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

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

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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