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Explore the intersection of art, math, and sculpture through computer-aided design. Discover the beauty of abstract geometric art and the intricate designs of Brent Collins. Experience the process of prototyping and fabrication, and learn how technology is revolutionizing the world of sculpture. Join us on this creative journey where computers and art collide.
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USF -- February 2001 Art, Math, and Sculpture Connecting Computers and Creativity Carlo H. Séquin University of California, Berkeley
My Professional Focus Computer-Aided Design • Design useful and beautiful objectswith the help of computers. • Develop (interactive) computer programsto 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. 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 My love for geometry and abstract sculptureemerged long long before I learned to play with computers. Thanks to:Alexander Calder, Naum Gabo,Max Bill, M.C. Escher, Frank Smullin, ...
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”
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 Mockup for the "Saddle Trefoil" Armature for the "Hyperbolic Heptagon" 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 marksand 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 withincontains the final sculpture. Example: “Vox Solis”
“Sculpture Generator I” Prototyping & Visualization tool forScherk-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
The Basic Element Scherk’s 2ndminimal surface 3-story tower,trimmed, thickened 180 degreesof twist added
Toroidal Warp into Collins Ring 8-story tower warped into a ring 360º twist added
Edge Treatment square, flat cut semi-circular bulging out
Embellishment of Basic Shape color texture background
=== VIDEO === • 6 min
A Simple Scherk-Collins Toroid • branches = 2 • storeys = 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 • storeys = 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 • storeys = 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 • storeys = 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 • storeys = 1 • height = 5 • flange = 1.00 • thickness = 0.06 • rim_bulge = 1.25 • warp = 180 • twist = 0 • azimuth = 0 • textr_tiles = e • detail = 12
Main Goal in Sculpture Generator I Real-time Interactive Speed ! • Can’t afford surface optimizationto obtain true minimal surfaces; • also, this would be aesthetically too limited. > Make closed-form hyperbolic approximation.
Base Geometry: One Scherk Story • Hyperbolic Slices ==> Triangle Strips • precomputed -- then warped into toroid
The Basic Saddle Element with surface normals
Hyperbolic Contour Lines On a straight tower and on a ring
How to Obtain a Real Sculpture ? • Prepare a set of cross-sectional blue printsat eaqually spaced height intervals,corresponding to the board thicknessthat 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 • One thick slicethru “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, 3/4of one board.
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)
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