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

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.

Computer-Aided Design I : Integrated Circuits: “RISC I” chip (1981)

Computer-Aided Design II : Mathematical Models “Granny Knot” Lattice Berkeley UniGrafix (1982)

Computer-Aided Design III : Buildings Soda Hall, CS Dept Computer-Aided Design III : Buildings Soda Hall, CS Dept. Berkeley (1992)

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

Computer-Aided Design V : Abstract Sculpture (virtual) (Since 1995)

Computer-Aided Design V : Abstract Sculpture (virtual) Scherk-Collins Tower

Computer-Aided Design V : Abstract Sculpture (virtual) Doubly-looped Scherk-Collins saddle-chain

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

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.

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

Brent Collins: Early Sculptures All photos by Phillip Geller

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”

Brent Collins: Stacked Saddles

Scherk’s 2nd Minimal Surface Normal “biped” saddles Generalization to higher-order saddles (monkey saddle)

“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 ?

Closing the Loop straight or twisted

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.

Brent Collins’ Prototyping Process Armature for the "Hyperbolic Heptagon" Mockup for the "Saddle Trefoil" Time-consuming ! (1-3 weeks)

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 !

Collins’ Fabrication Process Lamination process to make an overall shape that within contains the final sculpture. Example: “Vox Solis”

“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

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

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.

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)

“Hyperbolic Hexagon II” (wood)

Heptoroid ( from Sculpture Generator I ) Cross-eye stereo pair

Emergence of the “Heptoroid” (1) Assembly of the precut boards

Emergence of the “Heptoroid” (2) Forming a continuous smooth edge

Emergence of the “Heptoroid” (3) Smoothing the whole surface

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

Sculpture Design branches = 4 storeys = 11 height = 1.55 flange = 1.00 thickness = 0.06 rim_bulge = 1.00 warp = 330.00 twist = 247.50 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

Breckenridge Competition

FDM Maquette of Solar Arch 2nd place

We Can Try Again … in L.A.

… or in Washington D.C.

V-art Glass Scherk Tower with Monkey Saddles Jane Yen

SFF Maquettes of Future Sculptures Monkey- Saddle Cinquefoil

Various “Scherk-Collins” Sculptures

Fused Deposition Modeling (FDM)

Looking into the FDM Machine

Zooming into the FDM Machine

Séquin’s “Minimal Saddle Trefoil” Stereo-lithography master

Séquin’s “Minimal Saddle Trefoil” bronze cast, gold plated

Minimal Trefoils -- cast and finished by Steve Reinmuth

Brent Collins’ Trefoil

Family of Symmetrical Trefoils W=2 W=1 B=1 B=2 B=3 B=4

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

Exploring New Ideas Going twice around the loop ... Resulting in an interwoven structure.

9-story Intertwined Double Toroid Bronze investment casting from wax original made on 3D Systems’ “Thermojet”

Brent Collins: Pax Mundi

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.

SLIDE-UI for “Pax Mundi” Shapes

“Viae Globi” Family (Roads on a Sphere) L2 L3 L4 L5

Via Globi 3 (Stone) Wilmin Martono

Via Globi 5 (Wood) Wilmin Martono

Via Globi 5 (Gold) Wilmin Martono

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.

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.

Questions ? THE END

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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

Figure-8 Knot with C-Section

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 !