1. Art-in-Science (and Science-in-Art) Feb. 27, 2014 Carlo H. Séquin University of California, Berkeley Art of Minimal Energy (and of Maximal Beauty?)

2. Soap Films

3. Minimal Surfaces u The two principal curvatures (maximal and minimal) are of equal and opposite magnitude at every point of the surface!

4. 1980s: Brent Collins: Stacked Saddles

5. 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). u George K. Francis analyzed Collins’ work in terms of the knots formed by the rims and the topology of the spanning surfaces. He told Brent about minimal surfaces (1992).

6. 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

7. Brent Collins: Hyperbolic Hexagon Six balanced saddles in a circular ring. Inspired by the shape of a soap film suspended in a wire frame. = Deformed “Scherk Tower”.

8. Scherk’s 2 nd Minimal Surface (1834) u The central part of this is a “Scherk Tower.”

9. Generalizing the “Scherk Tower” Normal “biped” saddles Generalization to higher-order saddles (“Monkey saddle”) “Scherk Tower”

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

11. Sculpture Generator 1, GUI

12. Some of the Parameters in “SC1”

13. Generated Scherk-Collins Shapes

14. Base Geometry: One “Scherk Story” u Taylored hyperbolas, hugging a circle Hyperbolic Slices  Triangle Strips

15. Shapes from Sculpture Generator 1

16. Minimality and Aesthetics Are minimal surfaces the most beautiful shapes spanning a given edge configuration ?

17. 3 Monkey Saddles with 180º Twist Minimal surface spanning three (2,1) torus knots Maquette made with Sculpture Generator I

18. Rapid Prototyping: Fused Deposition Modeling (FDM)

19. Zooming into the FDM Machine Build Support Build Support

20. Some Scherk-Collins FDM Models

21. “Bonds of Friendship”

22. Slices through “Minimal Trefoil” 50%10%23%30% 45%5%20%27% 35%2%15%25%

23. First Collaborative Piece Brent Collins: “Hyperbolic Hexagon II” (1996)

24. 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”

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

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

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

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

29. Exploring New Ideas: W=2 u Going around the loop twice... … resulting in an interwoven structure. (cross-eye stereo pair)

30. 9-story Intertwined Double Toroid Bronze investment casting from wax original made on 3D Systems’ Thermojet

31. Extending the Paradigm: “Totem 3” Bronze Investment Cast

32. “Cohesion” SIGGRAPH’2003 Art Gallery

33. “Atomic Flower II” by Brent Collins Minimal surface in smooth edge (captured by John Sullivan)

34. Volution Surfaces (twisted shells) Costa Cube --- Dodeca-Vol Here, minimal surfaces seem aesthetically optimal.

35. Triply Periodic Minimal Surfaces Schoen’s F-RD Surface Brakke’s Pseudo Batwing modules Surface embedded in a cubic cell, 12 “quarter-circle” boundaries on cube faces

36. A Loop of 12 Quarter-Circles Simplest Spanning Surface: A Disk Minimal surface formed under those constraints

37. Higher-Genus Surfaces u Enhancing simple surfaces with extra tunnels / handles “Volution_0” “Volution_2” “Volution_4” A warped disk 2 tunnels 4 tunnels

38. Ken Brakke’s Surface Evolver u For creating constrained, optimized shapes Start with a crude polyhedral object Subdivide triangles Optimize vertices Repeat the process

39. Optimization Step u To minimize “Surface Area”: u move every vertex towards the equilibrium point where the area of nearest neighbor triangles (A v ) is minimal, i.e.: u move along logarithmic gradient of area:

40. “Volution_2” ( 2 tunnels = genus 2 ) Patina by Steve Reinmuth

41. “Volution” Surfaces (Séquin, 2003) “Volution 0” --- “Volution 5” Minimal surfaces of different genus.

42. “Volution’s Evolution”

43. An Unstable Equilibrium … will not last long!

44. Stable vs. Unstable Equilibria u Stable equilibrium is immune to small disturbances. u Unstable equilibrium will run away when disturbed. u Computer can help to keep a design perfectly balanced.

45. Fighting Tunnels u The two side by side tunnels are not a stable state. u If one gets slightly smaller, the pull of its higher curvature will get stronger, and it will tug even more strongly on the larger tunnel. u It will collapse to a zero-diameter and pinch off. u But in a computer we can add a constraint that keeps the two tunnels the same size!

46. Limitations of “Minimal Surfaces” u “Minimal Surface” - functional works well for large-area, edge-bounded surfaces. u But what should we do for closed manifolds ? u Spheres, tori, higher genus manifolds … cannot be modeled by minimal surfaces.  We need another functional !

47. Closed Soap-film Surfaces u Pressure differences:  Spherical shapes

48. Surface Bending Energy u Bending a thin (metal) plate increases it energy. u Integrating the total energy stored over the whole surface can serve as another measure for optimization:  Minimal Energy Surfaces (MES)

49. Minimum Energy Surfaces (MES) u Sphere, cones, cyclides, Clifford torus Lawson’s genus-5 surfaces:

50. Lawson Surfaces of Minimal Energy Genus 3 Genus 5 Genus 11 Shapes get worse for MES as we go to higher genus … 12 little legs [ … see models ! ]

51. A Better Optimization Functional? u Penalize change in curvature !  Minimum Variation Surfaces (MVS):  (d  1  de 1  2 + (d  2  de 2  2 dA  Spheres, Cones, Various Tori, Cyclides … u The Sphere now has a cost/penalty of zero!

52. Minimum-Variation Surfaces (MVS) u The most pleasing smooth surfaces… u Constrained only by topology, symmetry, size. Genus 3 D 4h Genus 5 OhOh

53. Comparison: MES   MVS (genus 4 surfaces) 

54. Comparison MES  MVS Things get worse for MES as we go to higher genus: Genus-5 MES MVS keep nice toroidal arms 3 holes pinch off

55. A sculpture done with “minimal energy” ?

56. 2003: “Whirled White Web”

57. A 10’x10’x12’ block of compacted snow Breckenridge, CO, 2003 – Day 0

58. Day 1: The “Monolith” Removing lots of snow …

59. End of Day 2: The Torus

60. Day 3, pm: Flanges, Holes

61. End of Day 4: Desired Geometry

62. Day 5, am: Surface Refinement

63. Official Team Photo

64. 12:40 pm -- 42° F

65. 12:41 pm -- 42° F

66. “WWW” Wins Silver Medal

67. Inauguration Sutardja Dai Hall 2/27/09

68. QUESTIONS ? ? http://www.cs.berkeley.edu/~sequin/TALKS/2014_Art_Energy.ppt