The study of minimal surfaces, also known as optimal geometry, is a branch of differential geometry, because the methods of differential calculus are applied to geometrical problems- One of the oldest questions here is: ``What is the surface of smallest area spanning a given contour?'' The question is nontrivial despite the fact that every physical soap film appears to know the answer. Unbordered minimal surfaces have the property that each point is the center of a small patch that behaves like a soap-film relative to its boundary contour. From the point of view of local geometry, a minimal surface is equivalently described as one that is equally bent in all directions so as to have zero average curvature, e.g., a saddle shape.
The field of minimal surfaces has been one of the success stories of mathematical visualization: insights gleaned from computer graphics tools have led directly to concrete results and theorems.
Previously unknown and certainly unexpected minimal surfaces were found by David Hoffman and his collaborators at GANG, the Center for Geometry, Analysis, Numerics, and Graphics at the University of Massachusetts in 1985. They first used their MESH computer graphics system to find these surfaces, and then later proved their existence with fully rigorous mathematics. This truly excited the minimal surface community and piqued their interest in computer graphics. In Figure 11, we show new surfaces recently given by Hoffman, Wei, and Karcher ; the one-hole surface is the first complete, properly embedded minimal surface of finite topology and infinite total curvature to be found since the discovery of the helicoid in the eighteenth century.
The ``Minimal Surfaces Team'' of the Geometry Center consists of mathematicians and applied mathematicians modeling equilibrium and growth shapes of surfaces such as occur in soap bubbles and crystals. It includes Jean Taylor of Rutgers, Fred Almgren of Princeton, Ken Brakke of Susquehanna University, and John Sullivan of the University of Minnesota. Recently, Brakke's Surface Evolver was instrumental in finding a counter-example to an 1887 conjecture of Lord Kelvin. A partition of space into equal volume cells was found with less interface area than one conjectured by Lord Kelvin to be minimal in 1887. The Surface Evolver has also been used outside of pure mathematics, by an engineer at Martin-Marietta to aid in the design of rocket-fuel tanks where surface tension is the only force available to guide fuel to the intake valve in low-gravity conditions (see Figure 10). Since the end of the 1980's, other groups in Berlin and Bonn working mostly in the area of minimal/optimal surfaces have developed systems tailored to their visualization needs that have led to significant results .