CS 285

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CS 285

• Analogies from 2D to 3D Exercises in
  Disciplined Creativity
• Carlo H. Séquin
• University of California, Berkeley

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Motivation 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 ?
• Whats a good solution to a problem ?
• How do we know when we are done ?

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Shockleys 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
  combinationsonly 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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Shockleys Model of Creativity

• ACOR
• Key Attributes
• Comparison Operators
• Orderly Relationships
• Quantum of conceptual ideas ?

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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.

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Geometric 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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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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The 2D Hilbert Curve
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Artists Use of the Hilbert Curve

• Helaman Ferguson, Umbilic Torus NC,silicon
  bronze, 27x27x9 in., SIGGRAPH86

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Design Problem 3D Hilbert Curve

• What are the plausible constraints ?
• 3D array of 2n x 2n x 2n 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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Construction of 3D Hilbert Curve
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Design 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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Student Solutions

• see foils ...

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More than One Solution !

• gtgtgt 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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The Borromean Rings in 3D

• Borromean Rings vs. Tangle of 3 Rings

No pair of rings interlock!
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Artists Realization of Bor. Tangle

• Genesis by John Robinson

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Artists 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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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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Artistic Tangle of 4 Triangles
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Abstract Interlock-Analysis

• How should the rings relate to one another ?

wraps around
3 loops ? cyclical relationship 4 loops ? no
symmetrical solution 5 loops ? every loop
encircles two others 4 loops ? has an
asymmetrical solution
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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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Parameter Adjustments in SLIDE
WIDTH
LENGTH
ROUND
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5-loop Tangle -- made with FDM
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Alan Holdens 4-loop Tangle
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Wood models Borrom. 4-loops

• see models...

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Other Tangles by Alan Holden

• 10 MutuallyInterlocking Triangles
• Use 30 edge-midpoints of dodecahedron.

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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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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 Bills 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 (?) -- Winks 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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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.

C2
S2
Mz
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My 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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My Preferred 3D Yin-Yang

• SLA parts

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Design Problem 3D Spiral
Logarithmic Spiral

• Do this in 3D !

Looking for a curveAsimovs 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 trianglesand edges for windows
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Spiral Surface Paper Model
CHS 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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Spiral Surface CAD Model for SFF
Jane Yen
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Conclusions

• 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.