CHSCHSUCBUCB USF -- February 2001USF -- February 2001

Art, Math, and SculptureConnecting Computers and Creativity

Carlo H. Séquin

University of California, Berkeley

CHSCHSUCBUCB My Professional FocusMy Professional Focus

Computer-Aided Design

Design useful and beautiful objectswith the help of computers.

Develop (interactive) computer programsto make these tasks easier.

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

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Computer-Aided Design II :Computer-Aided Design II :Mathematical ModelsMathematical Models

“Granny Knot” Lattice

Berkeley UniGrafix (1982)

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

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Computer-Aided Design IV : Computer-Aided Design IV : Mechanical PartsMechanical Parts

Design (1985) Realization (FDM) (2000)

Octahedral Gear

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

(Since 1995)

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

Scherk-Collins Tower

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

Doubly-looped Scherk-Collins saddle-chain

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Computer-Aided Design V : Computer-Aided Design V : Abstract Sculpture (real)Abstract Sculpture (real)

“Bonds of Friendship”(2001)

Fabricated by: Fused Deposition Modeling

Currently in S.F.:at Gallery 650,Delancy/Brannan

CHSCHSUCBUCB Roots of My Passion for SculptureRoots 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, ...

CHSCHSUCBUCB Leonardo -- Special IssueLeonardo -- Special Issue

On Knot-Spanning Surfaces: An Illustrated Essay on Topological ArtWith an Artist’s Statement by Brent Collins

George K. Francis with Brent Collins

CHSCHSUCBUCB Brent Collins: Early SculpturesBrent Collins: Early Sculptures

All photos by Phillip Geller

CHSCHSUCBUCB My Fascination with...My Fascination with...

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’ Abstract Geometric Art

CHSCHSUCBUCB Brent Collins: Stacked SaddlesBrent Collins: Stacked Saddles

CHSCHSUCBUCB Scherk’s 2nd Minimal SurfaceScherk’s 2nd Minimal Surface

Normal“biped”saddles

Generalization to higher-order saddles(monkey saddle)

CHSCHSUCBUCB ““Hyperbolic Hexagon” by B. CollinsHyperbolic 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 ?

CHSCHSUCBUCB Closing the LoopClosing the Loop

straight

or

twisted

CHSCHSUCBUCB Collins - Séquin CollaborationCollins - 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.

CHSCHSUCBUCB Brent Collins’ Prototyping ProcessBrent Collins’ Prototyping Process

Armature for the "Hyperbolic Heptagon"

Mockup for the "Saddle Trefoil"

Time-consuming ! (1-3 weeks)

CHSCHSUCBUCB Collins’ Fabrication ProcessCollins’ 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 !

CHSCHSUCBUCB Collins’ Fabrication ProcessCollins’ Fabrication Process

Lamination process to make an overall shape that withincontains the final sculpture.

Example: “Vox Solis”

CHSCHSUCBUCB ““Sculpture Generator I”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

CHSCHSUCBUCB Scherk-Collins SculpturesScherk-Collins Sculptures

CHSCHSUCBUCB The Basic ElementThe Basic Element

Scherk’s 2ndminimal surface

3-story tower,trimmed, thickened

180 degreesof twist added

CHSCHSUCBUCB Toroidal Warp into Collins RingToroidal Warp into Collins Ring

8-story tower warped into a ring 360º twist added

CHSCHSUCBUCB A Plethora of ShapesA Plethora of Shapes

CHSCHSUCBUCB Edge TreatmentEdge Treatment

square, flat cut semi-circular bulging out

CHSCHSUCBUCB Embellishment of Basic ShapeEmbellishment of Basic Shape

color backgroundtexture

CHSCHSUCBUCB === VIDEO === === VIDEO ===

6 min

CHSCHSUCBUCB A Simple Scherk-Collins ToroidA 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

CHSCHSUCBUCB Also a Scherk-Collins ToroidAlso 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

CHSCHSUCBUCB A Scherk Tower A Scherk Tower (on its side)(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

CHSCHSUCBUCB 1-story Scherk Tower1-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

CHSCHSUCBUCB 180º Arch = Half a Scherk Toroid180º 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

CHSCHSUCBUCB Main Goal in Sculpture Generator IMain 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.

