Dip a flat wire ring into a basin of soapy water. The ring comes out spanned by a taut, iridescent soap film in the form of a thin disk. Its area is smaller than it would be if the surface had peaks and valleys, or even small wrinkles. A clinging soap film invariably settles into the shape that mathematicians call a minimal surface. They can also imagine minimal surfaces that don’t exist in nature.
Consider a perfectly flat disk of soap film, for example, that extends so far over the horizon that its boundary can’t be seen. This two-dimensional plane is the simplest example of a minimal surface that is infinite in extent and not an endless repetition of some basic shape.