1. Meeting Alhambra, Granada 2003
Florida 1999
Meeting Alhambra, Granada 2003
Volution’s Evolution Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley Thanks to: Cathy Tao

2. Examples of Volution Sculptures
Volution_ Volution_5

3. Definition of Volution
Webster’s Dictionary: volution:
1) a spiral turn or twist
2) a whirl of a spiral shell
3) …

4. Outline Roots of the ideas for such elements
Systematic taxonomy of possible patterns
Evolution from simple disk to higher genus surfaces
Making those modules stackable
Aesthetics of minimal surfaces

5. Percy Hooper, NC State University, 1999
Roots: The “Iggle”
Percy Hooper, NC State University, 1999

6. Triply Periodic Minimal Surfaces
Schoen’s F-RD Surface Brakke’s Pseudo Batwing modules

7. Specific Definition of Volution Elements
Two sided surfaces
Embedded in a cube
Edge is formed by pairs of quarter circles on each cube face
Overall D2 symmetry  3 C2 rotational axes
Forms modular elements, stackable in 1, 2 or 3D

8. Image sent by Jeff Hrdlicka My virtual emulation
P. J. Stewart’s Surface
Image sent by Jeff Hrdlicka My virtual emulation

9. Basic sweep path Sculpture with morphing
Aurora (Séquin, 2001)
Basic sweep path Sculpture with morphing

10. Sweep-path used for Aurora Subdiv-surface in octahedron
The Underlying Theme
Sweep-path used for Aurora Subdiv-surface in octahedron

11. How Many Volution Elements Are There?
In how many ways can the edges be connected?
What kinds of saddles can be formed in between?
How can we build higher-order genus elements?
Let’s rotate some of the cube faces ...

12. All 32 Possible Edge Cycles
Drawn on the un-folded cube surfaces

13. The Different Edge-Cycle Patterns
1a, 1b are mirror images !

14. Characteristics of Edge Cycle Pattern
“4”
1 instance
4 ears in tetrahedral configuration
“3b”
2 instances
3-fold symmetrical Costa surface
“3a”
6 instances
1 trench plus 2 ears
“2b”
3 instances
2 trenches = Mace configuration
“2a”
12 instances
C-valley plus a single ear
“1c”
3-fold symmetrical Gabo curve
“1a,b”
Left- & right-handed Iggle curves
Survey of all 32 cases

15. Simplest Spanning Surface: A Disk

16. Spanning Surfaces for Two Edge-Cycles
Cylinder Space-diagonal tunnels Face-diagonal tunnels

17. Maximum Number of Edge Cycles: 4
Tetrahedral symmetry
Space-diagonal tunnels (like Schoen’s F-RD)

18. Breaking the Tetrahedral Symmetry
Rotate edge pattern on one cube face:
 Two of the four ears merge into trench
 3-cycle edge pattern, 6/32 occurrences

19. Another 3-cycle Configuration
6-edge ring separates two 3-edge cycles
Same as edge configuration of Costa surface
3-fold symmetry around cube diagonal  genus-2 Costa (each funnel splits into 3 tunnels)

20. A First 2-cycle Edge Pattern: “Mace”
Reminiscent of C.O. Perry’s Sculpture
Composed of two “Trenches”
D2d symmetry
Already seen some possible spanning surfaces …

21. Another 2-cycle Edge Pattern
Only D1d = C2h symmetry; (shows up 12 times)
Less obvious how to connect these edges with a spanning surface
Select some tunnels from space/face-diagonal sets
Maintain overall symmetry
Shapes are less attractive  not studied extensively

22. The Two Single-Cycle Edge Patterns
Iggle (2 mirror versions) Gabo 3 (with tunnels)

23. All Volution Surfaces Are Two-Sided
Disk is orientable, cuts volume of cube into 2 differently colored regions.
Tunnels can only be added thru such a region; They must connect equally colored surfaces.

