Double Soap Bubble

The double bubble, or double soap bubble, geometry has long posed a fascinating mathematical challenge: which shape for the surface of a double bubble encloses volume most efficiently?

When a double soap bubble forms, it consists of two spheres that intersect, with a soap film separating the chambers on each side. This separating wall connects with the surface of each sphere at a 120-degree angle. If the two spheres have equal size, then the separating film has the shape of a flat disk. If one sphere is larger, then the separating wall bulges into the larger space enclosed by the larger sphere.

The question of which geometric shape for a co-equal double bubbles maximizes the internal volume baffled mathemsticians for decades.