1. Schloss Dagstuhl, September 2014 Shape Representation Carlo H. Séquin University of California, Berkeley “LEGO Knot” and an Optimization Problem in a High-Dimensional Discrete Solution Space

2. Discussion Points: Shape representation issues at the start and conclusion of designing RP models u Focus on HCI difficulties and CAD problems, at the start and end of a design / modeling project: u How to get started?  How to get your ideas into the CAD system. u How to finish?  How to get your model properly 3D printed.

3. User-Guided Inverse 3D Modeling u Few designs start from scratch in a vacuum. Often there is a previous artifact that provides inspiration or may even be close enough so that some high-level redesign might be the most effective approach. Unfortunately there may be no CAD files available or they may be at such a low level (100’000 triangles) that it is not a good starting point for a major redesign. u “User-Guided Inverse 3D Modeling” is an approach to re-create a well-structured, high-level, parameterized, procedural description of some geometry very close to the inspirational artifact. Its hierarchical structure and the degree of its parameterization are imposed with some high-level instructions by the designer, so that the resulting description is most appropriate to make the intended design changes. u REF: http://www.cs.berkeley.edu/~sequin/UGI3DM/index.html

4. Another Issue... “LEGO ® ” Knots EECS Computer Science Division University of California, Berkeley Carlo H. Séquin

5. Inspiration: Henk van Putten “Borsalino” “Interaction” Sculptural forms put together from a few modular shapes

6. Geometry of the Borsalino u Just 2 geometrical components:  3 semi-circular end-caps (orange)  6 curved connectors, bending through 45º == a square cross section swept along 9 circular arcs.

7. The Wonders of Rapid-Prototyping Two modular components can form the Borsalino Connector R=2.4142 End-Cap R=1.0

8. Hands-on Sculpting André Eveline Lorenzo

9. Inspiration: Jon Krawczyk 303 2 nd Street, San Francisco

10. Inspiration: Paul Bloch “After Wright” (Guggenheim, NYC)

11. LEGO ® DUPLO u Match interface

12. More “User-Studies” (3 rd Gen.) Sienna (5) and Elise (7)

13. My Personal Quest: What kind of parts does it take to make Mathematical Knots with nice graceful curvature and smooth loop closure ?

14. Real Knots: Trefoil (3_1) u One custom-designed piece (magenta) for smooth closure D 3 symmetry

15. Trefoil Knot (3_1)

16. Real Knots: Figure-8 Knot (4_1) u Two new pieces (magenta, red) for smooth closure 4-fold glide symmetry

17. Figure-8 Knot (4_1)

18. Composition Problems in a Discrete Solution Space u Similar to the Zome-Tool Approximation: u Suppose we restrict ourselves to just using one single module! u Can we build elegant and symmetrical knots? Single-Module Knots

19. Richard Zawitz: Museum Tangle (1982) u Pliable UnKnot made from 18 quarter-torus segments

20. Naef Wooden Toys Caterpillar

21. Knots Made from ONE Module M. Zawidzki & K. Nishinari: u Problems: too many elements, lack of symmetry, self-intersections, bad loop closure.

22. Forming Closed Loops Is Difficult ! M. Zawidzki & K. Nishinari:

23. A First Try on a Figure-8 Knot u Composed of 4×10 wedge elements (4-fold symmetry) u Does not properly close! ( 6 DoF: x, y, z, 3 angles ) 12-gon profile

24. The Module Chosen u 16-gon cross section (finer control of azimuth) u 30°bending angle in module (fewer overall modules) u r/R = 0.3 (pipe radius / bending radius  “wiggle space”)

25. A First Batch of 20 Modules u Out of the “Uprint” FDM machine

26. Simplest “ModKnot”: K 3_1 Trefoil Knot: K 3_1; D 3 symmetry; uses 33 modules

27. Trefoil Knot Sculpture 33 parts

28. Simple “ModKnots”: K 4_1 Figure-8 Knot: K 4_1; S 4 symmetry; uses 40 modules Even with the computer plan, this is difficult to assemble!

29. Figure-8 Knot Sculpture 40 parts

30. Cinquefoil Knot: K 5_1 D 5 symmetry 50 modules

31. Mathematical Links Borromean Link

32. Forming Mathematical Knots and Links Challenging Computational Issue: What is the right algorithm to find the best solution for any such problem in its high-dimensional solution space ?