1 / 50

Exploring Volution: From Simple Surfaces to Higher-Genus Sculptures

Dive into the evolution of volution elements from basic surfaces to complex sculptures, exploring symmetries, morphing structures, and aesthetics. Understand the mathematics behind genus determination and prototyping models for study. Discover the importance of geometry and stability in minimal surfaces.

aschroeder
Download Presentation

Exploring Volution: From Simple Surfaces to Higher-Genus Sculptures

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Meeting Alhambra, Granada 2003 Volution’s EvolutionCarlo H. Séquin EECS Computer Science Division University of California, BerkeleyThanks to: Cathy Tao

  2. Examples of Volution Sculptures Volution_0 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. 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. P. J. Stewart’s Surface Image sent by Jeff Hrdlicka My virtual emulation

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

  10. 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. “4” 1 instance 4 ears in tetrahedral configuration “3b” 2 instances 3-fold symmetrical Costasurface “3a” 6 instances 1 trench plus 2 ears “2b” 3 instances 2 trenches = Maceconfiguration “2a” 12 instances C-valley plus a single ear “1c” 2 instances 3-fold symmetrical Gabocurve Left- & right-handed Igglecurves 6 instances “1a,b” Characteristics of Edge Cycle Pattern 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_0 Volution_1 Volution_2

  25. THICK: GENUS 2 THIN: GENUS 1 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:

  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 usingan 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. To fix this, the edges must be brought closer together. 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:

  35. Fix for Volution_5 • Bring edges closer by using hyper-quadrics instead of quarter circles: x2 + y2 = r2 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 timeto 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 • The most pleasing smooth surfaces… • Constrained only by topology, symmetry, size. D4h Oh Genus 3 Genus 5

  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) Minimal surface in three circles Sculpture constructed by hand

  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 ?

More Related