Plateau's Laws

Bubbles can meet only at angles of  (for three bubbles) and  (for four bubbles), where  is the supplementary angle of the tetrahedral dihedral angle. This was proved by Jean Taylor using measure theory to study area minimization. The double bubble is area minimizing, but it is not known if the triple bubble is also area minimizing. It is also unknown if empty chambers trapped inside can minimize area for bubbles.

REFERENCES:

Morgan, F. "Mathematicians, including Undergraduates, Look at Soap Bubbles." Amer. Math. Monthly 101, 343-351, 1994.

Taylor, J. E. "The Structure of Singularities in Soap-Bubble-Like and Soap-Film-Like Minimal Surfaces." Ann. Math. 103, 489-539, 1976.