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UCBUCB Microsoft MOSAIC Microsymposium 2000Microsoft MOSAIC Microsymposium 2000

Analysis and Synthesis of Intuitively Conceived Geometrical Art

Carlo H. Séquin

University of California, Berkeley

Brent Collins

Gower, Missouri

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UCBUCB Roots of My PassionRoots of My Passion

My love for geometry and abstract sculptureemerged long long before I learned to play with computers.

Acknowledgement to the artists who have strongly influenced me ...

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UCBUCB

Appreciation of Simple Appreciation of Simple Geometric Forms (`50s, `60s )Geometric Forms (`50s, `60s )

Alexander Calder,

Naum Gabo,

Max Bill:

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UCBUCB Intro to Computer Graphics (`70s)Intro to Computer Graphics (`70s)

Ken Knowlton, Lillian Schwarz (at Bell Labs)

Frank Smullin (at D.A.Conference):

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UCBUCB

Granny Granny Knot Knot LatticeLattice

Séquin 1982

BerkeleyUniGrafix

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UCBUCB Complex Compositions (‘80s)Complex Compositions (‘80s)

M. C. Escher,

Bruce Beasly,

George Rickey,

Alan Holden:

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UCBUCB Recent Interactions (`90s)Recent Interactions (`90s)

Charles Perry,

Helaman Ferguson,

Robert Longhurst:

• Brent Collins: Collaboration since 1995

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UCBUCB Leonardo FoilLeonardo Foil

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

George K. Francis with Brent Collins

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UCBUCB Early Brent Collins SculpturesEarly Brent Collins Sculptures

All photos by Phillip Geller

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UCBUCB ==> BRENT COLLINS’ Presentation==> BRENT COLLINS’ Presentation

Early work with saddles ...

through “Hyperbolic Hexagon”

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UCBUCB My Fascination with...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”

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UCBUCB Brent Collins: Stacked SaddlesBrent Collins: Stacked Saddles

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

Normal“biped”saddles

Generalization to higher-order saddles(monkey saddle)

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UCBUCB ““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 ?

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UCBUCB Closing the LoopClosing the Loop

straight

or

twisted

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UCBUCB 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.

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UCBUCB Brent Collins’ Prototyping ProcessBrent Collins’ Prototyping Process

Armature for the "Hyperbolic Heptagon"

Mockup for the "Saddle Trefoil"

Time-consuming ! (1-3 weeks)

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UCBUCB 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 !

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UCBUCB Collins’ Fabrication ProcessCollins’ Fabrication Process

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

Example: “Vox Solis”

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UCBUCB ““Sculpture Generator I”Sculpture Generator I”

Prototyping 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

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UCBUCB === VIDEO === === VIDEO ===

6 min

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UCBUCB Base Geometry: One Scherk StoryBase Geometry: One Scherk Story

Hyperbolic Slices ==> Triangle Strips

precomputed -- then warped into toroid

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UCBUCB Slices through the SculptureSlices through the Sculpture

One thick slicethru “Heptoroid”from which Brent can cut boards and assemble a rough shape.

Shown are top andbottom as well ascuts at 1/4, 1/2, 3/4of one board.

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UCBUCB 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)

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

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UCBUCB Heptoroid Heptoroid ( from Sculpture Generator I )( from Sculpture Generator I )

Cross-eye stereo pair

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

Assembly of the precut boards

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

Forming a continuous smooth edge

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

Smoothing the whole surface

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UCBUCB ==> BRENT COLLINS’ Presentation==> BRENT COLLINS’ Presentation

Hyperbolic Heptagon

Trefoils

Hyperbolic Hexagon II, and Heptoroid

Pax Mundi, Egg, Vox Solis

Atomic Flowers

Pieces with Positive Curvature

(Bronze casting will be described at MOSAIC talk)

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UCBUCB 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

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UCBUCB Sculpture DesignSculpture 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

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UCBUCB FDM Maquette of Solar ArchFDM Maquette of Solar Arch

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UCBUCB Breckenridge CompetitionBreckenridge Competition

2nd place

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

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UCBUCB … … or in Washington D.C.or in Washington D.C.

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UCBUCB V-artV-art

GlassScherkTowerwithMonkeySaddles

Jane Yen

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UCBUCB SFF Maquettes of Future SculpturesSFF Maquettes of Future Sculptures

Monkey-SaddleCinquefoil

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

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

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UCBUCB Fused Deposition Modeling (FDM)Fused Deposition Modeling (FDM)

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UCBUCB Fused Deposition Modeling (FDM)Fused Deposition Modeling (FDM)

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UCBUCB Looking into the FDM MachineLooking into the FDM Machine

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UCBUCB Zooming into the FDM MachineZooming into the FDM Machine

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

Stereo-lithography master

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

bronze cast, gold plated

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UCBUCB Brent’s TrefoilBrent’s Trefoil

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UCBUCB

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

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UCBUCB Family of Symmetrical TrefoilsFamily of Symmetrical Trefoils

W=2

W=1

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

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

W=1 W=2

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UCBUCB 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

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UCBUCB Exploring New IdeasExploring New Ideas

Going twice around the loop ...

