1. CHS UCB CS285 Sculpture Generator I Carlo H. Séquin University of California, Berkeley In Collaboration with Brent Collins Gower, Missouri

2. CHS UCB Brent Collins “Genesis” – Brent Collins at BRIDGES 2000

3. 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

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

5. 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 ? ==> Focus on “Chains of Saddles”

6. CHS UCB Brent Collins: Stacked Saddles

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

8. 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 ?

9. CHS UCB Closing the Loop straight or twisted

10. 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 !

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

12. 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 !

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

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

15. CHS UCB “Sculpture Generator I” Prototyping 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.

16. CHS UCB Generated Scherk-Collins Shapes

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

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

19. CHS UCB A Plethora of Shapes

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

21. CHS UCB Embellishment of Basic Shape colorbackgroundtexture

22. CHS UCB Sculpture Generator, GUI

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

24. 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.

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

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 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.

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

33. 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.

34. 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)

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

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

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

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

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

40. CHS UCB “Heptoroid”

41. 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

42. CHS UCB 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

43. CHS UCB FDM Maquette of Solar Arch

44. CHS UCB Breckenridge Competition u 2nd place

45. CHS UCB We Can Try Again … in L.A.

46. CHS UCB … or in Washington D.C.

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

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

49. CHS UCB Various “Scherk-Collins” Sculptures

50. CHS UCB Fused Deposition Modeling (FDM)

51. CHS UCB Zooming into the FDM Machine

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

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

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

55. CHS UCB Brent Collins’ Trefoil

56. CHS UCB New Possibilities Developing Parameterized, Procedurally Generated Sculpture Families

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

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

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

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

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

62. 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

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

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

65. 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 

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

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

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

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

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

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

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

73. CHS UCB SLIDE as a Design Tool u SLIDE originally a modular rendering tool. u Later enhanced to serve as a CAD tool: l Spline curves and surfaces l Morphing sweeps along such curves l 3D warping module (Sederberg, Rockwood) l Many types of subdivision surfaces u These are key elements for a 2 nd Generation Sculpture Generator

74. CHS UCB SLIDE-UI for Knot Generation

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

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

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

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

79. CHS UCB Conclusions (1)  Procedural thinking about some art object adds a new and promising dimension. It allows the artist to increase the complexity, precision, and optimality of a particular piece of art.  The computer must be seen as yet another “power-tool” at the artist’s disposition, -- supplementing the pneumatic chisel, the airbrush, and the welding machine.

80. 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.

81. 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 !

82. CHS UCB Conclusions (4) 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.

83. CHS UCB Questions ? THE END