1. ISAMA 2004
Artist’s Sketch, SIGGRAPH 2006, Boston, Carlo H. Séquin, EECS, U.C. Berkeley
Ling Xiao is an undergraduate student who worked with me on this project for the last 12 months. He did all the programming and fancy rendering for this talk
-- while at the same time he also worked on 3 other projects for 3 other professors. Unfortunately he cannot be here today because he is busy finishing up some of these other projects, preparing himself fro graduate school at Stanford, and starting a new company.
The title is somewhat cryptic – but to first order this is about …
Hilbert Cube 512

2. a “space-filling” curve
3D Hilbert Cube
a “space-filling” curve

3. A plane-filling Peano curve
The 2D Hilbert Curve (1891)
A plane-filling Peano curve
Fall 1983: CS Graduate Course: “Creative Geometric Modeling” Do This In 3 D !

4. Artist’s Use of the Hilbert Curve
ISAMA 2004
Artist’s Use of the Hilbert Curve
one conceivable answer might be this artwork by H.F. ... but still a 2D pattern, just on a more complicated 3D surface.
Helaman Ferguson, “Umbilic Torus NC” Silicon bronze, 27 x 27 x 9 in., SIGGRAPH’86

5. Construction of the 2D Hilbert Curve
ISAMA 2004
Construction of the 2D Hilbert Curve 1 2 3
To understand how we might truly carry this idea to 3D, take another look at the construction ...

6. “Do This in 3 D !” What are the plausible constraints ?
3D array of 2n x 2n x 2n vertices
Visit all vertices exactly once
Only nearest-neighbor connections
Fill “local” neighborhood first
Aim for self-similarity
Recursive formulation (for arbitrary n)

7. Construction of 3D Hilbert Curve

8. Construction of 3D Hilbert Curve
Use this element with proper orientation, mirroring.

9. Design Choices: 3D Hilbert Curve
ISAMA 2004
Design Choices: 3D Hilbert Curve
What are the things one might optimize ?
Maximal symmetry
Overall closed loop
No consecutive collinear segments
No (3 or 4 ?) coplanar segment sequence
others ... ?
 More than one acceptable solution !
Is the answer no fully predetermined, or is there some freedom of design ?

10. Typical Early Student Solution
Design Flaws:
2 collinear segments
less than maximal symmetry
4 coplanar segments
D. Garcia, and T. Eladi (1994)

11. Jane Yen: “Hilbert Radiator Pipe” (2000)
Flaws ( from a sculptor’s point of view ):
4 coplanar segments
Not a closed loop
Broken symmetry

12. ISAMA 2004
Time-Line, Background
David Hilbert, Construction of a 2D Peano curve (1891).
E. N. Gilbert, “Gray codes and the Paths on the N-Cube” Bell Syst. Tech. J. 37 (1958).
William J. Gilbert, “A Cube-filling Hilbert Curve” Mathematical Intelligencer 6(3) (1984).
C. H. Séquin, “Do This in 3D!” Graduate course assignments ( now).
Nelson Max, “Visualizing Hilbert Curves” (VIS’98); “Homage to Hilbert” computer-generated video.
C. H. Séquin, Plastic models (1998). 
C. H. Séquin, Metal Sculpture (2005). 
Have these students invented something completely new ?
1958 BSTJ  Coding theory !

13. Plastic Model (from FDM) (1998)
Support removal can be tedious, difficult !

14. Fused Deposition Modeling (FDM)
Plastic Filament
Support Filament
Heated Head, moving in x,y
Nozzles
Stage, moving vertically

15. The Next Level of Recursion
Presented a challenge to remove supports.
Resulted in a flimsy, spongy model.
Would like to have a more durable model in metal.

16. 2006: Metal Sculpture in Exhibit
ISAMA 2004
2006: Metal Sculpture in Exhibit
Finally got around to doing a metal sculpture last year using a special layered sintering process
Design:
closed loop
maximal symmetry
at most 3 coplanar segments

17. The Devil is in the Details !
ISAMA 2004
The Devil is in the Details !
Aesthetic design goals dominated.
Abandoned strict self-similar recursion.
Used a different lowest-level unit element.
Moved top-level connections to center.
Strict S4-symmetry could be obtained.
This solution could not have been found without computer-aided design tools.
To achieve the various aesthetic design goals, and some constraints imposed by the fabrication, strict self-similar recursion had to be abandoned.
To avoid 4 coplanar elements, the 8-segment Hilbert path, at the smallest unit level, was broken at one of the 4 mutually parallel segments.
To move the connections at the largest scaleto the center of the sculpture (needed for support; more on that later ...),half the smallest unit cubes were rotated 90°.
Overall strict S4-symmetry could be obtained.

