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Diamonds are rarities not just on earth but also mathematically. The crystal structure of diamond has two key distinguishing properties, notes mathematician Toshikazu Sunada of Meiji University in Japan. It has maximal symmetry, which means that its components cannot be rearranged to make it any more symmetrical than it is, and a strong isotropic property, which means that it looks the same when viewed from the direction of any edge. In the February Notices of the American Mathematical Society, Sunada finds that out of an infinite universe of crystals that can exist mathematically, just one other shares these properties with diamond. Whereas diamond is a web of hexagonal rings, its cousin is made of 10-sided rings.
Sunada had originally thought that no one had described this object before (which he had dubbed K4). But it turns out that “I rediscovered the crystal structure mathematically in rather an accidental way” while working on another problem, Sunada says. After his paper was published, chemists and crystallographers informed him that they had long known about the crystal, which was called (10,3)-a by A. F. Wells in 1977. Diamond's mathematical twin can exist in a slightly distorted form as an arrangement of silicon atoms in strontium silicide.
