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

2. My Background: Geometry ! u 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) u Tubular sculptures; u Apple II program for u calculating intersections.

7. Frank Smullin: u “ 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 u 4 tetrahedral links... u like a carbon atom... u can be assembled into a 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 ) Inspirational Model Generative Paradigm Computer Program Many New Models 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) Virtual Glass Scherk Tower with Monkey Saddles (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 u 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 u Boundary Representations u Meshes of small triangles defining surface

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

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

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

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

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

31. Knot Representation u Control Polygon of Trefoil Knot: 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 Then just drag this text file onto “KnotView-3D.exe”

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

33. Brent Collins’ Pax Mundi 1997: 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... u Already addressed that question in 1998: u Pax Mundi could not be done with Sculpture Generator I u Needed a more general program ! u Used the Berkeley SLIDE environment. u 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” u Approximation with quartic B-spline with 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 u Can be seen as: Amplitude modulated, 4-period Gabo curve

39. SLIDE-GUI for “Pax Mundi” Shapes Good combination of interactive 3D graphics and 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) Viae Globi Family (Roads on a Sphere) 2 3 4 5 periods

42. Via Globi 5 (Virtual Wood) Wilmin Martono

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

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

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 Wood Pax Mundi Sweep curve on a sphere Via Globi Framework In Slide Bronze Pax Mundi Inspirational Model Generative Paradigm Computer Program Many New Models Insight, Analysis Math, Geometry Selection, Design