CHSCHSUCBUCB Hyperbolic Cross SectionsHyperbolic Cross Sections

CHSCHSUCBUCB Base Geometry: One Scherk StoryBase Geometry: One Scherk Story

Hyperbolic Slices ==> Triangle Strips

precomputed -- then warped into toroid

CHSCHSUCBUCB The Basic Saddle ElementThe Basic Saddle Element

with surface normals

CHSCHSUCBUCB Hyperbolic Contour LinesHyperbolic Contour Lines

On a straight tower and on a ring

CHSCHSUCBUCB How to Obtain a Real Sculpture ?How to Obtain a Real Sculpture ?

Prepare a set of cross-sectional blue prints

at eaqually spaced height intervals,

corresponding to the board thickness

that Brent is using for the construction.

CHSCHSUCBUCB Slices through “Minimal Trefoil”Slices through “Minimal Trefoil”

50% 10%23%30%

45% 5%20%27%

35% 2%15%25%

CHSCHSUCBUCB Profiled Slice through the SculptureProfiled 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.

CHSCHSUCBUCB Our First “Joint” SculptureOur 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)

CHSCHSUCBUCB ““Hyperbolic Hexagon II” (wood)Hyperbolic Hexagon II” (wood)

CHSCHSUCBUCB Heptoroid Heptoroid ( from Sculpture Generator I )( from Sculpture Generator I )

Cross-eye stereo pair

CHSCHSUCBUCB Emergence of the “Heptoroid” (1)Emergence of the “Heptoroid” (1)

Assembly of the precut boards

CHSCHSUCBUCB Emergence of the “Heptoroid” (2)Emergence of the “Heptoroid” (2)

Forming a continuous smooth edge

CHSCHSUCBUCB Emergence of the “Heptoroid” (3)Emergence of the “Heptoroid” (3)

Smoothing the whole surface

CHSCHSUCBUCB Advantages of CAD of SculpturesAdvantages 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

CHSCHSUCBUCB Sculpture Design: “Solar Arch”Sculpture Design: “Solar Arch”

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

CHSCHSUCBUCB Competition in Breckenridge, COCompetition in Breckenridge, CO

CHSCHSUCBUCB SLA Maquette of “Solar Arch”SLA Maquette of “Solar Arch”

Back-lighting and photo by Philip Geller

CHSCHSUCBUCB FDM Maquette of Solar ArchFDM Maquette of Solar Arch

2nd place

CHSCHSUCBUCB We Can Try Again … in L.A.We Can Try Again … in L.A.

CHSCHSUCBUCB … … or in Washington D.C.or in Washington D.C.

CHSCHSUCBUCB V-artV-art

VirtualGlassScherkTowerwithMonkeySaddles

Jane Yen

CHSCHSUCBUCB SFF Maquettes of Future SculpturesSFF Maquettes of Future Sculptures

Monkey-SaddleCinquefoil

CHSCHSUCBUCB Various “Scherk-Collins” SculpturesVarious “Scherk-Collins” Sculptures

CHSCHSUCBUCB Fused Deposition Modeling (FDM)Fused Deposition Modeling (FDM)

CHSCHSUCBUCB Looking into the FDM MachineLooking into the FDM Machine

CHSCHSUCBUCB Zooming into the FDM MachineZooming into the FDM Machine

CHSCHSUCBUCB Séquin’s “Minimal Saddle Trefoil”Séquin’s “Minimal Saddle Trefoil”

Stereo-lithography master

CHSCHSUCBUCB Séquin’s “Minimal Saddle Trefoil”Séquin’s “Minimal Saddle Trefoil”

bronze cast, gold plated

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Minimal Trefoils -- cast and Minimal Trefoils -- cast and finished by Steve Reinmuthfinished by Steve Reinmuth

CHSCHSUCBUCB Brent Collins’ TrefoilBrent Collins’ Trefoil

CHSCHSUCBUCB Family of Symmetrical TrefoilsFamily of Symmetrical Trefoils

W=2

W=1

B=1 B=2 B=3 B=4

CHSCHSUCBUCB Higher-order Trefoils Higher-order Trefoils (4th order saddles)(4th order saddles)

W=1 W=2

CHSCHSUCBUCB Exploring New IdeasExploring New Ideas

Going around the loop twice ...

