E92 -- October 2002 Art, Math, and Sculpture Connecting Computers and Creativity Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

My Professional Path u 65-70: Basel: Physics – Experiment design u 70-76: Bell Labs: CCD – Circuit, chip design u 76-82: UCB: RISC – CPU architecture design u 80-??: Graphics – Modeling & Rendering SW u 82-90: VLSI CAD – CAD algorithm design u 87-94: Soda Hall – Building design, VR u 92-98: Architecture – ArchCAD tool design u 95-??: Mech. Eng. – Develop SIF, CyberCut u 96-??: Sculpture – Virtual Prototyping

My Professional Focus Computer-Aided Design u Design useful and beautiful objects with the help of computers. u Develop (interactive) computer programs to make these tasks easier.

CCD Camera, Bell Labs, 1972

Integrated Circuits: “RISC I”, 1981

Mathematical Models “Granny Knot” Lattice Berkeley UniGrafix (1982)

Buildings: Soda Hall, 1992

Mechanical Parts: “Octa-Gear” Design (1985) Realization (FDM) (2000) Octahedral Gear

Geometrical Sculpture (virtual) (Since 1995)

Geometrical Sculpture (real) “Cohesion” 2002

“Whirled White Web” u Design for the 2003 International Snow Sculpture Championship Breckenridge, CO, Jan.28 – Feb.2

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,...

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... u Beautiful symmetries u Graceful balance of saddle surfaces u Superb craftsmanship u Intriguing run of the edges u What type of knot is formed ? u Mystery: one-sided or two-sided ? ==> Focus on “Chains of Saddles” Brent Collins’ Abstract Geometric Art:

Brent Collins: Stacked Saddles

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

“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 ?

Closing the Loop straight or twisted

Collins - Séquin Collaboration u Discuss ideas on the phone u Exchange sketches u Vary the topological parameters u But how do you know whether it is beautiful ? Need visual feedback. u 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): u Take measurements from mock-up model, transfer parallel contours to 1” boards. u Roughly precut boards, leaving registration marks and contiguous pillars for gluing boards together. u Stack and glue together precut boards, remove auxiliary struts. u 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. u Slider control for this one shape-family, u Control of about 12 parameters. u Main goal: Speed for interactive editing. u Geometry part is about 5,000 lines of C; u 10,000 lines for display & user interface.

Sculpture Generator, GUI

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 towerwarped into a ring360º twist added

Edge Treatment square, flat cutsemi-circularbulging out

Embellishment of Basic Shape colorbackgroundtexture

A Simple Scherk-Collins Toroid u branches = 2 u storeys = 1 u height = 5.00 u flange = 1.00 u thickness = 0.10 u rim_bulge = 1.00 u warp = u twist = 90 u azimuth = 90 u textr_tiles = 3 u detail = 8

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

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

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

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

“Hyperbolic Hexagon II” (wood)

How to Obtain a Real Sculpture ? u 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%10%23%30% 45%5%20%27% 35%2%15%25%

Profiled Slice through the Sculpture u 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.

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” u Collaboration by Brent Collins & Carlo Séquin

Advantages of CAD of Sculptures u Exploration of a larger domain u Instant visualization of results u Eliminate need for prototyping u Create virtual reality pictures u Making more complex structures u Better optimization of chosen form u More precise implementation u Rapid prototyping of maquettes

Sculpture Design: “Solar Arch” u branches = 4 u storeys = 11 u height = 1.55 u flange = 1.00 u thickness = 0.06 u rim_bulge = 1.00 u warp = u twist = u azimuth = u mesh_tiles = 0 u textr_tiles = 1 u detail = 8 u bounding box: u xmax= 6.01, u ymax= 1.14, u zmax= 5.55, u xmin= -7.93, u ymin= -1.14, u zmin= -8.41

Competition in Breckenridge, CO

We Can Try Again … in L.A.

