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Although not physically perfect, a reasonable mathematical model for a bubble's shape is that it minimizes surface area subject to fixed volume.

A single floating bubble is usually a sphere, but bubbles only need to find local minima, not global minima. This makes more complicated bubble shapes possible.

In a YouTube video, a performer discusses making a cubical compartment inside a complicated bubble. Is this possible in a bubble with finitely-many compartments and no wires or other framework to provide additional boundary conditions?

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+1 because you reminded me of a very old adventure of Donald Duck in which he was saved from captivity (after having been captured by an exotic tribe venerating cubes!) by his nephews' ability to make cubical bubbles with chewing-gum. –  Georges Elencwajg Jan 21 '12 at 20:55
    
Once the bubble complex falls off his wire hoop, it looks like it remains (diffeomorphically) the same shape. Is that not an answer to your question? –  Rahul Jan 21 '12 at 21:51
    
@Rahul I don't understand, sorry. How does this address whether or not there can be a cubical bubble? –  Mark Eichenlaub Jan 21 '12 at 21:54
    
Sorry, maybe I don't understand the question. I thought you were asking whether there can be a cubical compartment inside a free-floating "bubble complex" with no external boundary conditions. What am I misunderstanding? –  Rahul Jan 21 '12 at 22:01
    
@Rahul Yup, that's what I wanted to know. I just don't really understand how your comment addressed the question. I think there is something about it I'm missing. (I'm not a mathematician.) Will Jagy's answer was pretty much what I wanted. –  Mark Eichenlaub Jan 21 '12 at 22:25
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up vote 4 down vote accepted

Such a bubble is not actually cubical. See my answer on Quora.

animation

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Good point about the angles along the edges and at the vertices. –  Will Jagy Nov 9 '12 at 21:34
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EDIT, November 2012: another answer points out, quite correctly, that any three bubble compartments in a multiple bubble meet at $120^\circ$ around an edge, and four compartments meet around a vertex in such a way that the angles between pairs of edges are about $109^\circ,$ to be exact $\arccos \frac{-1}{3}.$ I actually did my dissertation in minimal surfaces and I should have thought of this, but all that was a long time ago.

ORIGINAL: Yes, such a thing is possible. The real skill of the performer is in making the thing so quickly, real soap bubbles pop so quickly. Anyway, the rules for multiple bubbles are that any smooth part of bubble surface, either between one bubble and the outside or between two bubbles, be a surface of constant mean curvature. A flat square qualifies, it has mean curvature $0.$

A picture that is roughly what the performer creates: BRAKKE

More information: MEAN MINIMAL BUBBLE

There is not all that much work on specific clusters of small numbers of bubbles. One exception is the resolution of the double bubble CONJECTURE

EDIT: There is a nice short discussion in Soap Bubbles by C. V. Boys, pages 120-127, called "Composite Bubbles," where he discusses a bit about the curves where two bubbles meet. In the cube construction, there are seven separate regions containing air, and at the corners of the cube four regions meet.

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Great answer! Thanks for all the references. –  Mark Eichenlaub Jan 21 '12 at 22:46
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@MarkEichenlaub with some practice, you might be able to build a cubical framework of coated wire and blow this easier cube-within-a-cube: susqu.edu/brakke/evolver/examples/cubble/cubbleend.8.gif This is also color photograph 4.14(b) in the section Color Plates between numbered pages 46 and 47 of Cyril Isenberg, The Science of Soap Films and Soap Bubbles. –  Will Jagy Jan 21 '12 at 23:29
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