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

Soap Films

Minimal Surfaces u The two principal curvatures (maximal and minimal) are of equal and opposite magnitude at every point of the surface!

1980s: Brent Collins: Stacked Saddles

The Math in Collins’ Sculptures u Collins works with rulers and compasses; any math in his early work is intuitive. u He is inspired by nature, e.g. soap films (= minimal area surfaces). u George K. Francis analyzed Collins’ work in terms of the knots formed by the rims and the topology of the spanning surfaces. He told Brent about minimal surfaces (1992).

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

Brent Collins: Hyperbolic Hexagon Six balanced saddles in a circular ring. Inspired by the shape of a soap film suspended in a wire frame. = Deformed “Scherk Tower”.

Scherk’s 2 nd Minimal Surface (1834) u The central part of this is a “Scherk Tower.”

Generalizing the “Scherk Tower” Normal “biped” saddles Generalization to higher-order saddles (“Monkey saddle”) “Scherk Tower”

Closing the Loop straight or twisted “Scherk Tower”“Scherk-Collins Toroids”

Sculpture Generator 1, GUI

Some of the Parameters in “SC1”

Generated Scherk-Collins Shapes

Base Geometry: One “Scherk Story” u Taylored hyperbolas, hugging a circle Hyperbolic Slices  Triangle Strips

Shapes from Sculpture Generator 1

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

3 Monkey Saddles with 180º Twist Minimal surface spanning three (2,1) torus knots Maquette made with Sculpture Generator I

Rapid Prototyping: Fused Deposition Modeling (FDM)

Zooming into the FDM Machine Build Support Build Support

Some Scherk-Collins FDM Models

“Bonds of Friendship”

Slices through “Minimal Trefoil” 50%10%23%30% 45%5%20%27% 35%2%15%25%

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

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

Emergence of the Heptoroid (1) Assembly of the precut boards

Emergence of the Heptoroid (2) Forming a continuous smooth edge

Emergence of the Heptoroid (3) Smoothing the whole surface

The Finished Heptoroid u at Fermi Lab Art Gallery (1998).

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

9-story Intertwined Double Toroid Bronze investment casting from wax original made on 3D Systems’ Thermojet

Extending the Paradigm: “Totem 3” Bronze Investment Cast

“Cohesion” SIGGRAPH’2003 Art Gallery

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

Volution Surfaces (twisted shells) Costa Cube --- Dodeca-Vol Here, minimal surfaces seem aesthetically optimal.

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

A Loop of 12 Quarter-Circles Simplest Spanning Surface: A Disk Minimal surface formed under those constraints

Higher-Genus Surfaces u Enhancing simple surfaces with extra tunnels / handles “Volution_0” “Volution_2” “Volution_4” A warped disk 2 tunnels 4 tunnels

Ken Brakke’s Surface Evolver u For creating constrained, optimized shapes Start with a crude polyhedral object Subdivide triangles Optimize vertices Repeat the process

Optimization Step u To minimize “Surface Area”: u move every vertex towards the equilibrium point where the area of nearest neighbor triangles (A v ) is minimal, i.e.: u move along logarithmic gradient of area:

“Volution_2” ( 2 tunnels = genus 2 ) Patina by Steve Reinmuth

“Volution” Surfaces (Séquin, 2003) “Volution 0” --- “Volution 5” Minimal surfaces of different genus.

“Volution’s Evolution”

An Unstable Equilibrium … will not last long!

Stable vs. Unstable Equilibria u Stable equilibrium is immune to small disturbances. u Unstable equilibrium will run away when disturbed. u Computer can help to keep a design perfectly balanced.

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

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

Closed Soap-film Surfaces u Pressure differences:  Spherical shapes

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

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

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

A Better Optimization Functional? u Penalize change in curvature !  Minimum Variation Surfaces (MVS):  (d  1  de 1  2 + (d  2  de 2  2 dA  Spheres, Cones, Various Tori, Cyclides … u The Sphere now has a cost/penalty of zero!

Minimum-Variation Surfaces (MVS) u The most pleasing smooth surfaces… u Constrained only by topology, symmetry, size. Genus 3 D 4h Genus 5 OhOh

Comparison: MES   MVS (genus 4 surfaces) 

Comparison MES  MVS Things get worse for MES as we go to higher genus: Genus-5 MES MVS keep nice toroidal arms 3 holes pinch off

A sculpture done with “minimal energy” ?

2003: “Whirled White Web”

A 10’x10’x12’ block of compacted snow Breckenridge, CO, 2003 – Day 0

Day 1: The “Monolith” Removing lots of snow …

End of Day 2: The Torus

Day 3, pm: Flanges, Holes

End of Day 4: Desired Geometry

Day 5, am: Surface Refinement

Official Team Photo

12:40 pm -- 42° F

12:41 pm -- 42° F

“WWW” Wins Silver Medal

Inauguration Sutardja Dai Hall 2/27/09

QUESTIONS ? ?