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UCBUCB CS 285CS 285

Analogies from 2D to 3D

Exercises in Disciplined Creativity

Carlo H. Séquin

University of California, Berkeley

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UCBUCB Motivation — Puzzling QuestionsMotivation — Puzzling Questions

What is creativity ?

Where do novel ideas come from ?

Are there any truly novel ideas ?Or are they evolutionary developments, and just combinations of known ideas ?

How do we evaluate open-ended designs ?

What’s a good solution to a problem ?

How do we know when we are done ?

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UCBUCB Shockley’s Model of CreativityShockley’s Model of Creativity

We possess a pool of known ideas and models.

A generator randomly churns up some of these.

Multi-level filtering weeds out poor combinations;only a small fraction percolates to consciousness.

We critically analyze those ideas with left brain.

See diagram (from inside front cover of “Mechanics”)

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UCBUCB

“ACOR”: Key

Attributes

Comparison Operators

Orderly Relationships

= Quantum of conceptual ideas ?

Shockley’s Model of CreativityShockley’s Model of Creativity

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UCBUCB Human Mind vs. ComputerHuman Mind vs. Computer

The human mind has outstanding abilities for:

pattern recognition,

detecting similarities,

finding analogies,

making simplified mental models,

carrying solutions to other domains.

It is worthwhile (& possible) to train this skill.

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UCBUCB Geometric Design ExercisesGeometric Design Exercises

Good playground to demonstrate and exercise above skills.

Raises to a conscious level the many activities that go on when one is searching for a solution to an open-ended design problem.

Nicely combines the open, creative search processes of the right brain and the disciplined evaluation of the left brain.

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UCBUCB Selected ExamplesSelected Examples

Examples drawn from graduate courses in geometric modeling:

3D Hilbert Curve

Borromean Tangles

3D Yin-Yang

3D Spiral Surface

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UCBUCB The 2D Hilbert CurveThe 2D Hilbert Curve

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UCBUCB Artist’s Use of the Hilbert CurveArtist’s Use of the Hilbert Curve

Helaman Ferguson, Umbilic Torus NC,silicon bronze, 27x27x9 in., SIGGRAPH’86

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UCBUCB Design Problem: 3D Hilbert CurveDesign Problem: 3D Hilbert Curve

What are the plausible constraints ?

3D array of 2n x 2

n x 2

n vertices

Visit all vertices exactly once

Aim for self-similarity

No long-distance connections

Only nearest-neighbor connections

Recursive formulation (to go to arbitrary n)

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UCBUCB Construction of 3D Hilbert CurveConstruction of 3D Hilbert Curve

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UCBUCB Design Choices: 3D Hilbert CurveDesign Choices: 3D Hilbert Curve

What are the things one might optimize ?

Maximal symmetry

Overall closed loop

No consecutive collinear segments

No (3 or 4 ?) coplanar segment sequence

Closed-form recursive formulation

others ?

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UCBUCB == Student Solutions== Student Solutions

see foils ...

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UCBUCB = More than One Solution != More than One Solution !

>>> Compare wire models

What are the tradeoffs ?

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UCBUCB 3D Hilbert Curve -- 3rd Generation3D Hilbert Curve -- 3rd Generation

Programming,

Debugging,

Parameter adjustments,

Display

through SLIDE

(Jordan Smith)

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UCBUCB Hilbert_512 Radiator PipeHilbert_512 Radiator Pipe

Jane Yen

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UCBUCB 3D Hilbert Curve, Gen. 2 -- (FDM)3D Hilbert Curve, Gen. 2 -- (FDM)

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UCBUCB The Borromean RingsThe Borromean Rings

Borromean Rings vs. Tangle of 3 Rings

No pair of rings interlock!

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UCBUCB The Borromean Rings in 3DThe Borromean Rings in 3D

Borromean Rings vs. Tangle of 3 Rings

No pair of rings interlock!

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UCBUCB Artist’s Realization of Bor. TangleArtist’s Realization of Bor. Tangle

Genesis by John Robinson

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UCBUCB Artist’s Realization of Bor. TangleArtist’s Realization of Bor. Tangle

Creation by John Robinson

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UCBUCB Design Task: Borromean TanglesDesign Task: Borromean Tangles

Design a Borromean Tangle with 4 loops;

then with 5 and more loops …

What this might mean:

Symmetrically arrange N loops in space.

Study their interlocking patterns.

Form a tight configuration.

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UCBUCB Finding a “Tangle" with 4 LoopsFinding a “Tangle" with 4 Loops

Ignore whether the loops interlock or not.

How does one set out looking for a solution ?

Consider tetrahedral symmetry.

Place twelve vertices symmetrically.

Perhaps at mid-points of edges of a cube.

Connect them into triangles.

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UCBUCB Artistic Tangle of 4 TrianglesArtistic Tangle of 4 Triangles

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UCBUCB Abstract Interlock-AnalysisAbstract Interlock-Analysis

How should the rings relate to one another ?

3 loops: cyclical relationship 4 loops: no symmetrical solution5 loops: every loop encircles two others4 loops: has an asymmetrical solution

CD

AA

A A

B

BB

BC CCD

ED

= “wraps around”

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UCBUCB Construction of 5-loop TangleConstruction of 5-loop Tangle

Construction based on dodecahedron.

