Taking the Edge Off

摘要

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.