Using the calculus of variations, the Belgian mathematician Joseph Plateau deduced that the interfaces between adhering soap films represent localized minimal surfaces, which occupy minimum areas around a set of constrained geometric regions. Of course, this is not a global minimum, which would consist of a single spherical bubble. This Demonstration considers the case of three connected bubbles with variable radii, centered at , , and . You can also choose to see the details of Plateau's construction.
Every intersection of two or three bubbles makes angles of exactly 120°. These interfacial angles are maintained even if any of the radii are changed. The radii , , and of the three interface arcs lie on a straight line, as shown in the construction diagram. In turn, these are connected to the three centers by a network of straight lines. The interface arcs are given by relations similar to those for the focal lengths of thin lenses: , , .