Can there be a cubical bubble?

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?

Solution 1:

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.

Solution 2:

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