BUBBLES! (May 13, 2004)
Minimal Surfaces
An old question in mathematics concerns finding surfaces with a given fixed boundary which have the smallest surface area. For example, given a circle, a disc is the surface with minimal surface area bounded by the circle.
One way to investigate this problem is to build wire frames, dip them in a soap solution, and then look at the soap film formed in the wire frame. This soap film will be a minimum area surface.
Exercise
Try this with wire frames and note the soap films. Is it possible to get two different minimal surfaces with the same wire frame? Try to predict what you are going to get before you dip the wire frame. Make your own wire frames and investigate what sort of soap films you get. Notice what happens when three soap films meet along an edge.
In the 1830's, the Belgian physicist Joseph Plateau studied soap film in much the same way that we are. He observed that when three soap films meet along an edge they form angles of 120°, and when four of these edges meet at a point, they form angles of 109° 28'. This second observation wasn't mathematically proven until the 1970's. Both of these phenomena can be observed in the soap film obtained by dipping the tetrahedral wire frame in the soap solution. Not all of Plateau's experiments were so successful. In the name of science, he stared directly at the sun for 25 seconds which damaged his eyes and eventually caused him to lose his eyesight completely.
Double Bubbles
The sphere is the most efficient shape to enclose a given volume. That is, it is the surface which maximizes the ratio of volume to surface area. A dodecahedron enclosing the same volume as a sphere will have a greater surface area. This is why we can't blow dodecahedral bubbles.
Exercise
Try producing some double bubbles. Pay particular attention to the membrane between the two bubbles. Is it flat or curved?
Double bubbles formed from soap film always take the same shape. So, it seems likely that this is the most efficient shape to enclose two volumes. However, like Plateau's observations, this was only mathematically proven recently. In 1995, Joel Hass, Michael Hutchings, and Roger Schlafly proved that the double bubble was the most efficient way to enclose two equal volumes. Their proof involved using a computer to check many cases. Then in 2000, Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros proved the more general case without the use of computers.