Energy Minimizers and the Beauty of Soap Bubbles

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
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•  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 =σf A
free energy
•  More in general, total energy necessary to increase the area of a
A
soap film from zero to 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