It’s not magic, it’s math!
Originally shared by Richard Green
Double Bubble: Toil and Trouble
For about 100 years, an open problem in mathematics was the Double Bubble Conjecture. This asked the following question: do two bubbles that meet in the usual way enclose and separate two equal volumes of air in such a way as to use the least possible amount of surface area?
This easily stated question had been assumed to be true without proof as early as 1896. However, it resisted all efforts to prove it until 1995, when Hass, Hutchings and Schlafly showed conclusively, with the help of a computer, that the answer to the question is yes. Assuming that the bubbles are of equal volume, the authors showed that the separating boundary between the bubbles is a flat disc, meeting each bubble at an angle of 120 degrees, and that each of the bubbles stuck to this central disc is a piece of a perfect sphere. Furthermore, the authors showed that this is the most efficient way to enclose two equal volumes by using the least possible amount of surface area.
In 2000, Hutchings, Morgan, Ritoré and Ros announced that the answer to the question is still “yes”, even when the two bubbles involved are of different sizes. In this case, the separating boundary between the bubbles will not be flat, and the entire double bubble configuration will consist of three spherical caps, in such a way that the separating membrane meets each bubble at a 120 degree angle. As in the previously established case where the bubble volumes were equal, this is the most efficient way to enclose two volumes of air so as to minimize the surface area. The proof of the more general result was much more work than the equal volumes case, but the paper still only occupies 30 pages in the Annals of Mathematics (2002), and significantly, the argument does not rely on the aid of a computer.
Even more surprisingly, the double bubble conjecture was soon extended to four-dimensional bubbles by a team of four undergraduates (Reichardt, Heilmann, Lai and Spielman) under the direction of Frank Morgan, who was one of the authors of the second paper mentioned above. The conjecture is also known to be true in certain cases in dimensions even higher than four.
You can hear a four-part podcast series about the double bubble conjecture here (http://www.ams.org/samplings/mathmoments/mm103-bubbles-podcast). Frank Morgan is one of the people featured, and he gives very nice talks. Another of the speakers is James Sethian, who, together with Robert Saye, created this picture.
(Seen via the American Mathematical Society.)