Shapes of embedded minimal surfaces.

Surfaces that locally minimize area have been extensively used to model physical phenomena, including soap films, black holes, compound polymers, protein folding, etc. The mathematical field dates to the 1740s but has recently become an area of intense mathematical and scientific study, specifically in the areas of molecular engineering, materials science, and nanotechnology because of their many anticipated applications. In this work, we show that all minimal surfaces are built out of pieces of the surfaces in Figs. 1 and 2.