Put a handful of identical coins on the table, and push them around until they fit together as closely as possible. You will get a honeycomb pattern, or hexagonal lattice, in which each coin is tightly surrounded by six others (Fig. 1, overleaf). This experiment suggests two things: that the 'kissing number' in two dimensions is six, and that the hexagonal lattice is the most efficient way to pack circles. The Greeks could have proved the first statement with complete logical rigour, had they thought to do so. The second statement, though widely suspected, was not properly proved until 1940, which demonstrates how tricky this area of mathematics can be.
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