24. Higher-Genus Surfaces
Enhancing simple surfaces with extra tunnels / handles
Volution_ Volution_ Volution_2

25. Determining the Genus Tricky business ! ( Thanks to John Sullivan ! )
Process for surfaces: Close all “holes” (edge cycles) with disk-like patches.
Genus = maximum # of closed curves that do not completely divide the surface into two territories.
Need to distinguish: math surfaces  solid objects
Example: Disk With 1 Handle:
THICK: GENUS 2
THIN: GENUS 1

26. Genus =? Has 6 tunnels you can stick fingers through
Analyzed as a math surface: genus = 5
Analyzed as a solid object: genus = 10

27. Model Prototyping Draw polyhedral models in SLIDE:
parts only, use symmetry!
Smooth with subdivision techniques
Thicken with using an offset surface
Good for study of topology / symmetry
VOLUTION_1

28. Fused Deposition Modeling (FDM)

29. Zooming into the FDM Machine

30. Towards Minimal Surfaces
For sculptural elements: geometry matters!
Exact shape is important for aesthetics.
Minimal surfaces are a good starting point.
Does a minimal surface exist ?
Is it stable ?
 Use Brakke’ Surface Evolver
Is it the best solution ?

31. Classical Minimal Surfaces
Monkey saddle Costa surface
Scherk’s 2nd minimal surface

32. Unstable Minimal Surfaces
Example: Volution_0
Only stable on computer which strictly maintains starting symmetry.
In nature, a small disturbance would break symmetry; and the saddle would run away to one side.

33. Surfaces Without Equilibrium
Some surfaces don’t even have unstable balance points, they are just snapshots of run-away processes.
Fortunately, the smoothing and rounding occurs before the surface has run away too far from the desired shape; so they still look like minimal surfaces !
Run-away points
Balance point

34. Source of Run-away Force
The problem is that some edges connected by a spanning surface are too far apart for a catenoid tunnel to form between them:
To fix this, the edges must be brought closer together.

35. Fix for Volution_5
Bring edges closer by using hyper-quadrics instead of quarter circles:
x2 + y2 = r x4 + y4 = r4

36. A Struggle for Dominance
Even edges close enough to allow a stable catenoid, may still present a precarious balancing act:
Two side-by-side tunnels fight for dominance:
the narrower tunnel constricts ever more tightly,
until it pinches off, and then disappears !

37. Finding the Balance Point
If we balance the sizes of adjacent tunnels just right, they will stay stable for a long enough time to give the rest of the surface time to assume zero mean curvature (become a minimal surface).
Find balance point manually with a binary search.

38. Modular Building Blocks
Blocks are stackable, because edges match: They are all quarter circles.

39. Smooth Connections Between Blocks
We also would like G1 (tangent plane) – continuity:
Mirror surfaces:  surfaces must end normal on surface
C2 - connection:  surface must have straight inflection line
But we can no longer force edges to be quarter circles.  We loose full modularity!

40. Towards Full Modularity
For full modularity, we need to maintain the quarter-circle edge pattern.
For G1-continuity, we also want to force surfaces to end perpendicularly on the cube surfaces.
 This needs a higher-order functional: Could use Minimization of Bending Energy (this is an option in the Surface Evolver).
This would give us tangent continuity across seams.

41. The Ultimate Connection
For best aesthetics, we would like to have G2 (curvature)-continuous surfaces and seams.
If we want to keep modularity, we may have to specify zero curvature perpendicular to the cube surfaces.
To satisfy all three constraints: circles, normality, 0-curvature, we need an even higher-order functional!
MVS (Min.Var.Surf.) could do all that! 

42. Minimum-Variation Surfaces
Genus 3
D4h
Genus 5 Oh
The most pleasing smooth surfaces…
Constrained only by topology, symmetry, size.

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

44. “Tightest Saddle Trefoil” Séquin 1997
Shape generated with Sculpture Generator 1
Minimal surface spanning one (4,3) torus knots

45. “Whirled White Web” Séquin 2003
Maquette made with Sculpture Generator I
Minimal surface spanning three (2,1) torus knots

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

47. Surface by P. J. Stewart (J. Hrdlicka)
Sculpture constructed by hand
Minimal surface in three circles

48. For Volution Shapes, minimal surfaces seem to be aesthetically optimal

49. To Make a Piece of Art, It also Takes a Great Material Finish
PATINA BY STEVE REINMUTH

50. QUESTIONS ? DISCUSSION ?