… resulting in an interwoven structure.

CHSCHSUCBUCB 9-story Intertwined Double Toroid9-story Intertwined Double Toroid

Bronzeinvestment casting from wax original made on 3D Systems’“Thermojet”

CHSCHSUCBUCB Stepwise Expansion of HorizonStepwise 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

CHSCHSUCBUCB Brent Collins: Pax MundiBrent Collins: Pax Mundi

CHSCHSUCBUCB Keeping up with Brent ...Keeping up with Brent ...

A warped “Scherk tower” is not able to describe a shape like “Pax Mundi.”

Need a broader paradigm !

Use the SLIDE modeling environment(developed at U.C. Berkeley by J. Smith);it provides a nice combination of procedural modeling and interactivity.

CHSCHSUCBUCB SLIDESLIDE

SLIDE = Scene Language for Interactive Dynamic Environments

Developed as a modular rendering pipelinefor our introductory graphics course.

Primary Author: Jordan Smith

Based on OpenGL and Tcl/tk.

Good combination of interactive 3D graphicsand parameterizable procedural constructs.

CHSCHSUCBUCB SLIDE Example: Klein BottleSLIDE Example: Klein Bottle

Final Project CS 184, Nerius Landys & Shad Roundy

CHSCHSUCBUCB SLIDE Example Bug’s LifeSLIDE Example Bug’s Life

Final Project CS 184, David Cheng and James Chow

CHSCHSUCBUCB SLIDE as a Design ToolSLIDE as a Design Tool

SLIDE is being enhanced currentlyto serve as a front-end for CyberBuild.

Recently added:

Spline curves and surfaces

Morphing sweeps along such curves

3D warping module (Sederberg, Rockwood)

Many types of subdivision surfaces

These are key elements for Sculpture Generator II

CHSCHSUCBUCB 3D Hilbert Curves (FDM)3D Hilbert Curves (FDM)

Hilbert64 and Hilbert512

CHSCHSUCBUCB SLIDE-UI for Knot GenerationSLIDE-UI for Knot Generation

CHSCHSUCBUCB SLIDE-UI for “Pax Mundi” ShapesSLIDE-UI for “Pax Mundi” Shapes

CHSCHSUCBUCB ““Viae Globi” Family Viae Globi” Family (Roads on a Sphere)(Roads on a Sphere)

L2 L3 L4 L5

CHSCHSUCBUCB Via Globi 3 (Stone)Via Globi 3 (Stone)

Wilmin Martono

CHSCHSUCBUCB Via Globi 5 (Wood)Via Globi 5 (Wood)

Wilmin Martono

CHSCHSUCBUCB Via Globi 5 (Gold)Via Globi 5 (Gold)

Wilmin Martono

CHSCHSUCBUCB Figure-8 Knot with C-SectionFigure-8 Knot with C-Section

CHSCHSUCBUCB Conclusions (1)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 complexitythat manual techniques could not conquer.

CHSCHSUCBUCB Conclusions (2)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.

CHSCHSUCBUCB Questions ?Questions ?

THE ENDTHE END

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CHSCHSUCBUCB Conclusions (3)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 !

CHSCHSUCBUCB Some of the Parameters in “SC1”Some of the Parameters in “SC1”

CHSCHSUCBUCB AAAS 2001, San FranciscoAAAS 2001, San Francisco

Procedurally Defined Geometrical SculpturesProcedurally Defined Geometrical Sculptures

Carlo H. Séquin

University of California, Berkeley

Brent Collins

Gower, Missouri