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Art-in-Science (and Science-in-Art) Feb. 27, 2014. Art of Minimal Energy (and of Maximal Beauty?). Carlo H. Séquin University of California, Berkeley. Soap Films. Minimal Surfaces.
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Art-in-Science (and Science-in-Art) Feb. 27, 2014 Art of Minimal Energy (and of Maximal Beauty?) Carlo H. Séquin University of California, Berkeley
Minimal Surfaces • The two principal curvatures (maximal and minimal) are of equal and opposite magnitudeat every point of the surface!
The Math in Collins’ Sculptures • Collins works with rulers and compasses;any math in his early work is intuitive. • He is inspired by nature,e.g. soap films (= minimal area surfaces). • George K. Francis analyzed Collins’ workin terms of the knots formed by the rimsand 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 filmsuspended in a wire frame. = Deformed “Scherk Tower”.
Scherk’s 2nd Minimal Surface (1834) • 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”
Base Geometry: One “Scherk Story” • Taylored hyperbolas, hugging a circle Hyperbolic Slices Triangle Strips
Minimality and Aesthetics Are minimal surfaces the most beautiful shapes spanning a given edge configuration ?
3 Monkey Saddles with 180º Twist Maquette made with Sculpture Generator I Minimal surface spanning three (2,1) torus knots
Zooming into the FDM Machine Build Support Build Support
Slices through “Minimal Trefoil” 50% 30% 23% 10% 45% 27% 20% 5% 35% 25% 15% 2%
First Collaborative Piece Brent Collins: “Hyperbolic Hexagon II” (1996)
Profiled Slice through “Heptoroid” • One thick slicethru sculpture,from which Brent can cut boards and assemble a rough shape. • Traces represent: top and bottom,as well as cuts at 1/4, 1/2, 3/4of one board.
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 • at Fermi Lab Art Gallery (1998).
Exploring New Ideas: W=2 • Going around the loop twice ... … resulting in an interwoven structure. (cross-eye stereo pair)
9-story Intertwined Double Toroid Bronze investment casting fromwax original made on3D 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-VolHere, 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-CirclesSimplest Spanning Surface: A Disk Minimal surface formed under those constraints
Higher-Genus Surfaces • 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 • For creating constrained, optimized shapes Start with a crude polyhedral object Subdivide triangles Optimize vertices Repeat theprocess
Optimization Step • To minimize “Surface Area”: • move every vertex towards the equilibrium point where the area of nearest neighbor triangles (Av )is minimal, i.e.: • 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.
An Unstable Equilibrium … will not last long!
Stable vs. Unstable Equilibria • Stable equilibrium is immune to small disturbances. • Unstable equilibrium will run away when disturbed. • Computer can help to keep a design perfectly balanced.
Fighting Tunnels • The two side by side tunnels are not a stable state. • 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. • It will collapse to a zero-diameter and pinch off. • But in a computer we can add a constraintthat keeps the two tunnels the same size!
Limitations of “Minimal Surfaces” • “Minimal Surface” - functional works well forlarge-area, edge-bounded surfaces. • But what should we do for closed manifolds ? • Spheres, tori, higher genus manifolds … cannot be modeled by minimal surfaces. We need another functional !
Closed Soap-film Surfaces • Pressure differences: Spherical shapes
Surface Bending Energy • Bending a thin (metal) plate increases it energy. • Integrating the total energy stored over the whole surfacecan serve as another measure for optimization: Minimal Energy Surfaces (MES)
Minimum Energy Surfaces (MES) Lawson’s genus-5 surfaces: • Sphere, cones, cyclides, Clifford torus
Lawson Surfaces of Minimal Energy Genus 3 Genus 5 Genus 11 12littlelegs Shapes get worse for MES as we go to higher genus … [ … see models ! ]