1. USF -- February 2001 Art, Math, and Sculpture Connecting Computers and Creativity Carlo H. Séquin University of California, Berkeley

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... 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” Brent Collins’ Abstract Geometric Art

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. Scherk-Collins Sculptures

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

26. Toroidal Warp into Collins Ring 8-story towerwarped into a ring360º twist added

27. A Plethora of Shapes

28. Edge Treatment square, flat cutsemi-circularbulging out

29. Embellishment of Basic Shape colorbackgroundtexture

30. === VIDEO === u 6 min

31. A Simple Scherk-Collins Toroid u branches = 2 u storeys = 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

32. Also a Scherk-Collins Toroid u branches = 1 u storeys = 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

33. A Scherk Tower (on its side) u branches = 7 u storeys = 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

34. 1-story Scherk Tower u branches = 5 u storeys = 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

35. 180º Arch = Half a Scherk Toroid u branches = 8 u storeys = 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

36. 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. > Make closed-form hyperbolic approximation.

37. Hyperbolic Cross Sections

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

39. The Basic Saddle Element with surface normals

40. Hyperbolic Contour Lines On a straight tower and on a ring

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

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

43. 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.

44. 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)

45. “Hyperbolic Hexagon II” (wood)

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

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

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

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

50. 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

51. Sculpture Design: “Solar Arch” 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

52. Competition in Breckenridge, CO

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

54. FDM Maquette of Solar Arch u 2nd place

55. We Can Try Again … in L.A.

56. … or in Washington D.C.

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

58. SFF Maquettes of Future Sculptures Monkey- Saddle Cinquefoil

59. Various “Scherk-Collins” Sculptures

60. Fused Deposition Modeling (FDM)

61. Looking into the FDM Machine

62. Zooming into the FDM Machine

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

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

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

66. Brent Collins’ Trefoil

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

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

69. Exploring New Ideas u Going around the loop twice... … resulting in an interwoven structure.

70. 9-story Intertwined Double Toroid Bronze investment casting from wax original made on 3D Systems’ “Thermojet”

71. 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

72. Brent Collins: Pax Mundi

73. Keeping up with Brent... u A warped “Scherk tower” is not able to describe a shape like “Pax Mundi.” u Need a broader paradigm ! u Use the SLIDE modeling environment (developed at U.C. Berkeley by J. Smith); it provides a nice combination of procedural modeling and interactivity.

74. 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.

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

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

77. SLIDE as a Design Tool u SLIDE is being enhanced currently to serve as a front-end for CyberBuild. u Recently added: 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 Sculpture Generator II

78. 3D Hilbert Curves (FDM) Hilbert64 and Hilbert512

79. SLIDE-UI for Knot Generation

80. SLIDE-UI for “Pax Mundi” Shapes

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

82. Via Globi 3 (Stone) Wilmin Martono

83. Via Globi 5 (Wood) Wilmin Martono

84. Via Globi 5 (Gold) Wilmin Martono

85. Figure-8 Knot with C-Section

86. 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.

87. 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.

88. Questions ? THE END

89. ========= SPARE ========= =========================

90. 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 !

91. Some of the Parameters in “SC1”

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