Analogies from 2D to 3D

CS 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:
Two congruent parts
(Fall 1997)
J. Smith: Computer Model
A. Hsu: Clay Model
R. Hillaire: Acrylite Model
3D Yin-Yang (Robert Hillaire)
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
Perfection ? -- Cyclide solution
Yin-Yang Variants
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.
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.
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 !
3D Yin-Yang : Two mirror parts
 Stereolithography
models (Séquin 1999)
The 2D Hilbert Curve (1891)
A plane-filling Peano curve
Do This In 3 D !
Construction of 3D Hilbert Curve
Construction of 3D Hilbert Curve
Use this element with proper orientation, mirroring.
Typical Early Student Solution
Design Flaws:
D. Garcia, and T. Eladi (1994)
2 collinear segments
less than maximal
4 coplanar segments
Jane Yen: “Hilbert Radiator Pipe” (2000)
( from a sculptor’s
point of view ):
4 coplanar segments
Not a closed loop
Broken symmetry
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
others ... ?
 More than one acceptable solution !
Basic Element, Lowest Level
 not
this – but this
avoid 4 coplanar segments !
Plastic Model (from FDM) (1998)
Support removal can be tedious, difficult !
The 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 Exhibit
 closed loop
 maximal symmetry
 at most 3 coplanar segments