January 24, 2011
You’re familiar with partition numbers, even if you don’t recognize the term; even kindergartners know them. The partition of a number is all the ways that you can use integers to add up to that number. Start with 2. There is only one way to get there: 1 + 1. The number 3 has 2 partitions: 2 + 1 and 1 + 1 + 1. Four has 5 partitions: 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1. And so forth. But partition numbers get unwieldy pretty quickly. By the time you get to 100, there are more than 190,000,000 partitions. We’re well beyond elementary school math.
Mathematicians have been searching for the past couple of centuries for an easy way to calculate partition values. In the 18th century, Leonhard Euler developed a method that worked for the first 200 partition numbers. Solutions proposed in the early 20th century for larger partition numbers proved to be inexact or impossible to use. And the search continued.
The most recent mathematician to tackle the problem was Ken Ono at Emory University, who had a eureka moment while on a walk through the north Georgia woods with his post-doc Zach Kent. “We were standing on some huge rocks, where we could see out over this valley and hear the falls, when we realized partition numbers are fractal,” Ono says. “We both just started laughing.”
Fractals are a kind of geometric shape that looks incredibly complex but is actually composed of repeating patterns. Fractals are common in nature—snowflakes, broccoli, blood vessels—and as a mathematical concept they’ve been hauled into use for everything from seismology to music.
Ono and his team realized that these repeating patterns can also be found in partition numbers. “The sequences are all eventually periodic, and they repeat themselves over and over at precise intervals,” Ono says. That realization led them to an equation (all math leads to equations, it sometimes seems) that lets them calculate the number of partitions for any number.
The results of their studies will soon be published; a more detailed analysis is available at The Language of Bad Physics.
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