Artist’s Sketch, SIGGRAPH 2006, Boston, Artist’s Sketch, SIGGRAPH 2006, Boston, Carlo H. SCarlo H. Sééquin, EECS, U.C. Berkeleyquin, EECS, U.C. Berkeley

Hilbert Cube 512 Hilbert Cube 512

3D Hilbert Cube3D Hilbert Cube

a “space-filling” curvea “space-filling” curve

The 2D Hilbert Curve (1891)The 2D Hilbert Curve (1891)A plane-filling Peano curve

Fall 1983: CS Graduate Course: “Creative Geometric Modeling”

Do This In 3 D !

Artist’s Use of the Hilbert CurveArtist’s Use of the Hilbert Curve

Helaman Ferguson, “Umbilic Torus NC”Silicon bronze, 27 x 27 x 9 in., SIGGRAPH’86

Construction of the 2D Hilbert CurveConstruction of the 2D Hilbert Curve

112233

““Do This in 3 D !”Do This in 3 D !”

What are the plausible constraints ?

3D array of 2n x 2

n x 2

n vertices

Visit all vertices exactly once

Only nearest-neighbor connections

Fill “local” neighborhood first

Aim for self-similarity

Recursive formulation (for arbitrary n)

Construction of 3D Hilbert CurveConstruction of 3D Hilbert Curve

Construction of 3D Hilbert CurveConstruction of 3D Hilbert Curve

Use this element with proper orientation, mirroring.

Design Choices: 3D Hilbert CurveDesign 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 !

Typical Early Student SolutionTypical Early Student Solution

Design Flaws:

2 collinear segments

less than maximal symmetry

4 coplanar segments

D. Garcia, and T. Eladi (1994)

Jane Yen: “Jane Yen: “Hilbert Radiator PipeHilbert Radiator Pipe” ” (2000)(2000)

Flaws( from a sculptor’s point of view ):

4 coplanar segments

Not a closed loop

Broken symmetry

Time-Line, BackgroundTime-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 (1983 - 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).

Plastic Model (from FDM) (1998)Plastic Model (from FDM) (1998)

Support removal can be tedious, difficult !

SupportFilament

Nozzles

PlasticFilament

Heated Head, moving in x,y

Fused Deposition Modeling (FDM)Fused Deposition Modeling (FDM)

Stage, moving vertically

The Next Level of RecursionThe 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.

2006: Metal Sculpture in Exhibit2006: Metal Sculpture in Exhibit

Design:Design: closed loopclosed loop maximal symmetrymaximal symmetry at most 3 coplanar segmentsat most 3 coplanar segments

The Devil is in the Details !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.

Basic Element, Lowest LevelBasic Element, Lowest Level

not this – but this

avoid 4 coplanar segments !

Implementation ChallengesImplementation 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.

New Metal Sintering ProcessNew 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.

PROMETAL Printing ProcessPROMETAL 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.

Problems ...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 ...

Auxiliary Supports for “Green” PartAuxiliary Supports for “Green” Part

The Two Halves of the “Cubist Brain”The Two Halves of the “Cubist Brain”

View along a symmetry axis

Of Interest to Siggraph Attendees: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.

Design Problem: Design Problem: 3D Yin-Yang3D Yin-Yang

What this might mean ...

Subdivide a sphere into two halves.

“Do this in 3D !”

3D Yin-Yang Solutions 3D Yin-Yang Solutions (Fall 1997)(Fall 1997)

Amy Hsu:Clay Model

Robert Hillaire:Robert Hillaire:Acrylite Model

and these students are in good company ...

Max Bill’s Max Bill’s “Half-Sphere”“Half-Sphere”

Max Bill, Swiss (1908-1994)

“Hard Half of a Sphere”

Fused silica, 18 in. diameter

(1972).

Other, “More 3D” Partition SurfacesOther, “More 3D” Partition Surfaces

Smith Wink

Yin-Yang SymmetriesYin-Yang Symmetries

From the constraint that the two halves should be either identical or mirror images of one another, follow constraints for allowable dividing-surface symmetries.

C2 S2Mz

My Preferred 3D Yin-YangMy Preferred 3D Yin-Yang

Based entirely on cyclides (e.g., cone, horn torus),(All lines of principal curvatures are circles).

Implementation: Stereolithography (SLA).

Surprises !Surprises !

Should sphere be split into TRHEE parts ?

In Korea, the 3-part taeguk symbolizes heaven, earth and humanity.

And why not four, or more parts ... ?And why not four, or more parts ... ?

keep an open mind ...

Craig Schaffer “Craig Schaffer “5-fold Infinite Yin-Yang”5-fold Infinite Yin-Yang”

Black marble, 30 in. diameter

Toy: Yin Yang BallToy: Yin Yang Ball

(®2000)

Collaboration with Brent CollinsCollaboration with Brent Collins

“Genesis” – Brent Collins at BRIDGES 2000

““Hyperbolic Hexagon” by B. CollinsHyperbolic 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 ?

Closing the LoopClosing the Loop

straight

or

twisted

Sculpture Generator, GUI Sculpture Generator, GUI

““Hyperbolic Hexagon II” (wood)Hyperbolic Hexagon II” (wood)

Brent Collins

The Generative ProcessThe 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 !

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

Snowsculpting Championships 2003Snowsculpting Championships 2003