Artist

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Artists Sketch, SIGGRAPH 2006, Boston, Carlo H.
Séquin, EECS, U.C. Berkeley
Hilbert Cube 512
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3D Hilbert Cube

• a space-filling curve

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The 2D Hilbert Curve (1891)

• A plane-filling Peano curve

Fall 1983 CS Graduate Course Creative
Geometric ModelingDo This In 3 D !
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Artists Use of the Hilbert Curve
Helaman Ferguson, Umbilic Torus NCSilicon
bronze, 27 x 27 x 9 in., SIGGRAPH86
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Construction of the 2D Hilbert Curve
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2
3
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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)

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Construction of 3D Hilbert Curve
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Construction of 3D Hilbert Curve

• Use this element with proper orientation,
  mirroring.

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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 !

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Typical Early Student Solution

• Design Flaws
• 2 collinear segments
• less than maximal symmetry
• 4 coplanar segments

D. Garcia, and T. Eladi (1994)
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Jane Yen Hilbert Radiator Pipe (2000)

• Flaws( from a sculptors point of view )
• 4 coplanar segments
• Not a closed loop
• Broken symmetry

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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
  CurveMathematical Intelligencer 6(3) (1984).
• C. H. Séquin, Do This in 3D! Graduate course
  assignments (1983 - now).
• Nelson Max, Visualizing Hilbert Curves
  (VIS98)Homage to Hilbert computer-generated
  video.
• C. H. Séquin, Plastic models (1998). ?
• C. H. Séquin, Metal Sculpture (2005). ?

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Plastic Model (from FDM) (1998)

• Support removal can be tedious, difficult !

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Fused Deposition Modeling (FDM)
SupportFilament
PlasticFilament
Heated Head, moving in x,y
Nozzles
Stage, moving vertically
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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.

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2006 Metal Sculpture in Exhibit

• Design
• closed loop
• maximal symmetry
• at most 3 coplanar segments

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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.

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Basic Element, Lowest Level

• not this but this

avoid 4 coplanar segments !
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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.

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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.

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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 partstainless
  steel particles fuse,binder gets flushed
  out(hopefully in that order!) ? porous (50)
  stainless steel skeleton.
• Infiltrate with liquid bronze alloy?
  fully-dense composite.

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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 ...

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Auxiliary Supports for Green Part
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The Two Halves of the Cubist Brain

• View along a symmetry axis

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Of Interest to Siggraph Attendees

• New fabrication processallows 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
  processOpen-ended analogies ? intriguing
  results.
• ? Another example 3D Yin Yang. ? ? ?

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Design Problem 3D Yin-Yang
Do this in 3D !

• What this might mean ...
• Subdivide a sphere into two halves.

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3D Yin-Yang Solutions (Fall 1997)
Amy HsuClay Model
Robert HillaireAcrylite Model
and these students are in good company ...
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Max Bills Half-Sphere
Max Bill, Swiss (1908-1994) Hard Half of a
Sphere Fused silica, 18 in. diameter (1972).
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Other, More 3D Partition Surfaces

• Smith Wink

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Yin-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
S2
Mz
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My Preferred 3D Yin-Yang

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

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Surprises !

• Should sphere be split into TRHEE parts ?

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

• keep an open mind ...

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Craig Schaffer 5-fold Infinite Yin-Yang

• Black marble, 30 in. diameter

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Toy Yin Yang Ball
(2000)
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Collaboration with Brent Collins

• Genesis Brent Collins at BRIDGES 2000

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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 ?

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Closing the Loop
straight or twisted
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Sculpture Generator, GUI
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Hyperbolic Hexagon II (wood)
Brent Collins
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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
  artists creativity !

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Snowsculpting Championships 2003

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