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WILLIAMSTOWN, Mass., March 28, 2000 -- Four mathematicians have announced a mathematical proof of the Double Bubble Conjecture: that the familiar double soap bubble (see Figure 1 http://www.math.uiuc.edu/ ~jms/Images/double/) is the optimal shape for enclosing and separating two chambers of air.
In an address to the Undergraduate Mathematics Conference at the Rose-Hulman Institute of Technology in Indiana on Saturday (March 18), Frank Morgan of Williams College announced that he, Michael Hutchings of Stanford, and Manuel Ritori and Antonio Ros of Granada had finally proved that the double soap bubble had it right all along.
When two round soap bubbles come together, they form a double bubble (as on the right in Figure 1). Unless the two bubbles are the same size, the surface between them bows a bit into the larger bubble. The separating surface meets each of the two bubbles at 120 degrees.
This precise shape is now known to have less area than any other way to enclose and separate the same two volumes of air, even wild possibilities (as on the left in Figure 1), in which the second bubble wraps around the first, and a tiny separate part of the first wraps around the second. Such wild possibilities are shown to be unstable by a new argument which involves rotating different portions of the bubble around a carefully chosen axis at different rates.
The breakthrough came while Morgan was visiting Ritori and Ros at the University of Granada last spring. Their work is supported by the National Science Foundation and the Spanish scientific research foundation DGICYT.
In 1995 the special case of two equal bubbles was heralded as a major breakthrough when proved with the help of a computer by Hass, Hutchings, and Schlafly. The new general case involves more possibilities than computers can now handle. The new proof uses only ideas, pencil, and paper.
In an amazing postscript, a group of undergraduates has extended the theorem to 4-dimensional bubbles. The "SMALL" undergraduate research Geometry Group, consisting of Ben Reichardt of Stanford, Yuan Lai of MIT, and Cory Heilmann and Anita Spielman of Williams, working last summer at Williams under Morgan's direction, found a way to extend the proof to 4-space and certain cases in 5-space and above. Their work was part of the Research Experiences for Undergraduates sponsored by the National Science Foundation (and Williams College).
Contacts: Frank Morgan, Williams College, 413-597-2437, Frank.Morgan@williams.edu
Joel Hass, University of California at Davis, 530-752-1082, hass@math.ucdavis.edu
Mathematical research announcement at http://www.williams.edu/Mathematics/fmorgan/ann.html Preprint of whole mathematical paper at http://www.ugr.es/~ritore/bubble/bubble.html