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Artist’s Sketch, SIGGRAPH 2006, Boston, Carlo H. Séquin, EECS, U.C. Berkeley Hilbert Cube 512 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 Modeling” Do This In 3 D !

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Artist’s Use of the Hilbert Curve Helaman Ferguson, “Umbilic Torus NC” Silicon bronze, 27 x 27 x 9 in., SIGGRAPH’86

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Construction of the 2D Hilbert Curve 1 2 3

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

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Construction of 3D Hilbert Curve

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u Use this element with proper orientation, mirroring.

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Design Choices: 3D Hilbert Curve What are the things one might optimize ? u Maximal symmetry u Overall closed loop u No consecutive collinear segments u No (3 or 4 ?) coplanar segment sequence u others... ? More than one acceptable solution !

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

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

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

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

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

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

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2006: Metal Sculpture in Exhibit 2006: Metal Sculpture in ExhibitDesign: u closed loop u maximal symmetry u at most 3 coplanar segments

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The Devil is in the Details ! u Aesthetic design goals dominated. u Abandoned strict self-similar recursion. u Used a different lowest-level unit element. u Moved top-level connections to center. u Strict S 4 -symmetry could be obtained. This solution could not have been found without computer-aided design tools.

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

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

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New Metal Sintering Process u ProMetal is a division of The Ex One Company headquartered in Irwin, Pennsylvania USA. u 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 u Selectively, layer by layer, infiltrate metal powder with a binder (like “3D printing”). u Remove all un-bound metal powder. u Sinter the remaining “green” part; stainless steel particles fuse, binder gets flushed out (hopefully in that order!); porous (50%) stainless steel skeleton. u Infiltrate with liquid bronze alloy; fully-dense composite.

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Problems... u Green part is heavy, but not very strong. u My sculpture is a 320” inch long rod, 3/16 th ” thick, wound up in 4” cube, with no intermediate supports. u Green part needs additional supports !! We started with 12, but needed 36. u Finally these supports need to be removed again; put them near periphery for easy access. u But center also needs some supports (which would be hard to cut away); make these the permanent ones. u 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: u New fabrication process: allows to build things not previously possible. u Show the intricate design challenges behind a relatively simple sculpture. u What are its artistic merits ?... What associations does it raise ?... u Give you a glimpse of my creative process: Open-ended analogies intriguing results. Another example: 3D Yin Yang.

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

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3D Yin-Yang Solutions (Fall 1997) Amy Hsu: Clay Model Robert Hillaire: Robert Hillaire: Acrylite Model and these students are in good company...

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Max Bill’s “Half-Sphere” Max Bill, Swiss ( ) “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 u From the constraint that the two halves should be either identical or mirror images of one another, follow constraints for allowable dividing-surface symmetries. C2C2 S2S2 MzMz

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

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

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

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

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

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