Exploring the Potential of Minimal Surfaces

The structures designed by architect Frei Otto are as graceful and airy as spider webs. Translucent membranes, supported by steel-wire nets, reach out from tall masts. Anchors tie the fringes to the ground.

But these ethereal forms are also anchored in practical reality. Otto, working at the Institute for Lightweight Structures in Stuttgart, West Germany, wanted to use as little construction material as possible for enclosures that are easily put up, dismantled and moved.

So he drew his models from one of nature’s most parsimonious and elegant creations: soap films. They became one of Otto’s principal tools in designing exhibition halls, arenas and stadiums.

He starts with a plexiglass plate studded with thin rods of various heights. Drooping threads hanging loosely from post to post define rudimentary edges and ridges.

By dipping this configuration into a soap solution and gingerly withdrawing it, Otto magically transforms the ungainly contraption into a glistening, tent-like shape. The soap film, stretching out only as far as it must, pulls the threads taut to create a spectacular scalloped roof.

The soap-film models are then carefully photographed and measured. Solid miniatures are built and tested in wind tunnels to determine the potential impact of snow and wind.

Cable Out of Threads

Finally plans are drawn up, and construction begins, with sheets of synthetic material replacing the soap film and steel cables replacing the threads.

Many mathematicians have become interested in soap films, which provide excellent examples of the concept of minimal surfaces.

Gently poking the surface of a film stretched across a wire loop--of the type kids use to blow soap bubbles--always increases the film’s area. When the disturbance is gone, the soap film springs back to its original shape, again taking on the smallest possible area that spans the loop.

The basic physical principle underlying this behavior is a system’s tendency to seek a state of lowest energy. Stretching a soap film increases its surface energy. Work must be done to deform the film, in the same way that energy is needed to stretch a rubber band.

Whenever it can, a soap film seeks a shape that minimizes its surface energy; because its surface energy is proportional to its area, it automatically assumes the form of a minimal surface. Consequently, soap films can be used to solve mathematical problems like those that come up in Frei Otto’s design work.

Thin Flat Disks

A ring dipped into a basin of soapy water comes out spanned by an iridescent film in the form of a thin flat disk. Any other ring-bounded surface, whether barely wrinkled or strongly bulged, would have a larger area. The flat disk is the most miserly surface that a soap film spanning a circle can take on without showing some holes--the minimal surface defined by a circle.

When the ring is replaced by a twisted but still closed wire loop, the shape of the corresponding minimal surface is much less obvious. More than a century ago, the Belgian physicist Joseph Plateau spent years experimenting with liquid films and thinking about the forms that emerge.

From his experiments, Plateau concluded that nature always finds a way to mold a soap film so that it spans any loop of wire, no matter how bent.

Replacing the wire by a curve, or contour, and the soap film by a surface turns a specific physical observation into a broad mathematical question: Does at least one minimal surface--the mathematical version of a soap film--span every conceivable closed curve in space? That general question has become known as the Plateau problem, and it has perplexed mathematicians for decades.

Soap-film experiments show how convoluted this question can get. A loop of wire bent so that it looks like the outline of a pair of old-fashioned bulky earphones, for example, can come out displaying a soap film with any one of three different configurations.

Different Shapes Appear

Dipped twice into the same soapy liquid, a given wire frame may show two very different forms. It is even possible that a frame twisted in the right way would emerge with a different form every time it was dipped.

“Soap films provide a wonderful and accessible physical experiment that leads to many complicated mathematical problems,” says mathematician Anthony Tromba of UC Santa Cruz. “The problem is there. It wasn’t invented. Since it does exist, it stands as a challenge to the ingenuity and creativity of the mathematician.”

Although demonstrations with soap films and twisted wires reveal important clues about the possible behavior of minimal surfaces, mathematical questions cannot be settled by experiment. No set of experiments, no matter how extensive, can rule out the possibility that doing the experiment one more time would not lead to some new, unexpected result.

A mathematical proof is necessary. That usually entails writing down equations to specify curves and surfaces, then studying the equations to learn about the special features of these shapes.

Given the diversity of forms that a soap film or minimal surface can take on, mathematicians have also developed schemes for putting surfaces into different categories, depending on their various features.

One rough but useful classification comes out of a branch of geometry called topology, often described as rubber-sheet or plasticene geometry. In the strange world of topology, where distances have little meaning, a single-handled coffee mug and a doughnut are indistinguishable.

Topological Puzzles

Two geometrical forms are of the same topological type if one shape can be stretched, squeezed or twisted until it looks just like the other. Cutting and pasting or tearing are not allowed. This means that every point in one object finds a place in the other.

