University of California, Berkeley Florida 1999 CS285 Sculpture Generator I Carlo H. Séquin University of California, Berkeley In Collaboration with Brent Collins Gower, Missouri

Brent Collins “Genesis” – Brent Collins at BRIDGES 2000 Florida 1999 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

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

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. A key problem is making the sculpture look good from all sides !

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 Wood master pattern for sculpture Layered laminated main shape Example: “Vox Solis”

The Need for a Prototyping Tool Intriguing Conceptual Idea: “Turn the Hyperbolic Hexagon into a Heptagon” Goals: Give it more variety; Make different hole-angles visible from one direction by adding twist before closure. Results: Surface becomes single-sided; The four separate edges join into a torus knot. Question: Will it look beautiful ? What can we do if it doesn’ t ? Perhaps more twist can save it ? What is the best size for the holes, the flanges ? How can we find out quickly ?

“Sculpture Generator I” Prototyping 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.

Generated Scherk-Collins Shapes

The Basic Element Scherk’s 2nd minimal surface 3-story tower, trimmed, thickened 180 degrees of twist added

Toroidal Warp into Collins Ring 8-story tower warped into a ring 360º twist added

A Plethora of Shapes

Edge Treatment square, flat cut semi-circular bulging out

Embellishment of Basic Shape color texture background

Sculpture Generator, GUI

Some of the Parameters in “SC1”

Main Goal in Sculpture Generator I Real-time Interactive Speed ! Can’t afford surface optimization to obtain true minimal surfaces; also, this would be aesthetically too limited.  Use closed-form hyperbolic approximation.

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

A Simple Scherk-Collins Toroid Parameters: (genome) branches = 2 stories = 1 height = 5.00 flange = 1.00 thickness = 0.10 rim_bulge = 1.00 warp = 360.00 twist = 90 azimuth = 90 textr_tiles = 3 detail = 8

Also a Scherk-Collins Toroid branches = 1 stories = 5 height = 1.00 flange = 1.00 thickness = 0.04 rim_bulge = 1.01 warp = 360 twist = 900 azimuth = 90 textr_tiles = 1 detail = 20

A Scherk Tower (on its side) branches = 7 stories = 3 height = 0.2 flange = 1.00 thickness = 0.04 rim_bulge = 0 warp = 0 twist = 0 azimuth = 0 textr_tiles = 2 detail = 6

1-story Scherk Tower branches = 5 stories = 1 height = 1.35 flange = 1.00 thickness = 0.04 rim_bulge = 0 warp = 58.0 twist = 37.5 azimuth = 0 textr_tiles = 8 detail = 6

180º Arch = Half a Scherk Toroid branches = 8 stories = 1 height = 5 flange = 1.00 thickness = 0.06 rim_bulge = 1.25 warp = 180 twist = 0 azimuth = 0 textr_tiles = e detail = 12

How to Obtain a Real Sculpture ? Prepare a set of cross-sectional blue prints at equally spaced height intervals, corresponding to the board thickness that Brent is using for the construction.

Slices through “Minimal Trefoil” 50% 30% 23% 10% 45% 27% 20% 5% 35% 25% 15% 2%

Profiled Slice through the Sculpture Florida 1999 Profiled Slice through the Sculpture One thick slice thru “Heptoroid” from which Brent can cut boards and assemble a rough shape. Traces represent: 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) Florida 1999 “Hyperbolic Hexagon II” (wood) For whom I designed certain shapes on the computer which he then built oin wood. Brent Collins

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

“Heptoroid”

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

FDM Maquette of Solar Arch

Breckenridge Competition 2nd place

V-art Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

Fused Deposition Modeling (FDM)

SFF (Solid Free-form Fabrication) Monkey- Saddle Cinquefoil

Various “Scherk-Collins” Sculptures

Note: The computer becomes an amplifier / accelerator for the creative process.

Questions ? THE END