Robinson, Bonds of Friendship (1979)

  18. Borromean Rings: Link 632 John Robinson

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

  20. Tetrahedral Trefoil Tangle (FDM)

  21. A Loose Tangle of Trefoils

  22. Dodecahedral Pentafoil Cluster

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

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

  25. Split Trefoil (side view, closed)

  26. Split Trefoil (side view, open)

  27. Splitting Moebius Bands • Litho by FDM-model FDM-modelM.C.Escher thin, colored thick

  28. Split Moebius Trefoil (Séquin, 2003)

  29. “Knot Divided” Breckenridge, 2005

  30. Knotty Problem • How many crossings • does this “Not-Divided” Knot have ?

  31. 2.5D Celtic Knots – Basic Step

  32. Celtic Knot – Denser Configuration

  33. Celtic Knot – Second Iteration

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

  35. Knot Classification • What kind of knot is this ? • Can you just look it up in the knot tables ? • How do you find a projection that yields the minimum number of crossings ? • There is still no completely safe method to assure that two knots are the same.

  36. Project: “Beauty of Knots” • Find maximal symmetry in 3D for simple knots. Knot 41 and Knot 61

  37. Computer Representation of Knots String of piecewise-linear line segments. • Spline representation via its control polygon. But . . .

  38. Is the Control Polygon Representative? You may construct a nice knotted control polygon,and then find that the spline curve it defines is not knotted at all ! • A Problem:

  39. Unknot With Knotted Control-Polygon • Composite of two cubic Bézier curves

  40. Highly Knotted Control-Polygons • Use the previous configuration as a building block. • Cut open lower left joint between the 2 Bézier segments. • Small changes will keep the control polygons knotted. • Assemble several such constructs in a cyclic compound.

  41. Highly Knotted Control-Polygons • The Result: • Control polygon has 12 crossings. • Compound Bézier curve is still the unknot!

  42. An Intriguing Question: First guess: Probably NOT Variation-diminishing property of Bézier curves implies that a spline cannot “wiggle” more than its control polygon. • Can an un-knotted control polygon • produce a knotted spline curve ?

  43. Cubic Bézier and Its Control Polygon Two “entangled” curves With “non-entangled” control polygons Convex hull of control polygon Region where curve is “outside” of control polygon Cubic Bézier curve

  44. Two “Entangled” Bezier Segments “in 3D” • NOTE: The 2 control polygons are NOT entangled!

  45. The Building Block Two “entangled” curves With “non-entangled” control polygons

  46. Combining 4 such Entangled Units • Use several units …

  47. Control Polygons Are NOT Entangled … • Use several units …

  48. Can Be Reduced to the Chords

  49. This Is NOT a Knot !

  50. But This Is a Knot ! Knot 72

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