1. AAAS 2001, San Francisco Procedurally Defined Geometrical Sculptures Carlo H. Séquin University of California, Berkeley Brent Collins Gower, Missouri

2. My Professional Focus Computer-Aided Design u Design useful and beautiful objects with the help of computers. u Develop (interactive) computer programs to make these tasks easier.

3. Computer-Aided Design I : Integrated Circuits: “RISC I” chip (1981)

4. Computer-Aided Design II : Mathematical Models “Granny Knot” Lattice Berkeley UniGrafix (1982)

5. Computer-Aided Design III : Buildings Soda Hall, CS Dept. Berkeley (1992)

6. Computer-Aided Design IV : Mechanical Parts Design (1985) Realization (FDM) (2000) Octahedral Gear

7. Computer-Aided Design V : Abstract Sculpture (virtual) (Since 1995)

8. Computer-Aided Design V : Abstract Sculpture (virtual) Scherk-Collins Tower

9. Computer-Aided Design V : Abstract Sculpture (virtual) Doubly-looped Scherk-Collins saddle-chain

10. Computer-Aided Design V : Abstract Sculpture (real) “Bonds of Friendship” (2001) Fabricated by: Fused Deposition Modeling Currently in S.F.: at Gallery 650, Delancy/Brannan

11. Roots of My Passion for Sculpture My love for geometry and abstract sculpture emerged long long before I learned to play with computers. Thanks to: Alexander Calder, Naum Gabo, Max Bill, M.C. Escher, Frank Smullin,...

12. Leonardo -- Special Issue On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art With an Artist’s Statement by Brent Collins George K. Francis with Brent Collins

13. Brent Collins: Early Sculptures All photos by Phillip Geller

14. My Fascination with... Brent Collins’ Abstract Geometric Art u Beautiful symmetries u Graceful balance of the saddle surfaces u Superb craftsmanship u Intriguing run of the edges u What type of knot is formed ? u Mystery: one-sided or two-sided ? ==> Focus on “Chains of Saddles”

15. Brent Collins: Stacked Saddles

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

17. “Hyperbolic Hexagon” by B. Collins u 6 saddles in a ring u 6 holes passing through symmetry plane at ±45º u “wound up” 6-story Scherk tower u What would happen, l if we added more stories ? l or introduced a twist before closing the ring ?

18. Closing the Loop straight or twisted

19. Collins - Séquin Collaboration u Discuss ideas on the phone u Exchange sketches u Vary the topological parameters u But how do you know whether it is beautiful ? Need visual feedback. u Making models from paper strips is not good enough.

20. Brent Collins’ Prototyping Process Armature for the "Hyperbolic Heptagon" Mockup for the "Saddle Trefoil" Time-consuming ! (1-3 weeks)

21. Collins’ Fabrication Process Building the final sculpture (2-3 months): u Take measurements from mock-up model, transfer parallel contours to 1” boards. u Roughly precut boards, leaving registration marks and contiguous pillars for gluing boards together. u Stack and glue together precut boards, remove auxiliary struts. u Fine-tune overall shape, sand and polish the surface. A big investment of effort !

22. Collins’ Fabrication Process Lamination process to make an overall shape that within contains the final sculpture. Example: “Vox Solis”

23. “Sculpture Generator I” Prototyping & Visualization tool for Scherk-Collins Saddle-Chains. u Slider control for this one shape-family, u Control of about 12 parameters. u Main goal: Speed for interactive editing. u Geometry part is about 5,000 lines of C; u 10,000 lines for display & user interface. ==> VIDEO

24. === VIDEO === u 6 min

25. Base Geometry: One Scherk Story u Hyperbolic Slices ==> Triangle Strips u precomputed -- then warped into toroid

26. Slices through the Sculpture u One thick slice thru “Heptoroid” from which Brent can cut boards and assemble a rough shape. Shown are top and bottom as well as cuts at 1/4, 1/2, 3/4 of one board.

