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For someone who died at the age of 32 the largely self-taught Indian mathematician Srinivasa Ramanujan left behind an impressive legacy of insights into the theory of numbers—including many claims that he did not support with proof. One of his more enigmatic statements, made nearly a century ago, about counting the number of ways in which a number can be expressed as a sum, has now helped researchers find unexpected fractal structures in the landscape of counting.

Ramanujan's statement concerned the deceptively simple concept of partitions—the different ways in which a whole number can be subdivided into smaller numbers. Ken Ono of Emory University and his collaborators have now figured out new ways of counting all possible partitions, and found that the results form fractals—namely, structures in which patterns or shapes repeat identically at multiple different scales. "The fractal theory we've discovered completely answers Ramanujan's enigmatic statement," Ono says. The problems his team cracked were seen as holy grails of number theory, and its solutions may have repercussions throughout mathematics...

http://www.scientificamerican.com/article.cfm?id=mathematics-ramanujan