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

Examples of Volution Sculptures Volution_0 Volution_5

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

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

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

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

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

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

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

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

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 ...

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

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

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

Simplest Spanning Surface: A Disk

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

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

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

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)

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 …

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

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

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.

Higher-Genus Surfaces Enhancing simple surfaces with extra tunnels / handles Volution_0 Volution_1 Volution_2

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

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

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

Fused Deposition Modeling (FDM)

Zooming into the FDM Machine

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 ?

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

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.

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

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.

Fix for Volution_5 Bring edges closer by using hyper-quadrics instead of quarter circles: x2 + y2 = r2 x4 + y4 = r4

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 !

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.

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

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!

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.

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! 

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

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

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

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

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

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

For Volution Shapes, minimal surfaces seem to be aesthetically optimal

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

QUESTIONS ? DISCUSSION ?