FDM Maquette of Solar Arch

Various “Scherk-Collins” Sculptures

Fused Deposition Modeling (FDM)

Galapagos-6 in the Making

Galapagos-6 (v6)

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

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

Minimal Trefoils -- cast and finished by Steve Reinmuth

Brent Collins’ Trefoil

New Possibilities Developing Parameterized, Procedurally Generated Sculpture Families

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

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

Exploring New Ideas u Going around the loop twice... … resulting in an interwoven structure.

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

Stepwise Expansion of Horizon u Playing with many different shapes and u experimenting at the limit of the domain of the sculpture generator, u stimulates new ideas for alternative shapes and generating paradigms. Swiss Mountains

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

Inspiration: Brent Collins’ “Pax Mundi”

Keeping up with Brent... u A warped “Scherk tower” is not able to describe a shape like “Pax Mundi.” u Need a broader paradigm ! u Use the SLIDE modeling environment (developed at U.C. Berkeley by J. Smith); it provides a nice combination of procedural modeling and interactivity.

SLIDE SLIDE = Scene Language for Interactive Dynamic Environments Developed as a modular rendering pipeline for our introductory graphics course CS184. Primary Author: Jordan Smith u Based on OpenGL and Tcl/tk. u Good combination of interactive 3D graphics and parameterizable procedural constructs.

SLIDE Example: Klein Bottle Final Project CS 184, Nerius Landys & Shad Roundy

SLIDE Example Bug’s Life Final Project CS 184, David Cheng and James Chow

SLIDE as a Design Tool u SLIDE has been enhanced to serve as a procedural modeling (CAD) tool. u Recently added: l Spline curves and surfaces l Morphing sweeps along such curves l 3D warping module (Sederberg, Rockwood) l Many types of subdivision surfaces u These are key elements for “Sculpture Generator II”

Capturing the Paradigm The first task when trying to construct a generator for a new family of sculptures is to understand and define its underlying logic.

Sculptures by Naum Gabo Pathway on a sphere: Edge of surface is like seam of tennis ball; ==> 2-period Gabo curve.

2-period Gabo curve u Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used.

4-period Gabo curve Same construction as for as for 2-period curve

“Pax Mundi” Revisited u Can be seen as: Amplitude modulated, 4-period Gabo curve

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

Extending the Paradigm (again) Examples from Jane Yen’s Editor Program. This is a special purpose CAD program to draw nice loopy curves onto a sphere.  “Roads on a Sphere” or “Viae Globi” Circle Splines on the Sphere

“Maloja” ( FDM part ) u A very winding Swiss mountain pass road in the upper Engadin.

“Stelvio” u An even more convoluted alpine pass in Italy.

“Altamont” u Celebrating American multi-lane highways.

“Lombard” u A very famous crooked street in San Francisco u Note that I switched to a flat ribbon.

Who am I ? (1) Am I an Artist ? u What is “ART” these days ?? l Cute ideas l Emotional outpours l The obsession to be novel l The goal to shock and offend l A medium for political statements …

Who am I ? (2) I am a Designer -- and an Engineer ! l ( and proud of it. ) In design tasks you have: l Specified goals l Ways to evaluate a design l The drive to optimize your design

Sculpture Engineering The “Whirled White Web” is the result of such an activity. l It had to be based on a shared design with B. Collins l Complexity comparable to other successful designs l or, preferably, slightly higher l Must be executable in snow and ice l Should look dramatic, intriguing, pleasing … Lots of engineering work ahead: l Design plan of attack, using CAD and graphics l Prepare stencils to make implementation easier

Conclusions u Computers are becoming important tools – even in the field of art. u Virtual Prototyping can save time and can tackle sculptures of a complexity that manual techniques could not conquer. u 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.

Career Advice (1) u Find out what you really enjoy doing. u Find a job that pays you to do just that !

Career Advice (2) u Acquire solid technical foundations. u Stay flexible; keep learning new things. u Keep your eyes open for new opportunities.

Questions ? THE END