CS 285, Fall 2007CS 285, Fall 2007

Analogies from 2D to 3D

Carlo H. Séquin

University of California, Berkeley

Exercises to Stimulate Creative Thinking

Do this in 3D !

3D Yin-Yang Solutions:3D Yin-Yang Solutions:Two congruent partsTwo congruent parts(Fall 1997)(Fall 1997)

A. Hsu: Clay Model R. Hillaire: R. Hillaire: Acrylite Model

J. Smith: Computer Model

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

Max Bill’s SolutionMax Bill’s Solution

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

Perfection ? -- Cyclide solution

Yin-Yang VariantsYin-Yang Variants

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

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.

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

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 !

3D Yin-Yang 3D Yin-Yang :: Two mirror partsTwo mirror parts

Stereolithography models (Séquin 1999)

The 2D Hilbert Curve (1891)The 2D Hilbert Curve (1891)A plane-filling Peano curve

Do This In 3 D !

Construction of 3D Hilbert CurveConstruction of 3D Hilbert Curve

Construction of 3D Hilbert CurveConstruction of 3D Hilbert Curve

Use this element with proper orientation, mirroring.

Typical Early Student SolutionTypical Early Student Solution

Design Flaws:

2 collinear segments

less than maximal symmetry

4 coplanar segments

D. Garcia, and T. Eladi (1994)

Jane Yen: “Jane Yen: “Hilbert Radiator PipeHilbert Radiator Pipe” ” (2000)(2000)

Flaws( from a sculptor’s . point of view ):

4 coplanar segments

Not a closed loop

Broken symmetry

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

others ... ?

More than one acceptable solution !

Basic Element, Lowest LevelBasic Element, Lowest Level

not this – but this

avoid 4 coplanar segments !

Plastic Model (from FDM) (1998)Plastic Model (from FDM) (1998)

Support removal can be tedious, difficult !

The Next Level of RecursionThe Next Level of Recursion

Presented a challenge to remove supports.

Resulted in a flimsy, spongy model.

Would like to have a more durable model in metal.

2006: Metal Sculpture in Exhibit2006: Metal Sculpture in Exhibit

Design:Design: closed loopclosed loop maximal symmetrymaximal symmetry at most 3 coplanar segmentsat most 3 coplanar segments

CREATIVITYCREATIVITY

PLAY