Recipient of the 2006 AMS Award for Exemplary Department or Program
HMC

Soap Film

Soap film surfaces are examples of "minimal surfaces", or surfaces with zero mean curvature. The surface that spans a given boundary set is the one that minimizes the surface area. That is, among all possible surfaces that could span the wire frame, the one that appears is the one with minimal surface area. This is due to the surface tension in the film.

 

This beautiful fact is illustrated in the second picture, where a loop of string was embedded in the soap film surface. The region inside the loop was popped and the loop rapidly expanded to form a circle. A circle encloses the maximal area for a fixed perimeter, thus the complementary area (of the soap film surface) is minimized. This demo provides a fun physical proof that the soap film has minimal surface area (or that a circle encloses the maximum area if you want to assume surface tension minimizes surface area!).

 

If you want to make your own soap solution, the following recipe from Andrew Belmonte (PSU) works great:
2450 mL Water
500 mL Glycerol
50 mL Dawn soap

This is not good for blowing bubbles. The Glycerol thickens the film up to produce stable surfaces such as those show below.