Resulting in an interwoven structure.

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UCBUCB 9-story Intertwined Double Toroid9-story Intertwined Double Toroid

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

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UCBUCB Brent Collins: Pax MundiBrent Collins: Pax Mundi

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UCBUCB Keeping up with Brent ...Keeping up with Brent ...

Bent “Scherk towers” is not able to describe a shape like “Pax Mundi.”

Need a broader paradigm !

Use SLIDE modeling environment,it provides a nice combination of procedural modeling and interactivity.

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UCBUCB 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.

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UCBUCB SLIDE Example: Klein BottleSLIDE Example: Klein Bottle

Final Project CS 184, Nerius Landys & Shad Roundy

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UCBUCB SLIDE Example Bug’s LifeSLIDE Example Bug’s Life

Final Project CS 184, David Cheng and James Chow

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UCBUCB 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

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UCBUCB 3D Hilbert Curves (FDM)3D Hilbert Curves (FDM)

Hilbert64 and Hilbert512

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UCBUCB SLIDE-UI for Knot GenerationSLIDE-UI for Knot Generation

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

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UCBUCB Via Globi 5 (Wood)Via Globi 5 (Wood)

Wilmin Martono

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UCBUCB Via Globi 5 (Gold)Via Globi 5 (Gold)

Wilmin Martono

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UCBUCB Via Globi 3 (Stone)Via Globi 3 (Stone)

Wilmin Martono

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UCBUCB Via Globi FamilyVia Globi Family

L2 L3 L4 L5

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UCBUCB Conclusions 1Conclusions 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.

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UCBUCB Conclusions 2Conclusions 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.

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UCBUCB Conclusions 3Conclusions 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 !

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UCBUCB

========= SPARE ========= ========= SPARE ========= ==================================================

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UCBUCB Brent Collins: Spiral StructuresBrent Collins: Spiral Structures

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

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UCBUCB Preparation of Blue PrintsPreparation of Blue Prints

Cut geometrical shape into slices parallel to one axis:

Assemble several profile lines for each board,

and print out on separate sheet at 1:1 size.

==> (Examples from printer on wall)

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UCBUCB The role of the computerThe role of the computer

Eliminate need for prototyping

Exploration of larger domain

Making more complex structures

Better optimization

Precise blue prints

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UCBUCB SLA Maquette of “Solar Arch”SLA Maquette of “Solar Arch”

Back-lighting and photo by Philip Geller

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UCBUCB My Contorted Professional PathMy Contorted Professional Path

65-70: Basel: Physics – Experiment design

70-76: Bell Labs: CCD – Circuit, chip design

76-82: UCB: RISC – CPU architecture design

80-??: Graphics – Modeling & Rendering SW

82-90: VLSI CAD – CAD algorithm design

87-94: Soda Hall – Building design, VR

92-98: Architecture – ArchCAD tool design

95-??: Mech. Eng. – Develop SIF, CyberCut Sculpture – CAD, Virtual Prototyping

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UCBUCB ==Abstract==Abstract

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.Carlo Séquin is a professor at U.C. Berkeley, teachingcomputer graphics, geometric modeling, and computer aided design.Since 1995 he has been collaborating with Collins in the conceptionand design of intricate geometrical shapes that expand the originalwork of Collins. He has developed several procedural generatorprograms that recreate some of the shapes conceived by Collinsand can expand the basic concept in several possible directions.His interactive "sculpture generator" allows a quick perusal ofsome domain of the configuration space, and an optimization ofa given design along the axes of several parameters.Shapes of high artistic merits can then be sliced into 1-inch slabsand a corresponding set of cross sections printed out.Brent Collins uses these templates to precut a set of wood boardsand to preassemble a rough shape from which the final sculptureis then carved. In this way, the intuitive genius of the artist can beamplified by a computer tool that assists with the visualization and faithful reproduction of complex geometrical shapes.

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UCBUCB The Need for a Prototyping ToolThe 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 ?

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UCBUCB 5-loop Tangles (FDM)5-loop Tangles (FDM)

With and Without Overhang Supports

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UCBUCB 3D Yin-Yang (FDM)3D Yin-Yang (FDM)

The two complementary half-spheres