18. Basic Element, Lowest Level
not this – but this
avoid 4 coplanar segments !

19. Implementation Challenges
How to build this in metal ?
Impossible to get machine tool to inside;
Hard to cast; complex mold;
Fortunately, new process from X1 corp.

20. New Metal Sintering Process
ProMetal is a division of The Ex One Company headquartered in Irwin, Pennsylvania USA.
Ex One, known for innovative technologies, incorporates the ProMetal process to their line of products and services providing an advanced manufacturing solution.

21. PROMETAL Printing Process
Selectively, layer by layer, infiltrate metal powder with a binder (like “3D printing”).
Remove all un-bound metal powder.
Sinter the remaining “green” part; stainless steel particles fuse, binder gets flushed out (hopefully in that order!);  porous (50%) stainless steel skeleton.
Infiltrate with liquid bronze alloy;  fully-dense composite.

22. Problems ... Green part is heavy, but not very strong.
My sculpture is a 320” inch long rod, 3/16th” thick, wound up in 4” cube, with no intermediate supports.
Green part needs additional supports !! We started with 12, but needed 36.
Finally these supports need to be removed again;  put them near periphery for easy access.
But center also needs some supports (which would be hard to cut away);  make these the permanent ones.
This necessitated one more redesign ...

23. Auxiliary Supports for “Green” Part
ISAMA 2004
Auxiliary Supports for “Green” Part
fat green tubes near center are the permanent connections between the two halves of this “cubist brain”

24. The Two Halves of the “Cubist Brain”
View along a symmetry axis

25. Of Interest to Siggraph Attendees:
ISAMA 2004
Of Interest to Siggraph Attendees:
New fabrication process: allows to build things not previously possible.
Show the intricate design challenges behind a relatively simple sculpture.
What are its artistic merits ? What associations does it raise ? . . .
Give you a glimpse of my creative process: Open-ended analogies  intriguing results.
 Another example: 3D Yin Yang.   
Why do I think this sketch may be interesting to SIGGRAPH attendees ?
. . .
What are its artistic merits ? I find it an intriguing and rather pleasing form. Ideally good art speaks for itself and does not need lengthy explanations. Associations: A cubistic constructivist picture of the human brain ... Two halves loosely connected ...

26. Design Problem: 3D Yin-Yang
“Do this in 3D !”
What this might mean ...
Subdivide a sphere into two halves.

27. 3D Yin-Yang Solutions (Fall 1997)
ISAMA 2004
3D Yin-Yang Solutions (Fall 1997)
Amy Hsu: Clay Model
Perhaps the most obvious solution ...
Robert Hillaire: Acrylite Model
and these students are in good company ...

28. Max Bill’s “Half-Sphere”
Max Bill, Swiss ( )
“Hard Half of a Sphere”
Fused silica, 18 in. diameter
(1972).

29. Other, “More 3D” Partition Surfaces
Smith Wink

30. Yin-Yang Symmetries Mz C2 S2
From the constraint that the two halves should be either identical or mirror images of one another, follow constraints for allowable dividing-surface symmetries. Mz C2 S2

31. My Preferred 3D Yin-Yang
Based entirely on cyclides (e.g., cone, horn torus), (All lines of principal curvatures are circles).
Implementation: Stereolithography (SLA).

32. Surprises ! Should sphere be split into TRHEE parts ?
In Korea, the 3-part taeguk symbolizes heaven, earth and humanity.

33. And why not four, or more parts ... ?
keep an open mind ...

34. Craig Schaffer “5-fold Infinite Yin-Yang”
Black marble, 30 in. diameter

35. Toy: Yin Yang Ball
(®2000)

36. Collaboration with Brent Collins
ISAMA 2004
Collaboration with Brent Collins
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.
“Genesis” – Brent Collins at BRIDGES 2000

37. “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 ?

38. Closing the Loop
straight or
twisted

39. Sculpture Generator, GUI

40. “Hyperbolic Hexagon II” (wood)
ISAMA 2004
“Hyperbolic Hexagon II” (wood)
For whom I designed certain shapes on the computer which he then built oin wood.
Brent Collins

41. The Generative Process
Find the inherent constructive logic.
Devise an appropriate generative program.
Introduce sliders for crucial parameters.
Play with sliders to explore design space.
Reprogram to go outside current domain.
Think outside the box !
Many, many experiments . . .
 The computer becomes an amplifier for an artist’s creativity !

42. Snowsculpting Championships 2003
Silver Medal Winner: “Whirled White Web” (C. Séquin, S. Wagon, D. Schwalbe, B. Collins, S. Reinmuth)