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Bubbles can meet only at Angles of 120° (for two Bubbles) and
109.5° (for three Bubbles), where the exact value of 109.5° is the
Tetrahedral 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.
See also Bubble, Calculus of Variations, Double Bubble, Plateau's Problem
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.