CS 285 Analogies from 2D to 3D Exercises in Disciplined Creativity Carlo H. Séquin University of California, Berkeley

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 ?  What’s a good solution to a problem ?  How do we know when we are done ?

Shockley’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”)

“ACOR”:  Key Attributes  Comparison Operators  Orderly Relationships = Quantum of conceptual ideas ? Shockley’s Model of Creativity

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.

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.

Selected Examples Examples drawn from graduate courses in geometric modeling:  3D Hilbert Curve  Borromean Tangles  3D Yin-Yang  3D Spiral Surface

The 2D Hilbert Curve

Artist’s Use of the Hilbert Curve Helaman Ferguson, Umbilic Torus NC, silicon bronze, 27x27x9 in., SIGGRAPH’86

Design Problem: 3D Hilbert Curve What are the plausible constraints ?  3D array of 2 n 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)

Construction of 3D Hilbert Curve

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 ?

== Student Solutions  see foils...

= More than One Solution !  >>> Compare wire models  What are the tradeoffs ?

3D Hilbert Curve -- 3rd Generation  Programming,  Debugging,  Parameter adjustments,  Display through SLIDE (Jordan Smith)

Hilbert_512 Radiator Pipe Jane Yen

3D Hilbert Curve, Gen (FDM)

The Borromean Rings Borromean Rings vs. Tangle of 3 Rings No pair of rings interlock!

The Borromean Rings in 3D Borromean Rings vs. Tangle of 3 Rings No pair of rings interlock!

Artist’s Realization of Bor. Tangle Genesis by John Robinson

Artist’s Realization of Bor. Tangle Creation by John Robinson

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.

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.

Artistic Tangle of 4 Triangles

Abstract Interlock-Analysis How should the rings relate to one another ? 3 loops:  cyclical relationship 4 loops:  no symmetrical solution 5 loops:  every loop encircles two others 4 loops:  has an asymmetrical solution CD A A A A B B B B C C CD E D = “wraps around”

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 !

Parameter Adjustments in SLIDE WIDTH LENGTHROUND

5-loop Tangle -- made with FDM

Alan Holden’s 4-loop Tangle

Wood models: Borrom. 4-loops  see models...

Other Tangles by Alan Holden 10 Mutually Interlocking Triangles:  Use 30 edge- midpoints of dodecahedron.

More Tangle Models  6 pentagons in equatorial planes.  6 squares in offset planes  4 triangles in offset planes (wood models)  10 triangles

Introduction to the Yin-Yang  Religious symbol  Abstract 2D Geometry

Design Problem: 3D Yin-Yang What this might mean...  Subdivide a sphere into two halves. Do this in 3D !

3D Yin-Yang (Amy Hsu) Clay Model

3D Yin-Yang (Robert Hillaire)

Acrylite Model

Max Bill’s Solution

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

Yin-Yang Variants http//korea.insights.co.kr/symbol/sym_1.html

Yin-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.

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. C2C2 S2S2 MzMz

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 !

My Preferred 3D Yin-Yang  SLA parts

Design 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

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.

Pipe-cleaner Skeletons Three spirals and coordinate system Added surface triangles and edges for windows

Spiral Surface: Paper Model CHS 1999

Spiral Surface CAD Model

Jane Yen

Spiral Surface CAD Model Jane Yen

Spiral Surface CAD Model for SFF Jane Yen

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 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.