27. Our First “Joint” Sculpture Six monkey saddles in a ring with no twist (like Hyperbolic Hexagon) azimuth = –30°, flange 1.5 (aesthetics) size, thickness (fabrication consideration)

28. “Hyperbolic Hexagon II” (wood)

29. Heptoroid ( from Sculpture Generator I ) Cross-eye stereo pair

30. Emergence of the “Heptoroid” (1) Assembly of the precut boards

31. Emergence of the “Heptoroid” (2) Forming a continuous smooth edge

32. Emergence of the “Heptoroid” (3) Smoothing the whole surface

33. Advantages of CAD of Sculptures u Exploration of a larger domain u Instant visualization of results u Eliminate need for prototyping u Create virtual reality pictures u Making more complex structures u Better optimization of chosen form u More precise implementation u Rapid prototyping of maquettes

34. Sculpture Design u branches = 4 u storeys = 11 u height = 1.55 u flange = 1.00 u thickness = 0.06 u rim_bulge = 1.00 u warp = 330.00 u twist = 247.50 u azimuth = 56.25 u mesh_tiles = 0 u textr_tiles = 1 u detail = 8 u bounding box: u xmax= 6.01, u ymax= 1.14, u zmax= 5.55, u xmin= -7.93, u ymin= -1.14, u zmin= -8.41

35. Breckenridge Competition

36. FDM Maquette of Solar Arch u 2nd place

37. We Can Try Again … in L.A.

38. … or in Washington D.C.

39. V-art Glass Scherk Tower with Monkey Saddles Jane Yen

40. SFF Maquettes of Future Sculptures Monkey- Saddle Cinquefoil

41. Various “Scherk-Collins” Sculptures

42. Fused Deposition Modeling (FDM)

43. Looking into the FDM Machine

44. Zooming into the FDM Machine

45. Séquin’s “Minimal Saddle Trefoil” u Stereo- lithography master

46. Séquin’s “Minimal Saddle Trefoil” u bronze cast, gold plated

47. Minimal Trefoils -- cast and finished by Steve Reinmuth

48. Brent Collins’ Trefoil

49. Family of Symmetrical Trefoils W=2 W=1 B=1 B=2 B=3 B=4

50. Higher-order Trefoils (4th order saddles) W=1W=2

51. Exploring New Ideas u Going twice around the loop... Resulting in an interwoven structure.

52. 9-story Intertwined Double Toroid Bronze investment casting from wax original made on 3D Systems’ “Thermojet”

53. Brent Collins: Pax Mundi

54. Keeping up with Brent... u A bent “Scherk tower” is not able to describe a shape like “Pax Mundi.” u Need a broader paradigm ! u Use SLIDE modeling environment, it provides a nice combination of procedural modeling and interactivity.

55. SLIDE-UI for “Pax Mundi” Shapes

56. “Viae Globi” Family (Roads on a Sphere) L2 L3 L4 L5

57. Via Globi 3 (Stone) Wilmin Martono

58. Via Globi 5 (Wood) Wilmin Martono

59. Via Globi 5 (Gold) Wilmin Martono

60. Conclusions (1) u Interactive computer graphics is a novel (to artists) medium that can play an important role -- even for traditional artists. u Virtual Prototyping can save time and can tackle sculptures of a complexity that manual techniques could not conquer.

61. Conclusions (2) u The computer is not only a great visualization and prototyping tool, u It also is a generator for new ideas and u an amplifier for an artist’s inspiration.

62. Questions ? THE END

63. ========= SPARE ========= =========================

64. Stepwise Expansion of Horizon u Playing with many different shapes and u experimenting at the limit of the domain of the sculpture generator, u stimulates new ideas for alternative shapes and generating paradigms. Swiss Mountains

65. Figure-8 Knot with C-Section

66. Conclusions (3) u What makes a CAD tool productive for this kind of work ? l Not just “virtual clay,” l partly procedural; l fewer parameters that need to be set. l Keep things aligned, joined; l guarantee symmetry, regularity, l watertight surfaces. l Interactivity is crucial !