Crystal Math
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.

Choosiness for Cooperation
To explain why cooperation with nonrelatives arises and persists in populations, British researchers developed a computational model in which players have varying degrees of cooperativeness (a willingness to allow partners to accrue benefits at their own expense) and choosiness (a willingness to leave partners based on their teamwork). After each round of play, an individual receives a payoff that reflects the effort both teammates exert. Individuals do best, however, if they manage to get their partners to do most of the work. After the payoff, players can continue together to the next round or decide to divorce the other, in which case each would be randomly paired with another partner. Because players that are repeatedly divorced get slapped with greater costs than those that stick together, cooperation and choosiness rise in tandem when many rounds are played—the equivalent of long life spans. Choose the January 10 Nature for the complete payoff.

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