CHSUCB ZURICH, Aug. 2002ZURICH, Aug. 2002

ART -- MATH

Carlo H. Séquin

University of California, Berkeley

CHSUCB Focus of TalkFocus of Talk

How can we use the visualization power offered by computer graphics and by computer-controlled rapid prototyping for the design of geometrical sculptures?

CHSUCB OutlineOutline

Background (Why?)

Collaboration with Brent Collins

Parameterized Sculpture Families

Sculpture Optimization

CHSUCB I am a Designer …I am a Designer …

CHSUCB 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, ...

CHSUCB 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

CHSUCB Brent Collins: Early SculpturesBrent Collins: Early Sculptures

All photos by Phillip Geller

CHSUCB Collins’ Abstract Geometric ArtCollins’ 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 ?

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

Brent Collins

CHSUCB Brent Collins: Stacked SaddlesBrent Collins: Stacked Saddles

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

Normal“biped”saddles

Generalization to higher-order saddles(monkey saddle)

CHSUCB ““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 ?

CHSUCB Closing the LoopClosing the Loop

straight

or

twisted

CHSUCB 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.

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

CHSUCB Brent Collins’ Prototyping ProcessBrent Collins’ Prototyping Process

Armature for the "Hyperbolic Heptagon"

Mockup for the "Saddle Trefoil"

Time-consuming ! (1-3 weeks)

CHSUCB 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 !

CHSUCB Collins’ Fabrication ProcessCollins’ Fabrication Process

Example: “Vox Solis”

Layered laminated main shapeWood master pattern

for sculpture

CHSUCB Sculpture Generator, GUI Sculpture Generator, GUI

CHSUCB ““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.

CHSUCB Scherk-Collins SculpturesScherk-Collins Sculptures

CHSUCB The Basic ElementThe Basic Element

Scherk’s 2ndminimal surface

3-story tower,trimmed, thickened

180 degreesof twist added

CHSUCB Toroidal Warp into Collins RingToroidal Warp into Collins Ring

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

CHSUCB A Plethora of ShapesA Plethora of Shapes

CHSUCB Edge TreatmentEdge Treatment

square, flat cut semi-circular bulging out

CHSUCB Embellishment of Basic ShapeEmbellishment of Basic Shape

color backgroundtexture

CHSUCB A Simple Scherk-Collins ToroidA 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

CHSUCB Also a Scherk-Collins ToroidAlso 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

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

CHSUCB 1-story Scherk Tower1-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

CHSUCB 180º Arch = Half a Scherk Toroid180º 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

CHSUCB 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.

Use closed-form hyperbolic approximation.

CHSUCB V-artV-art

VirtualGlassScherkTowerwithMonkeySaddles

(Radiance 40 hours)

Jane Yen

CHSUCB How to Obtain a How to Obtain a RealReal Sculpture ? 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.

CHSUCB Slices through “Minimal Trefoil”Slices through “Minimal Trefoil”

50% 10%23%30%

45% 5%20%27%

35% 2%15%25%

CHSUCB 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.

CHSUCB 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)

CHSUCB Another Joint SculptureAnother Joint Sculpture

Heptoroid

CHSUCB Heptoroid Heptoroid ( from Sculpture Generator I )( from Sculpture Generator I )

Cross-eye stereo pair

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

Assembly of the precut boards

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

Forming a continuous smooth edge

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

Smoothing the whole surface

CHSUCB 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

CHSUCB SFF (Solid Free-form Fabrication)SFF (Solid Free-form Fabrication)

Monkey-SaddleCinquefoil

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

CHSUCB Fused Deposition Modeling (FDM)Fused Deposition Modeling (FDM)

CHSUCB Looking into the FDM MachineLooking into the FDM Machine

CHSUCB Zooming into the FDM MachineZooming into the FDM Machine

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

Stereo-lithography master

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

bronze cast, gold plated

CHSUCB

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

CHSUCB Brent Collins’ TrefoilBrent Collins’ Trefoil

CHSUCB Part IIIPart III

Developing Parameterized

Sculpture Families

CHSUCB Family of Symmetrical TrefoilsFamily of Symmetrical Trefoils

W=2

W=1

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

CHSUCB Close-up of Some TrefoilsClose-up of Some Trefoils

B=1 B=2 B=3

Varying the number of branches, the order of the saddles.

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

W=1 (Warp) W=2

CHSUCB Exploring New Ideas: W=2Exploring New Ideas: W=2

Going around the loop twice ...

… resulting in an interwoven structure.

CHSUCB 9-story Intertwined Double Toroid9-story Intertwined Double Toroid

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

CHSUCB 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

CHSUCB Note:Note:

The computer becomesan amplifier / acceleratorfor the creative process.

