Minimum Surface Area
Minimum Surface Area
Nearly everybody has, at some point, been fascinated by soap bubbles. A fun experiment is to dip a loop of wire into a soap solution and pull it out. As if by magic, a thin, transparent film of soap will form across the wire. Trying to predict what shape the soap film will be often yields a surprise. What makes the soap bubble take the shape of a perfect sphere, and what determines the shape of a soap film?
Soap Bubbles and Soap Films
Like many other physical systems, soap bubbles and soap films try to minimize energy. The energy in a soap bubble comes from the force of surface tension that holds it together and keeps it from popping. This energy is proportional to the area of the soap film.
Soap bubbles and films try to minimize their surface area.* A soap bubble takes the shape of the surface with the smallest area that encloses a certain volume of air. That surface is a sphere. (For example, the smallest surface that encloses 1 cubic inch of air is a sphere with a radius of 0.62 inches.) If a wire is dipped into a soap solution, the soap film takes the shape with the smallest area that will not require it to let go of the wire.
*Centuries ago, architects discovered that experimenting with soap bubbles could help them define the most economical form for the actual structure.
Shapes of Films. Although soap bubbles come in only one shape, soap films come in a staggering variety. A circular wire loop will produce a soap film that is a simple disk. A loop of wire folded into a tetrahedral frame will produce a soap film consisting of six triangular pieces that meet at a point in the middle. One might expect that a cubical frame would produce twelve flat sheets (one for each edge) that meet at a point. Twelve flat sheets are indeed the result, but surprisingly there is also a thirteenth sheet—a small square—in the middle.
In the 1830s, the Belgian physicist Joseph Plateau conducted such experiments and arrived at two general rules that govern the way sheets of soap films join together.
- Three sheets may join along an edge, and they must always form 120° angles with each other.
- Four edges and six sheets may join at a point (as in the tetrahedral configuration), and the edges must always form 109° angles with each other.
These are the only two possibilities. This explains why the cubical frame did not produce twelve triangles that met in the middle: Plateau's laws forced the film into a less symmetric arrangement where only six sheets meet at a time.
A minimal surface is a surface satisfying the zero mean curvature property. In a soap film, for example, for any curve in the surface that bends one way, there is another curve perpendicular to it that bends the other way. The reason is surface tension: Each of the curves pulls the surface in the direction that it bends, so in order for the surface to be stationary, the amount of bending in the two directions must cancel out. This is called the zero mean curvature property of soap films.
The zero mean curvature property prevents a soap film from ever closing up. Note that soap bubbles do not have the zero mean curvature property, because the surface tension in a soap bubble is counterbalanced by the air pressure of the air trapped inside.
Minimal surfaces have interesting properties, and some are so striking or attractive that they have been given names. The catenoid is the minimal surface bounded by two parallel circles; the helicoid resembles a strand of DNA; and Enneper's surface* is a beautiful, potato-chip shaped surface.
*A sculpture in the shape of Enneper's surface won the Peoples' Choice award at the 2000 snow sculpture contest in Breckenridge, Colorado.
With computer graphics, new kinds of minimal surfaces have been found that could not even be imagined before. For example, the Costa-Hoffman-Meeks surfaces, which have a stack of three or more circles as their boundary, were discovered in the 1980s.
Solving Area Minimization
Though Plateau's laws were easy to demonstrate experimentally, it took mathematicians over a century to prove that these were indeed the only possibilities.
The American mathematician Jean Taylor finally met the challenge in 1970. This has been a recurring pattern throughout the history of minimal surface theory: The area-minimization principle is easy to state, and the actual soap-film configurations are elegant and simple, but it is very difficult to prove that natural laws have solved the area-minimization problem correctly.
As another example, anyone who has blown soap bubbles has noticed that they sometimes join together in pairs, forming two spherical caps with a thin sheet between them. But the first proof that this configuration has the least area (the "double bubble conjecture") did not come out until 2000. The difficulty is that there are many other bizarre configurations, never seen in nature, that nevertheless obey Plateau's laws and therefore have to be ruled out on other grounds.
Applications of Minimization. Although many mathematicians study minimal surfaces for purely aesthetic reasons, the principle of energy minimization has many practical applications. For example, crystals minimize their surface energy as they grow, the same as soap bubbles. They do not grow into spheres, however, because their surface energy is direction-dependent. (It takes less energy to cut a crystal with the grain than against the grain.) Energy minimization also governs the shape of fluid in a capillary or the shape of a drop of water resting on a table. Although the shapes in all these problems are slightly different, the mathematical methods used to solve them are very similar.
Almgren, Frederick, and Jean Taylor. "Geometry of Soap Films and Soap Bubbles." Scientific American July (1976): 82–93.
Boys, Charles. Soap Bubbles. New York: Dover Publications, 1959.
Hoffman, David. "The Computer-Aided Discovery of New Embedded Minimal Surfaces." Mathematical Intelligencer 9, no. 3 (1987): 8–21.
Morgan, Frank. Geometric Measure Theory: A Beginner's Guide, 3rd ed. San Diego, CA: Academic Press, 2000.
———. "Proof of the Double Bubble Conjecture."American Mathematical Monthly 108 (2001): 193–205.
A MINIMAL SUMMARY
An area-minimizing surface is one that has the smallest possible area of any surface with the same boundary. Minimal surfaces might not be strictly area-minimizing: that is, there may be another surface with the same boundary and less area. However, minimal surfaces are always "locally" area-minimizing in the sense that any small change to the surface will increase its area.
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