1. CHS UCB ZURICH, Aug. 2002 ART -- MATH Carlo H. Séquin University of California, Berkeley

2. CHS UCB Focus of Talk u How can we use the visualization power offered by computer graphics and by computer-controlled rapid prototyping for the design of geometrical sculptures?

3. CHS UCB Outline u Background (Why?) u Collaboration with Brent Collins u Parameterized Sculpture Families u Sculpture Optimization

4. CHS UCB I am a Designer …

5. CHS UCB 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,...

6. CHS UCB 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

7. CHS UCB Brent Collins: Early Sculptures All photos by Phillip Geller

8. CHS UCB 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 ?

9. CHS UCB “Hyperbolic Hexagon II” (wood) Brent Collins

10. CHS UCB Brent Collins: Stacked Saddles

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

12. CHS UCB “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 ?

13. CHS UCB Closing the Loop straight or twisted

14. CHS UCB 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. u A key problem is making the sculpture look good from all sides !

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

16. CHS UCB 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 !

17. CHS UCB Collins’ Fabrication Process Example: “Vox Solis” Layered laminated main shape Wood master pattern for sculpture

18. CHS UCB Sculpture Generator, GUI

19. CHS UCB “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.

20. CHS UCB Scherk-Collins Sculptures

21. CHS UCB The Basic Element Scherk’s 2nd minimal surface 3-story tower, trimmed, thickened 180 degrees of twist added

22. CHS UCB Toroidal Warp into Collins Ring 8-story towerwarped into a ring360º twist added

23. CHS UCB A Plethora of Shapes

24. CHS UCB Edge Treatment square, flat cutsemi-circularbulging out

25. CHS UCB Embellishment of Basic Shape colorbackgroundtexture

26. CHS UCB A Simple Scherk-Collins Toroid Parameters: (genome) u branches = 2 u stories = 1 u height = 5.00 u flange = 1.00 u thickness = 0.10 u rim_bulge = 1.00 u warp = 360.00 u twist = 90 u azimuth = 90 u textr_tiles = 3 u detail = 8

27. CHS UCB Also a Scherk-Collins Toroid u branches = 1 u stories = 5 u height = 1.00 u flange = 1.00 u thickness = 0.04 u rim_bulge = 1.01 u warp = 360 u twist = 900 u azimuth = 90 u textr_tiles = 1 u detail = 20

28. CHS UCB A Scherk Tower (on its side) u branches = 7 u stories = 3 u height = 0.2 u flange = 1.00 u thickness = 0.04 u rim_bulge = 0 u warp = 0 u twist = 0 u azimuth = 0 u textr_tiles = 2 u detail = 6

29. CHS UCB 1-story Scherk Tower u branches = 5 u stories = 1 u height = 1.35 u flange = 1.00 u thickness = 0.04 u rim_bulge = 0 u warp = 58.0 u twist = 37.5 u azimuth = 0 u textr_tiles = 8 u detail = 6

30. CHS UCB 180º Arch = Half a Scherk Toroid u branches = 8 u stories = 1 u height = 5 u flange = 1.00 u thickness = 0.06 u rim_bulge = 1.25 u warp = 180 u twist = 0 u azimuth = 0 u textr_tiles = e u detail = 12

31. CHS UCB Main Goal in Sculpture Generator I Real-time Interactive Speed ! u Can’t afford surface optimization to obtain true minimal surfaces; u also, this would be aesthetically too limited.  Use closed-form hyperbolic approximation.

32. CHS UCB V-art Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

33. CHS UCB How to Obtain a Real Sculpture ? u Prepare a set of cross-sectional blue prints at equally spaced height intervals, corresponding to the board thickness that Brent is using for the construction.

34. CHS UCB Slices through “Minimal Trefoil” 50%10%23%30% 45%5%20%27% 35%2%15%25%

35. CHS UCB Profiled Slice through the Sculpture u One thick slice thru “Heptoroid” from which Brent can cut boards and assemble a rough shape. Traces represent: top and bottom, as well as cuts at 1/4, 1/2, 3/4 of one board.