CHSUCB

Inspiration: Inspiration:

Brent Collins’ Brent Collins’

“Pax Mundi”“Pax Mundi”

CHSUCB Keeping up with Brent ...Keeping up with Brent ...

Sculpture Generator I can only do warped Scherk towers,not able to describe a shape like Pax Mundi.

Need a more general approach ! Use the SLIDE modeling environment

(developed at U.C. Berkeley by J. Smith)to capture the paradigm of such a sculpturein a procedural form. Express it as a computer program

Insert parameters to change salient aspects / features of the sculpture

First: Need to understand what is going on

CHSUCB Sculptures by Naum GaboSculptures by Naum Gabo

Pathway on a sphere:

Edge of surface is like seam of tennis ball;

2-period Gabo curve.

CHSUCB 2-period Gabo curve2-period Gabo curve

Approximation with quartic B-splinewith 8 control points per period,but only 3 DOF are used.

CHSUCB 4-period Gabo curve4-period Gabo curve

Same construction as for as for 2-period curve

CHSUCB ““Pax Mundi” RevisitedPax Mundi” Revisited

Can be seen as:

Amplitude modulated, 4-period Gabo curve

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

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

L2 L3 L4 L5

CHSUCB Via Globi 3 (Stone)Via Globi 3 (Stone)

Wilmin Martono

CHSUCB Via Globi 5 (Wood)Via Globi 5 (Wood)

Wilmin Martono

CHSUCB Via Globi 5 (Gold)Via Globi 5 (Gold)

Wilmin Martono

CHSUCB Extending the ParadigmExtending the Paradigm

Try to Expand the Sculpture Family:

Aim for more highly convoluted paths,

maintain high degree of symmetry.

Need a better tool to draw on sphere …

CHSUCB Circle Splines on the SphereCircle Splines on the Sphere

Examples from Jane Yen’s Editor Program

CHSUCB Via Globi -- Virtual DesignVia Globi -- Virtual Design

Wilmin Martono

CHSUCB ““Maloja” -- FDM partMaloja” -- FDM part

A rather winding Swiss mountain pass road in the upper Engadin.

CHSUCB ““Stelvio”Stelvio”

An even more convoluted alpine pass in Italy.

CHSUCB ““Altamont”Altamont”

Celebrating American multi-lane highways.

CHSUCB ““Lombard”Lombard”

A very famous crooked street in San Francisco

Note that I switched to a flat ribbon.

CHSUCB Part IVPart IV

Using Virtual Shapes and Physical 3D Models

for Sculpture Optimization

CHSUCB Another Inspiration by B. CollinsAnother Inspiration by B. Collins

CHSUCB Collin’s Conceptual DesignCollin’s Conceptual Design

SWEEP CURVE (FOR DOUBLE CYLINDER)IS COMPOSED OF 4 IDENTICAL SEGMENTS,FOLLOWS THE SURFACE OF A SPHERE.

CHSUCB Reconstruction / Analysis (v1)Reconstruction / Analysis (v1)

AWKWARD ALIGNMENT

FROM THE FDM MACHINE

CHSUCB Further Explorations (v2: add twist)Further Explorations (v2: add twist)

CHSUCB A More Complex Design (v3)A More Complex Design (v3)

CHSUCB Verification with 3D Model (v4)Verification with 3D Model (v4)

GALAPAGOS-4 (SIDE VIEW)

CHSUCB Fine-tuned Final(?) Version (v5)Fine-tuned Final(?) Version (v5)

CHSUCB Galapagos-6 in the MakingGalapagos-6 in the Making

CHSUCB Galapagos-6 (v6)Galapagos-6 (v6)

CHSUCB Conclusions (1)Conclusions (1)

Virtual Design / Prototyping is a novel medium (to artists).

It can play an important role -- even for traditional sculptors:

it can save time and labor, and

allows to tackle sculptures of a complexitythat manual techniques could not conquer.

CHSUCB 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.

CHSUCB Conclusions (3)Conclusions (3)

Rapid prototyping (layered fabrication)must now be considered a new facetin the spectrum of MM technologies.

It provides tangible (high-quality haptic)output for objects with which usersmay want to interact.

Even for sculptures(intended primarily for visual enjoyment)the physical maquette disclosessubtle geometrical features that arenot visible in the virtual rendering.

CHSUCB AcknowledgementsAcknowledgements

Brent Collinsfor his inspiring artworkand many stimulating discussions.

Jordan Smith,Jane Yen,Human Meshkin,for developing some of the software modules that I am using in my work.

CHSUCB Questions ?Questions ?

THE ENDTHE END

CHSUCB

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CHSUCB 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 !

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