Plateau's problem is the general problem of determining the shape of the minimal surface constrained by a given boundary. It is named after Joseph Plateau who noticed that a handful of simple patterns seemed to completely describe the geometry of how soap bubbles fit together.
Plateau claimed that soap bubble surfaces always make contact in one of two ways: either three surfaces meet at 120° angles along a curve; or six surfaces meet at a vertex, forming angles of about 109°. For instance, in a cluster of bubbles, two intersecting bubbles (of possibly different sizes) will have a common dividing wall (the third surface) that meets the outer surfaces of the bubbles in 120° angles. On the other hand, the edges of the six soap-film faces that emerge within a tetrahedral wire frame, when dipped in a soapy solution, form angles of roughly 109° at a central vertex.
Until the American mathematician Jean Taylor came along in the mid-1970s, Plateau's patterns were just a set of empirical rules. However, as a follow-up to her doctoral thesis, Taylor was able to prove that Plateau's rules were a necessary consequence of the energy-minimizing principle – no other yet unobserved configurations were possible – thus settling a question that had been open for more than a century. The forces acting along the surface of a soap bubble all have the same magnitude in all directions. In crystals this is not the case (magnitudes of surface forces differ in different directions, though they may exhibit a grain, analogous to that in a piece of wood), but they still require the least energy to enclose a given volume. Minimal surfaces that model these conditions, like a cube with its corners chopped off or the bottom half of a cone mounted on a cylinder, are known as Wullf shapes, and provide fertile ground for mathematical study today.
1. Stuwe, M. Plateau's Problem and the Calculus of Variations. Princeton, NJ: Princeton University Press, 1989.