Energy Minimizers and the Beauty of Soap Bubbles
Sonia Kovalevsky High School Day October 28, 2011 Soap Films & Soap Bubbles: mathematics, biology, chemistry, physics… architecture
• Scientists have been experimenting with soap bubbles and films for hundreds of years • A = area of a soap film contained by a fixed boundary
σ f = film tension (constant in many cases) then energy available to alter the film shape is given by
F = σ f A free energy € • More in general, total energy necessary to increase the area of a soap film from zero to A: A F = ∫ σ f dA € 0 • Soap film is in equilibrium when free energy has a local minimum • Conclusion: soap films in equilibrium will have minimum area € …soap films in equilibrium will have minimum area…
Try it! Plateau Problem: Find a minimal surface with a given boundary • Empirical solutions to this problem may be obtained by dipping a model of the boundary into soapy water
Minimum surface contained by the edges of a tetrahedron Minimum surface formed by the twelve edges of a cubic framework
• Also, when we blow a soap bubble, the soapy surface stretches, when we stop the film tends toward equilibrium, and will form a minimal surface Bubbles at work! • Isoperimetric property: Circumferences are the least-perimeter way to enclose a given surface area
• Least-area way to enclose a given volume of air: sphere We see this all the time! When we blow into a soapy film, we fix a volume of air (the air that we blow), while the bubble soap forms the spherical surface which solves the minimization problem! • Least-area way to enclose and separate two given volumes of air?
? Double Bubble Conjecture: Is the familiar double soap bubble the solution?
TRUE!
Different Volumes Same Volumes
(Proved by an international team of mathematicians on 2000) J. Plateau discovered experimentally over hundred years ago that soap films contained by a framework always satisfy three geometrical conditions:
• Three smooth surfaces of a soap film intersect along a line • The angle between any two tangent planes to the intersecting surfaces, at any point along the line of intersection of three surface is 1200 • Four of the lines, each formed by the intersection of three surfaces, meet at a point and the angle between any pair of adjacent lines is 1090 28’ (angle whose cosine is -1/3)
Only in 1976, Jean E. Taylor was able to prove mathematically that the laws of Plateau are correct! Plateau’s principles at work! Plateau’s principles at work! Plateau’s principles at work!
• The Motorway Problem: what is the shortest path between four (or more) points? For example, how could we link a number of towns by the shortest road, water pipe, or cable? Motorway Problem: possible solutions…
(a) (d) & (e) same total length
(d) (e) (b) … shorter and shorter… but none is the smallest! (c) (f) Look at soap films!
Notice the 1200 angles! Experiments with soap bubbles have inspired also architects, notably German architect Otto Frei
• Montreal, Canada: German Pavilion, Expo ’67 Architect: Frei Otto Soap bubbles models were carefully constructed, photographed and measured. Solid models were then build and tested in wind tunnels • Munich, Germany: Olympic Games, 1972 Designed by Gunter Behnisch, with Otto Frei as roof design consultant • Beijing National Aquatics Center, 2008
References
• Cyril Isenberg, The Science of Soap Films and Soap Bubbles, Dover Publications, 1992 • Michele Emmer, Architecture and Mathematics: Soap Bubbles and Soap Films, pp. 53-65 in Nexus: Architecture and Mathematics, ed. Kim Williams, Edizioni dell’Erba, 1996 • Sami Polatoz, From Soap Bubbles to Technology, The Fountain Magazine, November-December 2008, Issue 66 • Frank Morgan, Double Bubble No More Trouble, Math Horizons, November 2000 • John Sullivan, http://torus.math.uiuc.edu/jms/Images/ • David Lovett, Demonstrating Science with Soap Films, Institute of Physics Publishing, 1994 • http://www.flickr.com/photos/scottgr/376657900/sizes/m/in/photostream/ • http://www.arch.mcgill.ca/prof/sijpkes/arch374/winter2001/tal/soapfilm/ final.htm • http://www.best-of-munich.com/olympic-park/img/munich-olympic-park.jpg • http://www.westland.net/expo67/map-docs/images/germany3.jpg • Beijing National Aquatic Center picture from Wikipedia