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.

**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.

© 1996-9

1999-05-25