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URAP, September 16, 2013

URAP, September 16, 2013. The Beauty of Knots. Carlo H. Séquin University of California, Berkeley. My Background: Geometry !. Descriptive Geometry – love since high school. Descriptive Geometry. 40 Years of Geometry and Design. CCD TV Camera Soda Hall.

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URAP, September 16, 2013

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  1. URAP, September 16, 2013 The Beauty of Knots Carlo H. Séquin University of California, Berkeley

  2. My Background: Geometry ! • Descriptive Geometry – love since high school

  3. Descriptive Geometry

  4. 40 Years of Geometry and Design CCD TV Camera Soda Hall RISC 1 Computer Chip Octa-Gear (Cyberbuild)

  5. More Recent Creations

  6. Frank Smullin (1943 – 1983) • Tubular sculptures; • Apple II program for • calculating intersections.

  7. Frank Smullin: • “ The Granny knot has more artistic merits than the square knot because it is more 3D;its ends stick out in tetrahedral fashion... ” Square Knot Granny Knot

  8. Granny Knot as a Building Block • 4 tetrahedral links ... • like a carbon atom ... • can be assembled intoa diamond-lattice ... ... leads to the “Granny-Knot-Lattice” 

  9. Granny Knot Lattice (1981)

  10. The Strands in the G.K.L.

  11. Capturing Geometry Procedurally Collaboration with sculptor Brent Collins: • “Hyperbolic Hexagon” 1994 • “Hyperbolic Hexagon II”, 1996 • “Heptoroid”, 1998

  12. The Process: (For Scherk-Collins Toroids) InspirationalModel GenerativeParadigm ComputerProgram Many NewModels Insight,Analysis Math,Geometry Selection,Design

  13. Brent Collins: Hyperbolic Hexagon

  14. Scherk’s 2nd Minimal Surface 2 planes: the central core 4 planes:bi-ped saddles 4-way saddles = “Scherk tower”

  15. Scherk’s 2nd Minimal Surface Normal “biped” saddles Generalization to higher-order saddles(monkey saddle) “Scherk Tower”

  16. V-art(1999) VirtualGlassScherkTowerwith MonkeySaddles(Radiance 40 hours) Jane Yen

  17. Closing the Loop straight or twisted “Scherk Tower” “Scherk-Collins Toroids”

  18. Sculpture Generator 1, GUI

  19. Shapes from Sculpture Generator 1

  20. Some of the Parameters in “SG1”

  21. The Finished Heptoroid • at Fermi Lab Art Gallery (1998).

  22. 2003: “Whirled White Web”

  23. Brent Collins and David Lynn

  24. Inauguration Sutardja Dai Hall 2/27/09

  25. Details of Internal Representation • Boundary Representations • Meshes of small triangles defining surface

  26. Base Geometry: One “Scherk Story” • Taylored hyperbolas, hugging a circle • Hyperbolic Slices  Triangle Strips

  27. The Basic Saddle Element with surface normals • precomputed -- then warped into toroid

  28. Shape Generation: • by stacking this basic hyperbolic element, • twisting that stack along z-axis, • bending (warping) it into an arch or loop.

  29. Knot Representations • Knot tables ! • A particular realization of an individual knotis just a closed space curve in 3D space. • It can be represented as a sequence of vertices: V0 (x,y,z); V1 (x,y,z) … • Connected with a poly-line for visualization.

  30. A Simple Tool to Display Knots • http://www.cs.berkeley.edu/~sequin/X/Knot-View/ B-Splines with their corresponding control-polygons

  31. Knot Representation 10.0 -2.0 4.0 -6.732 7.66 -4.0 -6.732 -7.66 4.0 10.0 2.0 -4.0 -3.268 9.66 4.0 -3.268 -9.66 -4.0 • Control Polygon of Trefoil Knot: Then just drag this text file onto “KnotView-3D.exe”

  32. Turning Knots into Sculptures • Define a cross-section and sweep it along the given 3D knot curve.

  33. Brent Collins’ Pax Mundi1997: wood, 30”diam. 2006: Commission from H&R Block, Kansas City to make a 70”diameter version in bronze. My task: to define the master geometry. CAD tools played important role.

  34. How to Model Pax Mundi ... • Already addressed that question in 1998: • Pax Mundicould not be done withSculpture Generator I • Needed a more general program ! • Used the Berkeley SLIDE environment. • First: Needed to find the basic paradigm   

  35. Sculptures by Naum Gabo Pathway on a sphere: Edge of surface is like seam of tennis- or base-ball;  2-period Gabo curve.

  36. 2-period “Gabo Curve” • Approximation with quartic B-splinewith 8 control points per period,but only 3 DOF are used (symmetry!).

  37. 4-period “Gabo Curve” Same construction as for as for 2-period curve

  38. Pax Mundi Revisited • Can be seen as:Amplitude modulated, 4-period Gabo curve

  39. SLIDE-GUI for “Pax Mundi” Shapes Good combination of interactive 3D graphicsand parameterizable procedural constructs.

  40. 2-period Gabo Sculpture Tennis ball – or baseball – seam used as sweep curve.

  41. Viae Globi Family (Roads on a Sphere) 2 3 4 5 periods

  42. Via Globi 5 (Virtual Wood) Wilmin Martono

  43. Sweep Curve Generator: Gabo Curves as B-splines Cross Section Fine Tuner: Paramererized shapes Sweep / Twist Controller Modularity of Gabo Sweep Generator

  44. How do we orient, move, scale, morph ...the cross section along the sweep path ? Sweep / Twist Control Natural orientationwith Frenet frame Torsion Minimization:Azimuth: tangential / normal 900° of twistadded.

  45. Extension: Free-form Curve on a Sphere Spherical Spline Path Editor (Jane Yen) Smooth interpolating curve through sparse data points

  46. Many Different Viae Globi Models

  47. Paradigm Extension:Sweep Path is no longer confined to a sphere! Music of the Spheres (Brent Collins)

  48. Allows Knotted Sweep Paths Chinese Button Knot

  49. Really Free-form 3D Space Curves Figure-8 knot

  50. The Process: Example: Pax Mundi Sweep curve on a sphere Via Globi Framework In Slide Wood Pax Mundi Bronze Pax Mundi InspirationalModel GenerativeParadigm ComputerProgram Many NewModels Insight,Analysis Math,Geometry Selection,Design

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