Minimal Energy and Maximal Beauty: Exploring Art-in-Science

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

2. Soap Films

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

4. 1980s: Brent Collins: Stacked Saddles

5. The Math in Collins’ Sculptures • Collins works with rulers and compasses;any math in his early work is intuitive. • He is inspired by nature,e.g. soap films (= minimal area surfaces). • George K. Francis analyzed Collins’ workin terms of the knots formed by the rimsand 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 filmsuspended in a wire frame. = Deformed “Scherk Tower”.

8. Scherk’s 2nd Minimal Surface (1834) • 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” • 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 Maquette made with Sculpture Generator I Minimal surface spanning three (2,1) torus knots

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% 30% 23% 10% 45% 27% 20% 5% 35% 25% 15% 2%

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

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

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 • at Fermi Lab Art Gallery (1998).

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

30. 9-story Intertwined Double Toroid Bronze investment casting fromwax original made on3D 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-VolHere, 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-CirclesSimplest Spanning Surface: A Disk Minimal surface formed under those constraints

37. Higher-Genus Surfaces • 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 • For creating constrained, optimized shapes Start with a crude polyhedral object Subdivide triangles Optimize vertices Repeat theprocess

39. Optimization Step • To minimize “Surface Area”: • move every vertex towards the equilibrium point where the area of nearest neighbor triangles (Av )is minimal, i.e.: • 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 • Stable equilibrium is immune to small disturbances. • Unstable equilibrium will run away when disturbed. • Computer can help to keep a design perfectly balanced.

45. Fighting Tunnels • The two side by side tunnels are not a stable state. • 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. • It will collapse to a zero-diameter and pinch off. • But in a computer we can add a constraintthat keeps the two tunnels the same size!

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

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

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

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

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