Meeting Alhambra, Granada 2003

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Meeting Alhambra, Granada 2003

• Volutions EvolutionCarlo H. Séquin
• EECS Computer Science Division
• University of California, BerkeleyThanks to
  Cathy Tao

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Examples of Volution Sculptures

• Volution_0 Volution_5

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Definition of Volution

• Websters Dictionary volution
• 1) a spiral turn or twist
• 2) a whirl of a spiral shell
• 3)

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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

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Roots The Iggle

• Percy Hooper, NC State University, 1999

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Triply Periodic Minimal Surfaces

• Schoens F-RD Surface Brakkes Pseudo
  Batwing modules

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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

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P. J. Stewarts Surface

• Image sent by Jeff Hrdlicka My virtual
  emulation

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Aurora (Séquin, 2001)

• Basic sweep path Sculpture with
  morphing

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The Underlying Theme

• Sweep-path used for Aurora Subdiv-surface in
  octahedron

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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?
• Lets rotate some of the cube faces ...

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All 32 Possible Edge Cycles

• Drawn on the un-folded cube surfaces

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The Different Edge-Cycle Patterns

• 1a, 1b are mirror images !

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Characteristics of Edge Cycle Pattern

• Survey of all 32 cases

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Simplest Spanning Surface A Disk
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Spanning Surfaces for Two Edge-Cycles

• Cylinder Space-diagonal tunnels
  Face-diagonal tunnels

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Maximum Number of Edge Cycles 4

• Tetrahedral symmetry
• Space-diagonal tunnels (like Schoens F-RD)

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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

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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)

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A First 2-cycle Edge Pattern Mace

• Reminiscent of C.O. Perrys Sculpture
• Composed of two Trenches
• D2d symmetry
• Already seen some possible spanning surfaces

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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

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The Two Single-Cycle Edge Patterns

• Iggle (2 mirror versions) Gabo 3
  (with tunnels)

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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
  regionThey must connect equally colored
  surfaces.

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Higher-Genus Surfaces

• Enhancing simple surfaces with extra tunnels /
  handles

Volution_0 Volution_1
Volution_2
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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

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Genus ?

• Has 6 tunnels you can stick fingers through
• Analyzed as a math surface genus 5
• Analyzed as a solid object genus 10

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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
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Fused Deposition Modeling (FDM)
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Zooming into the FDM Machine
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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 ?

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Classical Minimal Surfaces

• Monkey saddle
  Costa surface

Scherks 2nd minimal surface
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Unstable Minimal Surfaces

• Example Volution_0
• Only stable on computer which strictly maintains
  starting symmetry.
• In nature, a small disturbance would break
  symmetryand the saddle would run away to one
  side.

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Surfaces Without Equilibrium

• Some surfaces dont 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 shapeso they still look like minimal
  surfaces !

Run-away points
Balance point
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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

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Fix for Volution_5

• Bring edges closer by using hyper-quadrics
  instead of quarter circles
• x2 y2 r2
  x4 y4 r4

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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 !

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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.

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Modular Building Blocks

• Blocks are stackable, because edges match They
  are all quarter circles.

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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!

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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 functionalCould use
  Minimization of Bending Energy(this is an
  option in the Surface Evolver).
• This would give us tangent continuity across
  seams.

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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! ?

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Minimum-Variation Surfaces
D4h
Oh
Genus 3
Genus 5

• The most pleasing smooth surfaces
• Constrained only by topology, symmetry, size.

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Minimality and Aesthetics

• Are minimal surfaces the most beautiful shapes
  spanning a given edge configuration ?

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Tightest Saddle Trefoil Séquin 1997
Shape generated with Sculpture Generator 1
Minimal surface spanning one (4,3) torus knots
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Whirled White Web Séquin 2003
Minimal surface spanning three (2,1) torus knots
Maquette made with Sculpture Generator I
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Atomic Flower II by Brent Collins

• Minimal surface in smooth edge(captured by John
  Sullivan)

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Surface by P. J. Stewart (J. Hrdlicka)

• Minimal surface in three circles

Sculpture constructed by hand
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For Volution Shapes, minimal surfaces seem to be
aesthetically optimal
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To Make a Piece of Art,It also Takes a Great
Material Finish
PATINA BY STEVE REINMUTH
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QUESTIONS ? DISCUSSION ?