1. Microsoft MOSAIC Microsymposium 2000 Analysis and Synthesis of Intuitively Conceived Geometrical Art Carlo H. Séquin University of California, Berkeley Brent Collins Gower, Missouri

2. Roots of My Passion My love for geometry and abstract sculpture emerged long long before I learned to play with computers. Acknowledgement to the artists who have strongly influenced me...

3. Appreciation of Simple Geometric Forms (`50s, `60s ) ä Alexander Calder, ä Naum Gabo, ä Max Bill:

4. Intro to Computer Graphics (`70s) ä Ken Knowlton, Lillian Schwarz (at Bell Labs) ä Frank Smullin (at D.A.Conference):

5. Granny Knot Lattice Séquin 1982 Berkeley UniGrafix

6. Complex Compositions (‘80s) ä M. C. Escher, ä Bruce Beasly, ä George Rickey, ä Alan Holden:

7. Recent Interactions (`90s) ä Charles Perry, ä Helaman Ferguson, ä Robert Longhurst: Brent Collins: Collaboration since 1995

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

9. Early Brent Collins Sculptures All photos by Phillip Geller

10. ==> BRENT COLLINS’ Presentation ä Early work with saddles... ä through “Hyperbolic Hexagon”

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

12. Brent Collins: Stacked Saddles

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

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

15. Closing the Loop straight or twisted

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

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

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

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

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

21. === VIDEO === ä 6 min

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

23. Slices through the Sculpture ä 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.

24. 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)

25. “Hyperbolic Hexagon II” (wood)

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

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

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

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

30. ==> BRENT COLLINS’ Presentation ä Hyperbolic Heptagon ä Trefoils ä Hyperbolic Hexagon II, and Heptoroid ä Pax Mundi, Egg, Vox Solis ä Atomic Flowers ä Pieces with Positive Curvature (Bronze casting will be described at MOSAIC talk)

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

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

33. FDM Maquette of Solar Arch

34. Breckenridge Competition ä 2nd place

35. We Can Try Again … in L.A.

36. … or in Washington D.C.

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

38. SFF Maquettes of Future Sculptures Monkey- Saddle Cinquefoil

39. Various “Scherk-Collins” Sculptures

40. Figure-8 Knot with C-Section

41. Fused Deposition Modeling (FDM)

43. Looking into the FDM Machine

44. Zooming into the FDM Machine

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

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

47. Brent’s Trefoil

48. Minimal Trefoils cast and finished by Steve Reinmuth

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

52. Exploring New Ideas ä Going twice around the loop... Resulting in an interwoven structure.

53. 9-story Intertwined Double Toroid Bronze investment casting from wax original made on 3D Systems “Thermojet”

54. Brent Collins: Pax Mundi

55. Keeping up with Brent... ä Bent “Scherk towers” is not able to describe a shape like “Pax Mundi.” ä Need a broader paradigm ! ä Use SLIDE modeling environment, it provides a nice combination of procedural modeling and interactivity.

56. SLIDE SLIDE = Scene Language for Interactive Dynamic Environments Developed as a modular rendering pipeline for our introductory graphics course. Primary Author: Jordan Smith ä Based on OpenGL and Tcl/tk. ä Good combination of interactive 3D graphics and parameterizable procedural constructs.

57. SLIDE Example: Klein Bottle Final Project CS 184, Nerius Landys & Shad Roundy

58. SLIDE Example Bug’s Life Final Project CS 184, David Cheng and James Chow

59. SLIDE as a Design Tool ä SLIDE is being enhanced currently to serve as a front-end for CyberBuild. ä Recently added: ä Spline curves and surfaces ä Morphing sweeps along such curves ä 3D warping module (Sederberg, Rockwood) ä Many types of subdivision surfaces ä These are key elements for Sculpture Generator II

60. 3D Hilbert Curves (FDM) Hilbert64 and Hilbert512

61. SLIDE-UI for Knot Generation

62. SLIDE-UI for “Pax Mundi” Shapes

63. Via Globi 5 (Wood) Wilmin Martono

64. Via Globi 5 (Gold) Wilmin Martono

65. Via Globi 3 (Stone) Wilmin Martono

66. Via Globi Family L2 L3 L4 L5

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

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

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

70. ========= SPARE ========= =========================

71. Brent Collins: Spiral Structures

72. Some of the Parameters in “SC1”

73. Preparation of Blue Prints ä Cut geometrical shape into slices parallel to one axis: ä Assemble several profile lines for each board, ä and print out on separate sheet at 1:1 size. ä ==> (Examples from printer on wall)

74. The role of the computer ä Eliminate need for prototyping ä Exploration of larger domain ä Making more complex structures ä Better optimization ä Precise blue prints

75. SLA Maquette of “Solar Arch” Back-lighting and photo by Philip Geller

76. My Contorted Professional Path ä 65-70: Basel: Physics – Experiment design ä 70-76: Bell Labs: CCD – Circuit, chip design ä 76-82: UCB: RISC – CPU architecture design ä 80-??: Graphics – Modeling & Rendering SW ä 82-90: VLSI CAD – CAD algorithm design ä 87-94: Soda Hall – Building design, VR ä 92-98 : Architecture – ArchCAD tool design ä 95-??: Mech. Eng. – Develop SIF, CyberCut Sculpture – CAD, Virtual Prototyping

77. ==Abstract Brent Collins is a professional artist living in Gower, MO, who has been carving abstract geometrical structures from solid wood blocks or from laminated assemblies. Many of his sculptures comprise minimal surfaces which form an intricate composition of tunnels and saddles. Carlo Séquin is a professor at U.C. Berkeley, teaching computer graphics, geometric modeling, and computer aided design. Since 1995 he has been collaborating with Collins in the conception and design of intricate geometrical shapes that expand the original work of Collins. He has developed several procedural generator programs that recreate some of the shapes conceived by Collins and can expand the basic concept in several possible directions. His interactive "sculpture generator" allows a quick perusal of some domain of the configuration space, and an optimization of a given design along the axes of several parameters. Shapes of high artistic merits can then be sliced into 1-inch slabs and a corresponding set of cross sections printed out. Brent Collins uses these templates to precut a set of wood boards and to preassemble a rough shape from which the final sculpture is then carved. In this way, the intuitive genius of the artist can be amplified by a computer tool that assists with the visualization and faithful reproduction of complex geometrical shapes.

78. The Need for a Prototyping Tool ä Intriguing Conceptual Idea: Turn the Hyperbolic Hexagon into a Heptagon ä Goals: Give it more variety; ä Make different hole-angles visible from one direction by adding twist before closure. ä Results: Surface becomes single-sided; ä The four separate edges join into a torus knot. ä Question: Will it look beautiful ? ä What can we do if it doesn’ t ? ä Perhaps more twist can save it ? ä What is the best size for the holes, the flanges ? ä How can we find out quickly ?

79. 5-loop Tangles (FDM) With and Without Overhang Supports

80. 3D Yin-Yang (FDM) The two complementary half-spheres