1. CS 285, Fall 2007 Analogies from 2D to 3D Carlo H. Séquin University of California, Berkeley

2. Exercises to Stimulate Creative Thinking Do this in 3D !

3. 3D Yin-Yang Solutions: Two congruent parts (Fall 1997) A. Hsu: Clay Model R. Hillaire: R. Hillaire: Acrylite Model J. Smith: Computer Model

4. 3D Yin-Yang (Robert Hillaire)

5. Max Bill’s Solution

6. Many Solutions for 3D Yin-Yang u Most popular: -- Max Bill solution u Unexpected: -- Splitting sphere in 3 parts u Hoped for: -- Semi-circle sweep solutions u Machinable: -- Torus solution u Perfection ? -- Cyclide solution

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

8. 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.

9. Yin-Yang Symmetries u 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

10. My Preferred 3D Yin-Yang The Cyclide Solution: u Yin-Yang is built from cyclides only ! What are cyclides ? u Spheres, Cylinders, Cones, and all kinds of Tori (Horn tori, spindel tory). u Principal lines of curvature are circles. u Minumum curvature variation property !

11. 3D Yin-Yang : Two mirror parts  Stereolithography models (Séquin 1999)

12. The 2D Hilbert Curve (1891) A plane-filling Peano curve Do This In 3 D !

13. Construction of 3D Hilbert Curve

14. u Use this element with proper orientation, mirroring.

15. Typical Early Student Solution Design Flaws: u 2 collinear segments u less than maximal symmetry u 4 coplanar segments D. Garcia, and T. Eladi (1994)

16. Jane Yen: “Hilbert Radiator Pipe” (2000) Flaws ( from a sculptor’s. point of view ): u 4 coplanar segments u Not a closed loop u Broken symmetry

17. Design Choices: 3D Hilbert Curve What are the things one might optimize ? u Maximal symmetry u Overall closed loop u No consecutive collinear segments u No (3 or 4 ?) coplanar segment sequence u others... ?  More than one acceptable solution !

18. Basic Element, Lowest Level u not this – but this avoid 4 coplanar segments !

19. Plastic Model (from FDM) (1998) u Support removal can be tedious, difficult !

20. The Next Level of Recursion u Presented a challenge to remove supports. u Resulted in a flimsy, spongy model. u Would like to have a more durable model in metal.

21. 2006: Metal Sculpture in Exhibit 2006: Metal Sculpture in ExhibitDesign: u closed loop u maximal symmetry u at most 3 coplanar segments

22. CREATIVITY PLAY