36. CHS UCB 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)

37. CHS UCB Another Joint Sculpture u Heptoroid

38. CHS UCB Heptoroid ( from Sculpture Generator I ) Cross-eye stereo pair

39. CHS UCB Emergence of the “Heptoroid” (1) Assembly of the precut boards

40. CHS UCB Emergence of the “Heptoroid” (2) Forming a continuous smooth edge

41. CHS UCB Emergence of the “Heptoroid” (3) Smoothing the whole surface

42. CHS UCB 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

43. CHS UCB SFF (Solid Free-form Fabrication) Monkey- Saddle Cinquefoil

44. CHS UCB Various “Scherk-Collins” Sculptures

45. CHS UCB Fused Deposition Modeling (FDM)

46. CHS UCB Looking into the FDM Machine

47. CHS UCB Zooming into the FDM Machine

48. CHS UCB Séquin’s “Minimal Saddle Trefoil” u Stereo- lithography master

49. CHS UCB Séquin’s “Minimal Saddle Trefoil” u bronze cast, gold plated

50. CHS UCB Minimal Trefoils -- cast and finished by Steve Reinmuth

51. CHS UCB Brent Collins’ Trefoil

52. CHS UCB Part III Developing Parameterized Sculpture Families

53. CHS UCB Family of Symmetrical Trefoils W=2 W=1 B=1 B=2 B=3 B=4

54. CHS UCB Close-up of Some Trefoils B=1 B=2 B=3 Varying the number of branches, the order of the saddles.

55. CHS UCB Higher-order Trefoils (4th order saddles) W=1 (Warp)W=2 

56. CHS UCB Exploring New Ideas: W=2 u Going around the loop twice... … resulting in an interwoven structure.

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

58. CHS UCB 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

59. CHS UCB Note: The computer becomes an amplifier / accelerator for the creative process.

60. CHS UCB Inspiration: Brent Collins’ “Pax Mundi”

61. CHS UCB Keeping up with Brent... u Sculpture Generator I can only do warped Scherk towers, not able to describe a shape like Pax Mundi. u Need a more general approach ! u Use the SLIDE modeling environment (developed at U.C. Berkeley by J. Smith) to capture the paradigm of such a sculpture in a procedural form. l Express it as a computer program l Insert parameters to change salient aspects / features of the sculpture l First: Need to understand what is going on 

62. CHS UCB Sculptures by Naum Gabo Pathway on a sphere: Edge of surface is like seam of tennis ball;  2-period Gabo curve.

63. CHS UCB 2-period Gabo curve u Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used.

64. CHS UCB 4-period Gabo curve Same construction as for as for 2-period curve

65. CHS UCB “Pax Mundi” Revisited u Can be seen as: Amplitude modulated, 4-period Gabo curve

66. CHS UCB SLIDE-UI for “Pax Mundi” Shapes

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

68. CHS UCB Via Globi 3 (Stone) Wilmin Martono

69. CHS UCB Via Globi 5 (Wood) Wilmin Martono

70. CHS UCB Via Globi 5 (Gold) Wilmin Martono

71. CHS UCB Extending the Paradigm Try to Expand the Sculpture Family: u Aim for more highly convoluted paths, u maintain high degree of symmetry. u Need a better tool to draw on sphere …

72. CHS UCB Circle Splines on the Sphere Examples from Jane Yen’s Editor Program

73. CHS UCB Via Globi -- Virtual Design Wilmin Martono

74. CHS UCB “Maloja” -- FDM part u A rather winding Swiss mountain pass road in the upper Engadin.

75. CHS UCB “Stelvio” u An even more convoluted alpine pass in Italy.

76. CHS UCB “Altamont” u Celebrating American multi-lane highways.

77. CHS UCB “Lombard” u A very famous crooked street in San Francisco u Note that I switched to a flat ribbon.

78. CHS UCB Part IV Using Virtual Shapes and Physical 3D Models for Sculpture Optimization

79. CHS UCB Another Inspiration by B. Collins

80. CHS UCB Collin’s Conceptual Design SWEEP CURVE (FOR DOUBLE CYLINDER) IS COMPOSED OF 4 IDENTICAL SEGMENTS, FOLLOWS THE SURFACE OF A SPHERE.

81. CHS UCB Reconstruction / Analysis (v1) AWKWARD ALIGNMENT FROM THE FDM MACHINE

82. CHS UCB Further Explorations (v2: add twist)

83. CHS UCB A More Complex Design (v3)

84. CHS UCB Verification with 3D Model (v4) GALAPAGOS-4 (SIDE VIEW)

85. CHS UCB Fine-tuned Final(?) Version (v5)

86. CHS UCB Galapagos-6 in the Making

87. CHS UCB Galapagos-6 (v6)

88. CHS UCB Conclusions (1) u Virtual Design / Prototyping is a novel medium (to artists). u It can play an important role -- even for traditional sculptors: u it can save time and labor, and u allows to tackle sculptures of a complexity that manual techniques could not conquer.

89. CHS UCB 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.

90. CHS UCB Conclusions (3) u Rapid prototyping (layered fabrication) must now be considered a new facet in the spectrum of MM technologies. u It provides tangible (high-quality haptic) output for objects with which users may want to interact. u Even for sculptures (intended primarily for visual enjoyment) the physical maquette discloses subtle geometrical features that are not visible in the virtual rendering.

91. CHS UCB Acknowledgements u Brent Collins for his inspiring artwork and many stimulating discussions. u Jordan Smith, Jane Yen, Human Meshkin, for developing some of the software modules that I am using in my work.

92. CHS UCB Questions ? THE END

93. CHS UCB ========= SPARE ========= =========================

94. CHS UCB 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 !

95. CHS UCB Some of the Parameters in “SC1”