In a fascinating postscript, a group of four undergraduates doing research last summer under Morgan's direction managed to extend the double bubble result to four dimensions, and, with some extra conditions, to five and higher dimensions as well. Their proof rests on arguments about rotations and stability. In order to be sure that the genuine and only minimal surface was the familiar double bubble, the existence of a nonstandard minimizer had to be definitively ruled out - and that is what a team of fourįrank Morgan of Williams College, Massachusetts, Michael Hutchings of Stanford, and Manuel Ritori and Antonio Ros of Granada proved their result using only pencil and paper, despite the fact that the earlier proof (in 1995) of the special case when both volumes are the same was computer-aided, and very long. Here the larger region is broken into two components, one a tiny ring, like a torus,Īround the other region, which is also torus-like.Īlthough noone had managed to find a minimal surface among these strange configurations, or "non-standard bubbles", mathematicians need more than this to conclude that none exists. These configurations are never seen in nature, and can only be seen using computer modelling. The trouble was that, although the familiar double bubble configuration certainly satisfied all the known conditions, so did some other, considerably stranger, configurations. For example, the surface would have to be rotationally symmetric about a line, and consist of surfaces meeting in threes at 120 o angles along curves. Some facts about any minimal surface enclosing and separating two volumes have been known for some time, however. But progress on the two-volume problem - "How small can a surface be while enclosing and separating two given volumes?" - has been slower. The minimal surface enclosing a single volume is a sphere, as was asserted by Archimedes and proved by Schwarz in 1884.
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