Plateau's Laws
Bubbles can meet only at angles of 120 degrees (for three bubbles) and cos^(-1)(-1/3) approx 109 degrees28^'16^('') (for four bubbles), where cos^(-1)(-1/3) 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 n>=3 bubbles.
See also
Bubble, Calculus of Variations, Double Bubble, Minimal Surface, Plateau's ProblemExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Plateau's LawsCite this as:
Weisstein, Eric W. "Plateau's Laws." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PlateausLaws.html