A line and a circle, for instance, aren’t topologically the same because a circle must be torn before points on the circle can be mapped onto a line. A doughnut and a coffee cup are topologically equivalent because it’s possible to imagine expanding the coffee cup’s handle while shrinking its cup until all that’s left is a ring.

On the other hand, there’s no smooth way to transform an ordinary juice glass into a doughnut without punching a hole in the glass. By this reasoning, the surfaces of a sphere, a bowl and a coin all belong to one topological class while a doughnut and a coffee mug belong to another.

The disk-like soap film clinging to a ring emerging from soapy water is just one small piece of an idealized mathematical object called the plane. In some ways, the plane is like a giant soap film that extends so far over the horizon that the boundary can’t be seen. Like a soap film, it is also a minimal surface.

Two rings dipped in a soap solution may each come out with a disk-like film. But a soap film connecting the two rings may also emerge, creating a minimal surface that looks like a pinched cylinder, smoothly taken in at its waist. That hourglass form is the central piece of another infinite minimal surface called a catenoid. Its two open mouths, one at either end, reach out infinitely into space.

Back to Soap Films

A loose coil of wire twisted into the form of a helix supports a spiraling soap film. Extending its ends creates an infinite slide that would whirl any plummeting rider along its surface into a dizzy spin. This infinite minimal surface is called a helicoid.

The plane, catenoid and helicoid are special minimal surfaces sharing two important features: Because they are, roughly speaking, without boundaries, they are called complete minimal surfaces; and since none of them folds back and intersects itself, they are said to be embedded minimal surfaces.

Until recently, the plane, catenoid and helicoid were the only known examples of complete, embedded minimal surfaces of finite topology. Topologically, both the plane and helicoid can be modeled on a hollow sphere punctured by a single hole, whereas the catenoid resembles a twice-punctured sphere.

However, topologists had reasons to suspect that perhaps more than three such surfaces actually existed. The trouble was that potential candidates for this minimal-surface hall of fame typically were expressed in complicated equations that masked more than they revealed. This inside information had to be unlocked before the surface’s true nature could be seen.

A few years ago, a Brazilian graduate student, Celso Costa, managed to prove that a particularly thorny set of equations represented an infinite minimal surface.

Extends to Infinity

Topologically, Costa’s surface can be modeled on a chocolate-covered doughnut from which three bites have been taken. The bites puncture the surface and indicate that the surface, when deformed, can extend to infinity in three places.

But the question remained: Did the surface intersect itself? Mathematicians David Hoffman and Bill Meeks of the University of Massachusetts then took up the quest. Hoffman’s plan was to use a computer to find numerical values for the surface coordinates and then draw pictures of its Swiss-cheese core. These “snapshots” might catch the equations in revealing poses.

The first pictures were full of surprises. The surface appeared to be free of self-intersections, and it seemed to have a high degree of symmetry. “Extended staring,” in Hoffman’s words, led him to see that the surface consisted of eight identical pieces that fit together to make up the whole figure.

The entranced mathematicians started to explore the surface graphically, rotating it, examining it segment by segment. Many views later, the true form of this minimal surface began to emerge. And it was strikingly beautiful. The figure had the splendid elegance of a gracefully spinning ballerina flinging out her full skirt so that it whirled parallel to the ground.

Gentle waves undulated along the skirt’s hem. Two holes pierced the skirt’s lower surface and joined to form one catenoid that swept upward. Another pair of holes, set at right angles to the first pair, led from the top of the skirt downward into the second catenoid.

Revealing Symmetries

The symmetries revealed in pictures of the figure provided Hoffman and Meeks with just the tips they needed to analyze the equations. That, in turn, led to a mathematical proof that the surface, indeed, was the first complete, embedded minimal surface of a finite topology to be found in nearly 200 years. (Its predecessors, the catenoid and helicoid, had been discovered in the 18th Century.) Meeks and Hoffman soon went on to demonstrate the existence of whole families of new minimal surfaces.

Those families have more than mathematical or aesthetic interest; area-minimizing surfaces often occur in physical and biological systems, especially at the boundaries between materials.

These newly discovered forms may serve as useful models for understanding, as one biologist suggests, the shape of developing embryos or for the structure of certain polymers.

A dental surgeon has suggested that such a shape could be used in bone implants for securing false teeth. An implant designed with lots of holes and a least-area surface would minimize contact with bone while still ensuring a strong bond.

Bit by bit, the strangely beautiful new world of minimal surfaces is emerging. Mathematicians probing complex equations are coming up with fixtures for a surreal plumbing-supply store: contorted tubes, helicoids with tunnels, punctured basins. Some forms blossom into bizarre flowers, while others seem ready to fly off the pages of a science fiction novel.

Computer imaging is proving to be an indispensable tool that opens up for exploration a hitherto unseen realm of geometric forms.