Trailing after double bubbles

From Washington, D.C., at the Joint Mathematics Meetings

A standard double bubble enclosing two unequal volumes of space. John Sullivan/UIUC

Predicting the geometric shapes of bubble clusters can lead to surprisingly difficult problems.

In 1995, mathematicians finally proved that the so-called standard double bubble, familiar to any soap-bubble enthusiast, represents the least surface area when the two bubble volumes are equal (SN: 8/12/95, p. 101).

Such a two-chambered structure triumphs over any other possible geometric form as the most efficient way of enclosing and separating two equal volumes of space.

Now, a proof of the double-bubble conjecture for the case where the two volumes are unequal appears within reach, says Frank Morgan of Williams College in Williamstown, Mass.

In a standard double bubble, two bubbles share a disk-shaped wall. This divider meets the individual bubbles’ walls at an angle of 120 degrees.

If the bubbles are of equal size, the interface is flat. If one bubble is larger than the other, the rounded surface of the boundary film bulges into the bigger bubble.

Soap bubbles naturally assume such a configuration. Proving that no other structure would be stable has long stymied mathematicians. There are many possibilities to consider, Morgan notes, including one bubble wrapping around the other like an inner tube snugly fitted around a giant peanut.

Morgan, Michael L. Hutchings of Stanford University, and Manuel Ritoré and Antonio Ros of the University of Granada in Spain have now developed an efficient method for checking alternative configurations to establish whether they are unstable or fail to beat the standard double bubble as the most economical configuration. Mathematicians may also be able to generalize this approach to determine, for example, the optimal configuration of double bubbles in four-dimensional space.

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