Visualization of Soap Bubble Geometries
Leonardo: Journal of the International Society for the Arts, Sciences and Technology
The authors discuss mathematical soap bubble problems and a new technique for generating computer graphics of bubble clusters. The rendering program is based on Fresnel's equations and produces both the colored interference patterns of reflected light and the Fresnel effect of varying transparency. A single soap bubble possesses an exquisite perfection of form. Soap bubbles are lovely physical manifestations of simple geometric relationships created by the principles of area minimization. Our
... minimization. Our goals in studying soap bubble problems are both to better understand problems of area minimization and to use such problems as test cases for the computation and visualization of geometric structures which arise in other optimization problems. In this article we report in particular on new techniques for displaying soap bubble geometries; these techniques incorporate both colored interference patterns and the Fresnel effect of decreased transparency at oblique angles. Minimal surface forms. A collection of surfaces, interfaces, or membranes is called a 'minimal surface form' when it has assumed a geometric configuration of least area among those into which it can readily deform. Of course there must be some constraints in the problem to keep the configuration from collapsing completely. Typical constraints might be a fixed boundary wire the surface must span, or a volume it must enclose. The sphere, the shape of a single soap bubble, seems the simplest minimal surface form; it has least area among all surfaces which enclose the same volume. Minimal surface forms arise not only in the surface tension phenomena of liquids and thin films, such as soap bubbles, but also in grain boundaries in metals, in radiolarian skeletons, in close packing problems, in immiscible liquids in equilibrium, in sorting of embryonic tissues, in design, in art, and in mathematics  .