Three Coalescing Soap Bubbles

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: , , .

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Reference
[1] C. V. Boys, "Soap-Bubbles and the Forces Which Mould Them," London: Society for Promoting Christian Knowledge, 1896. www.gutenberg.org/files/33370/33370-h/33370-h.htm.
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