Group the 20 vertices into 5 groups of 4,

to yield 5 rectangles,which pairwise do not interlock !

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UCBUCB Parameter Adjustments in SLIDEParameter Adjustments in SLIDE

WIDTH

LENGTH ROUND

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UCBUCB 5-loop Tangle -- made with FDM5-loop Tangle -- made with FDM

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UCBUCB Alan Holden’s 4-loop TangleAlan Holden’s 4-loop Tangle

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UCBUCB Wood models: Borrom. 4-loopsWood models: Borrom. 4-loops

see models...

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UCBUCB Other Tangles by Alan HoldenOther Tangles by Alan Holden

10 MutuallyInterlocking Triangles:

Use 30 edge-midpoints of dodecahedron.

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UCBUCB More Tangle ModelsMore Tangle Models

6 pentagons in equatorial planes.

6 squares in offset planes

4 triangles in offset planes (wood models)

10 triangles

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UCBUCB Introduction to the Yin-YangIntroduction to the Yin-Yang

Religious symbol

Abstract 2D Geometry

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UCBUCB Design Problem: 3D Yin-YangDesign Problem: 3D Yin-Yang

What this might mean ...

Subdivide a sphere into two halves.

Do this in 3D !

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UCBUCB 3D Yin-Yang (Amy Hsu)3D Yin-Yang (Amy Hsu)

Clay Model

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UCBUCB 3D Yin-Yang (Robert Hillaire)3D Yin-Yang (Robert Hillaire)

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UCBUCB 3D Yin-Yang (Robert Hillaire)3D Yin-Yang (Robert Hillaire)

Acrylite Model

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UCBUCB Max Bill’s SolutionMax Bill’s Solution

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UCBUCB Many Solutions for 3D Yin-YangMany Solutions for 3D Yin-Yang

Most popular: -- Max Bill solution

Unexpected: -- Splitting sphere in 3 parts

Hoped for: -- Semi-circle sweep solutions

Machinable: -- Torus solution

Earliest (?) -- Wink’s solution

Perfection ? -- Cyclide solution

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UCBUCB Yin-Yang VariantsYin-Yang Variants

http//korea.insights.co.kr/symbol/sym_1.html

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UCBUCB Yin-Yang VariantsYin-Yang Variants

http//korea.insights.co.kr/symbol/sym_1.html

The three-part t'aeguk symbolizes heaven, earth, and humanity. Each part is separate but the three parts exist in unity and are equal in value. As the yin and yang of the Supreme Ultimate merge and make a perfect circle, so do heaven, earth and humanity create the universe. Therefore the Supreme Ultimate and the three-part t'aeguk both symbolize the universe.

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UCBUCB Yin-Yang SymmetriesYin-Yang Symmetries

From the constraint that the two halves should be either identical or mirror images of one another, follow constraints for allowable dividing-surface symmetries.

C2 S2Mz

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UCBUCB My Preferred 3D Yin-YangMy Preferred 3D Yin-Yang

The Cyclide Solution:

Yin-Yang is built from cyclides only !

What are cyclides ?

Spheres, Cylinders, Cones, andall kinds of Tori (Horn tori, spindel tory).

Principal lines of curvature are circles.

Minumum curvature variation property !

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UCBUCB My Preferred 3D Yin-YangMy Preferred 3D Yin-Yang

SLA parts

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UCBUCB Design Problem: 3D SpiralDesign Problem: 3D Spiral

Do this in 3D !

Logarithmic Spiral

But we are looking for a surface ! Not just a spiral roll of paper ! Should be spirally in all 3 dimensions. Ideally: if cut with 3 perpendicular planes, spirals should show on all three of them !

Looking for a curve:Asimov’s Grand Tour

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UCBUCB Searching for a Spiral SurfaceSearching for a Spiral Surface

Steps taken:

Thinking, sketching (not too effective);

Pipe-cleaner skeleton of spirals in 3D;

Connecting the surface (need holes!);

Construct spidery paper model;

CAD modeling of one fundamental domain;

Virtual images with shading;

Physical 3D model with FDM.

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UCBUCB Pipe-cleaner SkeletonsPipe-cleaner Skeletons

Three spirals and coordinate system

Added surface trianglesand edges for windows

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UCBUCB Spiral Surface: Paper ModelSpiral Surface: Paper Model

CHS 1999

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UCBUCB Spiral Surface CAD ModelSpiral Surface CAD Model

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UCBUCB Spiral Surface CAD ModelSpiral Surface CAD Model

Jane Yen

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UCBUCB Spiral Surface CAD ModelSpiral Surface CAD Model

Jane Yen

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UCBUCB Spiral Surface CAD Model for SFFSpiral Surface CAD Model for SFF

Jane Yen

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UCBUCB ConclusionsConclusions

Examples of dialectic design process:

Multi-”media” thinking and experimentation for finding creative solutions to open-ended design problems;

“Ping-pong” action between idea generationand checking them for their usefulness;

Synergy between intuitive associations and analytical reasoning.

Forming bridges between art and logic,i.e., between the right brain and left brain.