The crystalline face of soap films.

The crystalline face of soap films

The surface of a glistening soap film tightly stretched across a closed wire loop, of the type kids use to blow soap bubbles, is the smallest possible area that can span the loop. That minimal surface is a reflection of the soap film's tendency to seek to state of lowest energy. A soap bubble is spherical for the same reason. Any other closed shape of the same volume would have a higher surface energy.

Mathematician Jean E. Taylor of Rutgers University in New Brunswick, N.J., has extended this idea to crystals (SN: 1/31/87, p.76). The key difference is that whereas soap films and soap bubbles have a uniform surface energy, crystals do not. Different crystal faces may have different surface energies. In that case, the shape of a single crystal -- the analog of a single soap bubble -- is no longer necessarily spherical and may show flat faces.

Using her theory, Taylor can now compute and display the different types of minimal surfaces that a crystal surface assumes within a given boundary -- the solid analogs of soap films confined within a certain wire loop. In the computer-generated illustrations, the figure on the left shows a single crystal's equilibrium shape. The figure on the right shows one of the possible minimal surfaces that fits within the yellow boundary marked on the crystal. Together, the pair of crystal images is the equivalent of a soap bubble next to a soap film on a wire frame.

Taylor is using such images to study how changes in the crystal shape affect the forms of corresponding minimal surfaces. For example, she has noticed that whenever a crystal's smoothly curved surface meets a flat surface, the corresponding minimal surface has a cusp. Her computer experiments so far support this conjecture.

"I'm hoping that computation will be a useful tool for seeing which conjectures appear to be correct and for generating other conjectures," Taylor says. "There are a lot of phenomena and not very many theorems to explain them."