Touching Soap Films

An Introduction to Minimal Surfaces
By Hermann Karcher and Konrad Polthier

Introduction
Plateau
History
Visualization
Architecture
Crystallography
Weierstraß
Properties 1
Properties 2
Properties 3
Symmetry
Alteration
Periodic
Handles
Production
Scenes 1
Scenes 2
Scenes 3
Results
Exhibition
Numerics
References
Web Links

Crystallographic Models and Zeolithes

Fig 5. Inside a triply periodic minimal surface of Schwarz. with skeletal graph.

Contrary to a soap bubble which encloses under pressure a certain volume of air, a soap film is in equilibrium and has the same pressure on both of its sides. Infinite soap films (without self intersections) therefore divide space into two regions and can be seen as the interface between them. This natural separating property can be observed in a number of physical models: a group in Amherst, Massachusetts under the polymer chemist Edwin L. Thomas have made experiments about the equilibrium state of two mixed long-chained polymers.

Different to oil and water which separate one above the other in a clean way, these polymers generate complicated interweaving 3-dimensional structures. Since the investigators had no visible spatial model for the common boundary surface of these two components, they were at first unable to interpret the scanning electron projection micrographs they obtained. A comparison with images of triply periodic minimal surfaces revealed an astounding similarity between these two kinds of surfaces that had arisen in such very different ways and helped for a better understanding of those structures.

Zeolithe crystals consist of a skeleton of silicon, aluminum and oxygen atoms, and the remaining space is filled with crystal water. During careful heating the water evaporates and leaves a highly porous crystal skeleton which is used as an ion exchanger, used as a molecular sieve, and used during oil cracking. It turns out that the tetrahedral building units of sodalith has the form of a minimal surface of Schwarz. Similar connections were found for other zeolithes.

An amazing similarity was found during the investigation of electric fields in crystal grids between minimal surfaces and zero potential fields where point charges are located at the grid points of the crystal. In these studies minimal surfaces seem to play merely the role of prototypes of potential spatial structures. The real world applications are usually more complicated than the clean mathematical models.