1. Procedurally Defined Geometrical Sculptures
Florida 1999
AAAS 2001, San Francisco
Procedurally Defined Geometrical Sculptures
Carlo H. Séquin
University of California, Berkeley
Brent Collins
Gower, Missouri

2. Computer-Aided Design
My Professional Focus
Computer-Aided Design
Design useful and beautiful objects with the help of computers.
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
Computer-Aided Design III : Buildings Soda Hall, CS Dept. Berkeley (1992)

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

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
Florida 1999
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, ...
Before I talk about my collaboration with Brent Collins,
I would like to tell you where I am coming from artistically,
and give credit to the artists who had a formative influence on me.
Here is a quick overview, and then I will go into some more detail.

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
Beautiful symmetries
Graceful balance of the saddle surfaces
Superb craftsmanship
Intriguing run of the edges
What type of knot is formed ?
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
6 saddles in a ring
6 holes passing through symmetry plane at ±45º
“wound up” 6-story Scherk tower
What would happen,
if we added more stories ?
or introduced a twist before closing the ring ?

18. Closing the Loop
straight or
twisted

19. Collins - Séquin Collaboration
Discuss ideas on the phone
Exchange sketches
Vary the topological parameters
But how do you know whether it is beautiful ? Need visual feedback.
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):
Take measurements from mock-up model, transfer parallel contours to 1” boards.
Roughly precut boards, leaving registration marks and contiguous pillars for gluing boards together.
Stack and glue together precut boards, remove auxiliary struts.
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.
Slider control for this one shape-family,
Control of about 12 parameters.
Main goal: Speed for interactive editing.
Geometry part is about 5,000 lines of C;
10,000 lines for display & user interface.
==> VIDEO

24. === VIDEO ===
6 min

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

26. Slices through the Sculpture
Florida 1999
Slices through the Sculpture
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.
From these Collins will precut boards
then assemble the complete shape
and fine tune and polish it.

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

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

35. Breckenridge Competition

36. FDM Maquette of Solar Arch
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”
Stereo-lithography master

46. Séquin’s “Minimal Saddle Trefoil”
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= B= B= B=4

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

51. Exploring New Ideas 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 ...
A bent “Scherk tower” is not able to describe a shape like “Pax Mundi.”
Need a broader paradigm !
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)
L L L 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)
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.

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

62. Questions ?
THE END

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

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

65. Figure-8 Knot with C-Section

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