## Double Bubble Minimizes: Interval Computations Help in Solving a Long-Standing Geometric Problem

It is well known that of all surfaces surrounding an area with a given volume V, the sphere has the smallest area. This result explains, e.g., why a soap bubble tends to become a sphere. More than a hundred years ago, the Belgian physicist J. Plateaux asked a similar question: what is the least area surface enclosing two equal volumes? Physical experiments with bubbles seem to indicate that the desired least area surface is a "double bubble", a surface formed by two spheres (separated by a flat disk) that meet along a circle at an angle of 120 degrees. However, until 1995, it was not clear whether this is really the desired least area surface. Several other surfaces ("torus bubbles") have been proposed whose areas are pretty close to the area of the double bubble.

The theorem that double bubble really minimizes was recently proven by Joel Hass from Department of Mathematics, University of California at Davis (email hass@math.ucdavis.edu) and Roger Schlafly from the Real Software Co. (rschlafly@attmail.com). First, they proved that the desired surface is either a double bubble or a torus bubble, and then used interval computations (as well as other ingenious numerical techniques) to prove that for all possible values of parameters, the area of the torus bubble exceeds the area of the double bubble described above.

This result was mentioned in a popular magazine Discover as one of the main scientific achievements of the year.

This application of interval mathematics not only provides a solution to a long-standing mathematical problem; the authors also describe potential practical applications, one of the them: to the design of the lightest possible double fuel tanks for rockets.

A popular descripiton of the Double Bubble solution and of the role played by interval computation has appeared in American Scientist, September-October issue, 1996. The result has been published (w/o detailed proofs) in a paper

J. Hass, M. Hutchings, and R. Schlafly, The Double Bubble Conjecture, Electronic Research Announcements of the American Mathe. Society, 1995, Vol. 1, pp. 98-102.

A preprint with a full proof is available from the authors; it can also be accessed from Hass's homepage.