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Taking the Edge Off

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Science  05 Sep 2008:
Vol. 321, Issue 5894, pp. 1283b
DOI: 10.1126/science.321.5894.1283b


Math has a lot to say about packing things together. The abstract problem of cramming, for example, equal-sized circles into a larger square has applications as far-flung as error-correcting codes for digital communications and the physics of granular materials such as sand. But what if the square has no edges? A quartet of researchers has shown how packing works in such a borderless space.


The space in question is a torus, a shape like the surface of an inner tube. To topologists, a torus is equivalent to a parallelogram with its opposite edges glued together. On the unfurled, flattened-out torus map, anything leaving on one side immediately reenters from the other, as in many video games. William Dickinson of Grand Valley State University in Allendale, Michigan, and undergraduates Daniel Guillot of Louisiana State University, Baton Rouge, Anna Castelaz of the University of North Carolina, Asheville, and Sandi Xhumari of Grand Valley have spent the past two summers studying circle packings in tori.

Because a torus has no boundary, the circles are constrained only by one another—just as they would be on a patch of regularly repeating patterned wallpaper. Dickinson and students classified the graphs that can result when lines are drawn connecting centers of tangent circles (red lines in the figure, below), then set to work analyzing which ones lead to the densest packings (i.e., packings with circles of the largest possible radius). For five circles—the first truly challenging case—they found 20 different ways the circles could be arranged on the torus.

They applied the theory to two particular tori: the “square” torus formed by connecting opposite edges of a square, and the “triangular” torus, which starts from a rhombus with a 60-degree angle. Guillot and Castelaz found the best five-circle packing for the triangular torus last summer (2007), and Xhumari did the same for six circles this summer. Together, the ideas they developed enabled Dickinson to nail down the densest packing for five circles on the square torus. It occupies π/4 or 78.5% of the square torus, as compared with 71.1% on the triangular torus (see figure).


To cover a torus with circles, researchers studied how to pack them into a square or rhombus whose opposite edges are connected.


“In general, it is very difficult to prove that a particular packing is optimal,” says Ronald Graham, a circle-packing expert at the University of California, San Diego. Working without boundaries may make proofs easier to come by, he thinks, “but that is just an impression.”

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