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CHS
UCB
CS 285
Analogies from 2D to 3D
Exercises in Disciplined Creativity
Carlo H. Séquin
University of California, Berkeley
Motivation — Puzzling Questions
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UCB

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 ?
Shockley’s Model of Creativity
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
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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Shockley’s Model of Creativity
UCB
“ACOR”:

Key
Attributes

Comparison
Operators

Orderly
Relationships
= Quantum of
conceptual
ideas ?
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UCB
Human 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.
CHS
Geometric Design Exercises
UCB

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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UCB
Selected Examples
Examples drawn from
graduate courses in geometric modeling:
 3D
Hilbert Curve
 Borromean
Tangles
 3D
Yin-Yang
 3D
Spiral Surface
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UCB
The 2D Hilbert Curve
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Artist’s Use of the Hilbert Curve
Helaman Ferguson, Umbilic Torus NC,
silicon bronze, 27x27x9 in., SIGGRAPH’86
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Design Problem: 3D Hilbert Curve
What are the plausible constraints ?
n
n
n

3D array of 2 x 2 x 2 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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UCB
Construction of 3D Hilbert Curve
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Design Choices: 3D Hilbert Curve
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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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UCB
== Student Solutions

see foils ...
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= More than One Solution !

>>> Compare wire models

What are the tradeoffs ?
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3D Hilbert Curve -- 3rd Generation

Programming,

Debugging,

Parameter
adjustments,

Display
through SLIDE
(Jordan Smith)
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Hilbert_512 Radiator Pipe
Jane Yen
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3D Hilbert Curve, Gen. 2 -- (FDM)
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The Borromean Rings
Borromean Rings vs. Tangle of 3 Rings
No pair of rings interlock!
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UCB
The Borromean Rings in 3D
Borromean Rings vs. Tangle of 3 Rings
No pair of rings interlock!
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UCB
Artist’s Realization of Bor. Tangle
Genesis by John Robinson
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Artist’s Realization of Bor. Tangle
Creation by John Robinson
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Design 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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UCB
Finding 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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UCB
Artistic Tangle of 4 Triangles
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Abstract Interlock-Analysis
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How should the rings relate to one another ?
A
A
A
B
A
E
C
B
D
B
D
C
D
C
= “wraps around”
3 loops: 
4 loops: 
5 loops: 
4 loops: 
cyclical relationship
no symmetrical solution
every loop encircles two others
has an asymmetrical solution
C
B
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Construction 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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UCB
Parameter Adjustments in SLIDE
WIDTH
LENGTH
ROUND
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5-loop Tangle -- made with FDM
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UCB
Alan Holden’s 4-loop Tangle
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Wood models: Borrom. 4-loops

see models...
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Other Tangles by Alan Holden
10 Mutually
Interlocking
Triangles:

Use 30 edgemidpoints of
dodecahedron.
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UCB
More 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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UCB
Introduction to the Yin-Yang

Religious symbol

Abstract 2D Geometry
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Design Problem: 3D Yin-Yang
Do this in 3D !
What this might mean ...

Subdivide a sphere into two halves.
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3D Yin-Yang (Amy Hsu)
Clay Model
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3D Yin-Yang (Robert Hillaire)
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3D Yin-Yang (Robert Hillaire)
Acrylite Model
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Max Bill’s Solution
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Many 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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Yin-Yang Variants
http//korea.insights.co.kr/symbol/sym_1.html
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Yin-Yang Variants
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.
http//korea.insights.co.kr/symbol/sym_1.html
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UCB
Yin-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.
Mz
C2
S2
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UCB
My Preferred 3D Yin-Yang
The Cyclide Solution:

Yin-Yang is built from cyclides only !
What are cyclides ?

Spheres, Cylinders, Cones, and
all kinds of Tori (Horn tori, spindel tory).

Principal lines of curvature are circles.

Minumum curvature variation property !
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UCB
My Preferred 3D Yin-Yang

SLA parts
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UCB
Design Problem: 3D Spiral
Logarithmic Spiral
Do this in 3D !
Looking for a curve:
Asimov’s Grand Tour
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 !
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Searching 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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Pipe-cleaner Skeletons
Three spirals and
coordinate system
Added surface triangles
and edges for windows
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Spiral Surface: Paper Model
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1999
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Spiral Surface CAD Model
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Spiral Surface CAD Model
Jane Yen
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Spiral Surface CAD Model
Jane Yen
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UCB
Spiral Surface CAD Model for SFF
Jane Yen
CHS
Conclusions
UCB
Examples of dialectic design process:

Multi-”media” thinking and experimentation
for finding creative solutions
to open-ended design problems;

“Ping-pong” action between idea generation
and 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.