1. LASER, December 10, 2014 Carlo H. Séquin University of California, Berkeley "Hyperbolic Hexagon" to "Evolving Trefoil“ my 20-year collaboration with Brent Collins

2. Basel, Switzerland M N G

3. Jakob Bernoulli (1654 ‒ 1705) Logarithmic Spiral

4. Leonhard Euler (1707 ‒ 1783) Imaginary Numbers

5. Geometry in every assignment... CCD TV Camera (1973) Soda Hall (1992) RISC 1 MicroChip (1982) Octa-Gear (2000)

6. Recent Designs and Models

7. Leonardo -- Special Issue On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art With an Artist’s Statement by Brent Collins George K. Francis with Brent Collins

8. More Sculptures by Brent Collins Photos by Phillip Geller

9. The Math in Collins’ Sculptures u Collins works with rulers and compasses; any math in his early work is intuitive. u He is inspired by nature, e.g. soap films (= minimal area surfaces) (& minimal bending, too). u George Francis analyzed Collins’ sculptures in terms of the knots formed by their rims and the topology of the spanning surfaces. u Let’s look at the local geometry found often in these tunnel & saddle surfaces:  

10. Scherk’s 2 nd Minimal Surface (1834) Normal “biped” saddles “Scherk Tower” 2 planes:  bi-ped saddles

11. “Hyperbolic Hexagon” by Brent Collins u 6 saddles in a ring... u = “wound up” (toroidal) 6-story Scherk tower. u 6 holes passing through symmetry plane at ±45º u Discussion: What if … l we added more stories ? l or introduced a twist before closing the ring ?

12. Closing the Loop straight or twisted “Scherk Tower”“Scherk-Collins Toroids”

13. Brent Collins’ Prototyping Process Armature for the "Hyperbolic Heptagon” (7 Scherk stories) Mockup for the "Saddle Trefoil" Time-consuming ! (1-2 weeks) Bees wax

14. Sculpture Generator 1, GUI

15. Shapes from Sculpture Generator 1

16. Brent Collins, 1997 “Hyperbolic Hexagon II”

17. Collins’ Fabrication Process Example: Vox Solis Layered laminated main shape Wood master pattern for sculpture

18. u One thick slice thru sculpture, from which Brent can cut boards and assemble a rough shape. u Traces represent: top and bottom, as well as cuts at 1/4, 1/2, 3/4 of one board. Profiled Slice through “Heptoroid”

19. Emergence of the Heptoroid (1) Assembly of the precut boards

20. Emergence of the Heptoroid (2) Forming a continuous smooth edge

21. Emergence of the Heptoroid (3) Smoothing the whole surface

22. The Finished Heptoroid u at Fermi Lab Art Gallery (1998).

23. 2003: “Whirled White Web”

24. 12:40 pm -- 42° F

25. 12:41 pm -- 42° F

26. “WWW” Wins Silver Medal

27. 12-Story Scherk-Collins Toroid u branches = 4 u storeys = 11 u height = 1.55 u flange = 1.00 u thickness = 0.06 u rim_bulge = 1.00 u warp = 330.00 u twist = 247.50 u azimuth = 56.25 u mesh_tiles = 0 u textr_tiles = 1 u detail = 8 u bounding box: u xmax= 6.01, u ymax= 1.14, u zmax= 5.55, u xmin= -7.93, u ymin= -1.14, u zmin= -8.41

28. 12 Signs of the Zodiac

29. David Lynn, Nova Blue Studio Arts u http://sites.google.com/site/novabluestudioarts/ http://sites.google.com/site/novabluestudioarts/

30. Master Module for “Millennium Arch”

31. Fabrication of “Millennium Arch” The mold for the key moduleA polyester segment cast

32. Two Times Three Modules

33. Merging the Two Half-Circles

35. Brent Collins and David Lynn

36. “Millennium Arch” by Night

37. Millennium Man Vitruvian Man by Leonardo Millennium Man Vitruvian Man by Leonardo

38. Concept of “Evolving Trefoil” New: Let a hole-saddle chain follow an arbitrary space curve!

39. Sculpture Generator #2

40. An Evolution of Ideas and Forms

41. 3 of 6 Maquette Modules (FDM) (after cleaning away support)

42. Full-Size Master Module and Mold Milled foam core coated with clay About 20 fiberglass pieces bolted together

43. Formation of a Main Sculpture Module

44. Sealed-off Mold (Catalyzed Matrix Inside)

45. Fabricate 6 Identical Pieces u 6 identical pieces had to be fabricated in this labor-intensive way!

46. Three Units at Nova Blue Studio Arts

48. Installation at MWSU, Feb. 2013

56. V-art (1999) Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

58. Yet Another Medium: Stone “The Three Pillars of Engineering” Math – Materials – Physics(Science) Sponsored by Paul Suciu (EECS alum)

59. Spring, 2012

64. “Pax Mundi” by Brent Collins (1996, 2007) Team effort: Brent Collins, Steve Reinmuth, Carlo Séquin

65. SLIDE-GUI for “Pax Mundi” Shapes Good combination of interactive 3D graphics and parameterizable procedural constructs.

66. Assembly of Music of the Spheres “Music of the Spheres”

67. With My Heartfelt Thanks... Steve Reinmuth Brent Collins