Double Bubbles And Presidential Elections

Old Double Pen Challenge

What is the most efficient shape for two identical adjacent pens if any shapes are allowed, not just rectangles?


Chuck Gahr and Jan Smit found the most efficient shape: two overlapping circular pens, separated by a straight line as in Figure 1. The three fences meet at equal angles of 120 degrees. (This shape is better than one circle divided by a straight line through the middle.)

This fact was proved by a group of undergraduate students at Williams College in Williamstown, Mass., in 1990.

Smit pointed out that the double pen looks like the familiar double soap bubble, in which the bubbles also always meet at angles of 120 degrees.

Every double soap bubble, whether or not the two bubbles are the same size, is probably the most efficient shape for enclosing those two given volumes of air, but despite much effort, mathematicians have been unable to decide for sure.

Recent work has uncovered some weird alternative shapes, like the crazy double bubble computer simulation, in which the second region wraps around the first like an innertube around a beach ball. So far these new ones have turned out to be less efficient and too unstable to be made as physical soap bubbles.

In a breakthrough last year, Joel Hass (at the University of California, Davis) and Roger Schlafly (of Real Software, in Santa Cruz, Calif.) announced a computer proof that the familiar double soap bubble is best when the two volumes are the same size. Their work built on the earlier two-dimensional work of the undergraduates.

More on eclipses

Aubrey Dunne points out that worldwide there are more solar eclipses than lunar eclipses (although a typical fixed observer sees more lunar eclipses, because solar eclipses are visible only in a relatively narrow band, while lunar eclipses are visible anywhere on the side of the Earth facing the moon).

US Presidential Election Challenge Question

Assume that there are just two candidates and say half the population in each state votes. What is the fewest number of votes with which one could be elected president? What if there are three candidates?

* Send answers, comments, and new questions to:

Math Chat

Bronfman Science Center

Williams College

Williamstown, MA 01267

or by e-mail to:

The best submissions will receive a copy of the book Flatland, which helps explain higher dimensions. The challenge question answer and winner will be announced in the next column in two weeks.

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