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Tuesday, January 25, 2011

Partition values are fractal

Ken Ono of Emory University and his colleagues have solved a centuries old problem in mathematics: how to predict and make sense of partition values. The mathematicians have discovered that partition values are fractal in nature. Perhaps that means as little to you as it did to me upon first reading. Let me explain.

Any positive integer will equal one or more added smaller integers. The number of ways that the original integer can be created by adding is that integer’s partition value. For example, 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1. There are five ways to come up with the number 4 by adding, so there are 5 partitions of 4. The partition value of 4 is 5. Partition values form the basis of our adding and counting system.

Sounds simple enough right? Well, while it may be easy to figure out partition values for very small integers this rapidly becomes impossible. The number 100 has more than 190,000,000 partitions! Before 1937, no one had been able to accurately predict partition values for integers above 200. Since then, extremely tedious methods have been designed, but they generally require adding infinitely many numbers with infinitely many decimal points. Mathematicians wanted a finite formula for the problem.

Enter Ono, who had a mathematical epiphany while hiking with his student Zach Kent. The two of them realized that partition values are fractal in nature. In other words, partition values form part of a repeating pattern. This knowledge allowed the mathematicians to devise an algebraic formula for calculating partition values. I haven't included that formula here, but you can find it by following the links if you're so inclined.

Emory mathematician Ken Ono.

Photo by Carol Clark.

Okay, I admit that this story still may not mean that much to many of you. Probably only mathematicians can truly appreciate the scope of this discovery. Nevertheless, I think it’s a fascinating glimpse into the kinds of inquiry some people are exploring.