WE鈥橵E all seen them, yet since the early 19th century no one could explain
why they are the way they are. Now, at last, four mathematicians have proved
that double soap bubbles could not be any other shape.
The mathematics of bubbles got going in the 1830s, when the physicist Joseph
Plateau began dipping wire frames into soap solution. He was astounded by the
results, and despite years of research, many of his observations still lack
rigorous explanations.
An especially notorious case, known as the 鈥渄ouble bubble conjecture鈥�,
concerns the angles at which the surfaces of two bubbles meet when they
coalesce, and the actual shapes they form. The interface separating two
bubbles鈥攚hich is itself a portion of a sphere鈥攂ends a little into
the bigger partner. Plateau observed that where three soap-film surfaces meet,
they do so at an angle of 120掳 to each other.
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A single soap bubble is also a 鈥渕inimal surface鈥濃�攁 surface whose area
is as small as possible when holding a given amount of air, for instance.
According to the double bubble conjecture, the 120掳 angle is the only
configuration allowed for two bubbles, and every two-bubble shape does indeed
contain the smallest surface area possible. The challenge for mathematicians has
been to prove it.
In 1995, Joel Hass at the University of California, Davis, and Roger Schlafly
at the computer firm Real Software in Santa Cruz proved the conjecture for two
equal bubbles. But showing this is also the case for every other possible
combination of bubbles has proved elusive. Now four mathematicians, led by Frank
Morgan of Williams College in Williamstown, Massachusetts, have polished off the
problem.
It鈥檚 far from obvious that there are no alternatives that have a smaller
surface area. For example, one bubble might form a doughnut while the second
fits through it like a dumb-bell. Five years ago, Hass and
Schlafly found a way to rule out such bizarre alternatives for equal bubbles.
For this they needed to use powerful computers that could work out the solutions
to over 200 000 different integrals.
The new, complete proof can deal with unequal bubbles and involves far more
possibilities than any computer could ever hope to calculate. Yet it uses no
technology more advanced than pencil and paper. Morgan and his colleagues
presented the landmark proof in a lecture at the Rose-Hulman Institute of
Technology in Indiana last week.
The subject has already moved on. Under Morgan鈥檚 direction, a group of
undergraduates have extended the result to bubbles that exist in
four-dimensional space, and are making inroads into five dimensions.