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One such surface that I know of is unstable. I saw this demonstrated with soap films at a museum many years ago. It is generated using a wireframe that is in the form of the outer edges of a cube. The internal surfaces that are not directly on the 6 outer faces of the cube, but they have an instability and near the center they will continuously change as they seek a central meeting point that cannot be reached. Is there some maths to account for this?
In the catenary surface between two parallel loops, the surface is obviously not of minimum area but presumable is of minimum surface energy. For a specified wire hoop diameter and their (parallel) spacing, is there an algebraic expression for the minimum diameter of the surface between the two hoops?