## 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.

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