1. BID Lunch, February 25, 2014 “LEGO ® ” Knots EECS Computer Science Division University of California, Berkeley Carlo H. Séquin Michelle Galemmo

2. The Bridges Conference u Mathematical Connections in Art, Music, and Science. u Annually held in July/August since 1998 (4-5 days). u Attracts: mathematicians, scientists, artists, educators, musicians, writers, dancers, weavers, model builders, and computer scientists. u Conference Venues: l Formal paper presentations: regular, short, invited/plenary. l Also: workshops, art gallery, and informal show-and-tell/sell. l Evenings: concerts, theater, movies (with a math connection).

3. Bridges in Many Wonderful Places u Cities: Winfield, Towson, London, Coimbra, Granada, Donostia, Leeuwarden, Enschede, Pecs, …

4. My Favorite Annual Conference: 2014 u BRIDGES Art …

5. My First 20 Bridges Papers

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

7. 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.

8. The Wonders of Rapid-Prototyping Two modular components can form the Borsalino C R=2.4142 E R=1.0

9. Playing with those Two Components

10. The “Loose” Borsalino Two larger types of parts required: 3 end-caps; 6 connectors.

11. Rapid Prototyping with FDM

12. A Look Into the FDM Machine Galapagos 6 sculpture in progress 2 NOZZLES

13. Inexpensive FDM Machine u Afinia-H-Series 3D printer u Has only ONE type of filament. u Support structures: not desirable: l are hard to remove, l leave part surface scarred. u Limited cantilevering is possible to form overhangs and bridges: ( 45º max).

14. SUPPORT Fabrication Issues: Optimize Build Orientation Horizontal tube (flat): filled with support Tube on edge: may tip over, gets squashed Vertical tube: support at flanges Can we orient part so no supports are needed? (All surfaces must be steeper than 45º) SUPPORT

15. Square flanges Tapered flanges Upside-down support no support no support Fabrication Issues: Flanges

16. Fabrication Issues: Curved Connectors Ignore Problem Built-in Wall Extended Taper Chosen Solution! No face steeper than 45º !

17. Fabrication Issues: End-Caps Form a 45º slanted “cathedral ceiling” internally: needs support needs no support

18. Needs some support for central arch ! We wanted to keep inner tube open: Use scaffolding and take the trouble of doing some clean-up! Fabrication Issues: Enlarged End-Caps

19. Parts Catalog So Far u 2 types of end-caps; 3 curved connectors

20. What can we do with those 5 Parts? u Wild and crazy “snakes”, e.g., a Hilbert curve; u Emulation of other Henk van Putten sculptures …

21. Emulations Interaction

22. Many More Possibilities … u Mix and match … D C B A

23. 2 End-Caps plus 3 Connector Types... allow us to build these Twisted 2-Lobe Borsalinos The mathematics does not really work out quite right; they are off by 6%, 15%, and 6% in the radius ratios.

24. Flipped-end Borsalino Put extension between connectors move backward stretched move up, forward needs diagonal end-cap move forward move backward

25. Flipped-end Borsalino

26. The New Rhombic End-cap … … yields new possibilities:

27. Rhombic Borsalinos u We just need to make a new connector part: bending again through 45º, but in diagonal direction! u This Borsalino now has a loose enough geometry, so that there is room to introduce twisted legs:

28. Twisted Connector Pieces Enlarged connector pieces with 45º twist:  results in two different pieces (azimuth!)  4 pairs make a nice twisted ring (360º )

29. Twisted Borsalinos 1 twisted branch; 3 twisted branches.

30. Triply Twisted Rhombic Borsalino u Sculpture!

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

32. Helical Pieces Another useful component!

33. Spiral Sculptures u Using the helical pieces

34. A Look Behind the Scene u It does not really close smoothly!

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

36. “Coccoon” u Return path through the center of the helix.

37. Inspiration: Bruce Beasley Autodesk exhibition, December 2013

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

39. “Pas de Deux”

40. Real LEGO ® Knots ? u Beginning of the table of knots …  This is all you have seen so far. It is not really a knot! Unknot Trefoil Figure-8

41. Real Knots: Trefoil (3_1) u One new piece (magenta) for smooth closure

42. Trefoil Knot

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

44. Figure-8 Knot

45. LEGO ® DUPLO u Match interface

46. LEGO ® DUPLO Just playing around...

47. LEGO ® DUPLO Borromean Link Hopf Link

48. What Is the Right Interface ? u Open tubes or u LEGO nibs ?

49. Making Sculptures Glow …

50. u testing

51. Making Sculptures Glow …

52. Hands-on Sculpting Xmas break 2013

53. Branching Out ? u Junction pieces  arbitrary graphs

54. “Organic” Looking Objects Trees or corrals ?

55. Making Graph Structures Tetrahedral graph: 4 valence-3 vertices

56. Next Attempt: More compact, but... u This is not the tetrahedral graph! Not this: But this:

57. Tetra Graph u OK tetrahedral graph. u But needed one extra custom-made piece!

58. Other Attempts ? What about the cubical edge graph ?

59. Cubical Edge Graph: 2 nd Attempt u New stub placement in square frame: l Place 2 Y-component close together, l pointing in opposite directions. Put two such frames on top of one another and connect pairs of stubs with the rhombic end-caps.

60. Cubical Edge Graph -- Solution u Stack the two frames in an offset manner, so that the connecting arcs run at a 45º angle.

61. Cubical Edge Graph as a Sculpture

62. Max Bill Sculpture, and beyond... u What is needed to emulate this?

63. But there are other ways of “branching out”...   

64. BID Lunch, February 25, 2014 Tria-Tubes EECS Computer Science Division University of California, Berkeley Michelle Galemmo Carlo H. Séquin

65. A New Profile u All sculptures shown were based on a square cross section: u What will happen if we try a different profile, e.g. a triangular one ?

66. Designing a “Tri-Borsalino” u This is the tightest turn without self-intersections of a circular tube containing a triangular cross section in an arbitrary azimuth (angle*) orientation. u Two different alignments result in two different end-caps: *

67. Dimensioning the TRIA-TUBES

68. Two Triangular End-Caps Type #1 Type #2

69. Two “Tri-Borsalinos” u We keep the 3D space curve of the classical Borsalino as the sweep path for our new triangular cross section. u The two end-caps produce two different Tri-Borsalinos. u The connector pieces bending through 45º need to have a twist of 15º to make smooth connections.

70. Our Initial Parts List u 2 types of end-caps: u Curved connectors: for type #1 for type #2 Connector sleeve

71. Some Possible Assemblies: Using just End-Caps two hexagonal rings and a triangular loop

72. Some Possible Assemblies: from only curved connector pieces A variety of twisted Möbius rings

73. What Next ? u More twisted connectors? u Twisted end-caps? u Straight extension pieces for flip-over Borsalino?  

74. Tri-Borsalino Derivatives u r = 0.57735 u Extend the link (by √8) between the connectors. u R(endcap) = sqrt(4/3) = 2r = 1.1547 = cos(30)*4/3

75. Conclusions LEGO ® -Knots plus Tria-Tubes: u An ever expanding modular system to do hands-on geometrical sculpture for people who do not want to do math or touch a computer. u The result is a mixture of: -- creative design decisions and -- practical fabrication considerations.