As a graduate student in mathematics, I often found that lectures by visiting speakers exercised my eyelids more than my brain. I'd struggle to understand the subject for five minutes, fail, then struggle to stay awake for 55 minutes longer. But one talk was decidedly different. The speaker walked in, emptied his pockets of a large quantity of ball bearings, which rolled with a tremendous clatter all over the desk at the front of the room, and asked, "What's the best way to pack these things together?"
The speaker was Neil Sloane of Bell Laboratories (now AT&T Research), and his question—how to pack balls together in the densest possible way—was one of the oldest unsolved problems in mathematics. In 1611, the German physicist Johannes Kepler stated what he felt to be the obvious solution: You make a triangular array, then fit another layer into the interstices between the balls in the first layer, and so on. In this arrangement, called the face-centered cubic lattice, just over 74 percent of the volume of the space is taken up by balls, and 26 percent by the spaces between the balls. Kepler never even tried to prove that this was the densest packing. But later mathematicians questioned his assumption, now called the Kepler Conjecture. For all Sloane knew, his ball bearings might one day settle into a configuration with only 25 percent empty space.
Read full article from The Proof Is in the Packing » American Scientist
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