## Joel Hass solves the 2,000-year old Double Bubble Problem

Professor Joel Hass, in collaboration with Roger Schlafly, a mathematician who is currently president of Real Software, proved that Nature's most obvious double bubble attains the minimum surface area that encloses two equal volumes. It was known to the ancient Greeks that the perfect sphere is the optimal shape to contain a given volume, although a mathematically rigorous proof was only given in the end of the last century (by Schwartz). When one considers enclosing two separate volumes, the problem becomes considerably more difficult. This past summer, Professor Hass and his collaborator finally arrived at a rigorous proof, aided by computer calculations.

The news spread around the world very quickly. This breakthrough on a problem with a 2,000 year history has been reported in newspapers and such major magazines as Science News, The Economist, and Discover.

The year 1994 will be remembered forever in the mathematics community for the proof of Fermat's Last Theorem. In the same sense, the double bubble is the winner of 1995. Mathematics is vigorously active and alive!

A torus bubble, left, and a double bubble, right. Hass and Schlafly calculated the surface area of such bubbles using a computer, and found that the double bubble has smaller area than any other when the enclosed volumes are the same.
(Color graphics courtesy of John Sullivan, University of Minnesota)

The following is the official news release of the University.
(M.M.)

UC Davis News Release
August 3, 1995

## Double Bubbles Designed by Nature Are Best Containers, Mathematicians Say

Of all the possible shapes in the world, the "double bubble" is the most efficient at enclosing two equal volumes, say two mathematicians from the University of California, Davis, and Real Software in Santa Cruz, who report having solved this problem that began 2,000 years ago.

"There are infinitely many possible shapes for enclosing volumes - cubes, inner tubes, cell walls, gas tanks," says UC Davis mathematics professor Joel Hass. "As it turns out, nature's soap bubbles are the best."

Using a computer, Hass and colleague Roger Schlafly, president of Real Software, proved that two spherical bubbles optimally attached to each other require the least surface area necessary to enclose two equal volumes. Hass is presenting the results on Sunday, Aug. 6, at a special session on soap bubble geometry at the 1995 Burlington Mathfest in Burlington, Vt.

The joint national summer meeting is sponsored by the American Mathematical Society and Mathematical Association of America.

The double bubble is familiar to children who have played with bubbles. It can be made by forcing together two bubbles until they conglomerate into a compound bubble, with a flat wall separating two spherical pieces.

The mathematicians' findings may lead to practical applications, especially where efficient containment is important, Hass says. For example, engineers might use the double bubble to minimize the weight of a satellite tank that needs to hold two liquids that cannot mix, such as one gallon of liquid oxygen and one gallon of liquid hydrogen. However, the real usefulness of the double-bubble solution is in the new mathematical techniques it introduces, Hass says, which have potential applications to other problems in geometry and global optimization theory. Global optimization refers to problems that seek the best possible ways to maximize results, such as attaining the biggest profits, or to minimize results, such as using the least materials.

For as long as balloons have been inflated and bubbles blown, it has been recognized that the round sphere is the most efficient shape enclosing a given volume. Mathematicians have studied this problem since the time of ancient Greece and have given many partial mathematical proofs of this fact. It wasn't until 1884 that the efficiency of a single round bubble was fully proved to the satisfaction of the rigorous, exact standards of mathematics.

Until this summer, the most efficient shape enclosing two equal volumes remained uncertain. The double-bubble problem had languished until about five years ago, when a group of undergraduates working with mathematics professor Frank Morgan at Williams College took up a mathematical study of soap bubbles. By then, mathematicians assumed double bubbles minimized surface area, but it soon became apparent that no one knew for sure.

An undergraduate in the program, Michael Hutchings, now a graduate student at Harvard University, narrowed the possible solutions, making the problem more manageable. Other groundwork for the final solution came earlier from mathematics professors Fred Almgren of Princeton University, Jean Taylor of Rutgers University and Brian White of Stanford University. The problem was now narrowed down to two possible families of bubbles, the standard double bubble and the "torus" bubble, which is donut-shaped.

Last year, during a calm stretch between rapids while kayaking down the south fork of the American River in Northern California, Hass and Schlafly had the idea of trying a computer on the problem. Now, mathematicians don't normally use computers to obtain mathematical proofs, Hass says, because computers tend to make slight errors in doing calculations. "Computers usually keep about 16 decimal places of a number, and for things like modeling an airplane that's fine," Hass says. "But it's fatal for mathematical proofs, which need to be proven exactly, not roughly."

However, the researchers found a way to make it work. They reduced the double-bubble problem to 200,260 calculations, which the computer could run in about 20 minutes. The final solution describes two identical bubbles that meet at 120-degree angles and share a disk-shaped wall whose radius is equal to one-half the the square root of 3 times the radius of the sphere - or about 87 percent of the radius of the sphere.

Mathematicians did not expect that a geometrical problem of this type would be solved on a computer, Hass says, because the problem admits infinitely many possible solutions, while a computer can only do a finite number of calculations. The proof involves an exhaustive comparison of all possible minimizing surfaces, but only after narrowing down the possible shapes to make the search manageable. The computation is arranged so that a finite set of calculations can analyze an infinite number of surfaces at once.

"It is a remarkable 2,000-year story, including the ancient Greek geometry of Euclid and Archimedes, curved-space geometry of Gauss and Riemann, space-age singular geometry, recent advances by undergraduate students and the final computer triumph by Hass and Schlafly," says Morgan, who added a chapter on soap bubbles to the 1995 edition of his book, "Geometric Measure Theory." Morgan organized the special soap bubble geometry session for the Mathfest.