Eureka! math and a walk in the woods

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At least since Plato's day, the debate over whether numbers have an essence beyond their denominated value has raged.

At the macroscopic level, mathematics obviously correspond to physical properties, or engineered outcomes - that bridge, that integrated circuit, that aircraft - would depend on chance to work.

Theoretical physicist-cum-Anglican priest John Polkinghorne has said that if you have some ugly equations, "you've almost certainly got it wrong." He's argued that the one thing all mathematicians hold in common is an appreciation for the economy and beauty of the compact expression - pleasing aesthetics seem to be a feature of mathematical truth.

But to understand the deep properties of these numbers, two other mathematicians might just suggest a walk in the woods.

A eureka moment happened in September, when [Emory University mathematician Ken] Ono and Zach Kent [a postdoc colleague at Emory] were hiking to Tallulah Falls in northern Georgia. As they walked through the woods, noticing patterns in clumps of trees, Ono and Kent began thinking about what it would be like to 'walk' through partition numbers.

'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.'

The term fractal was invented in 1980 by Benoit Mandelbrot, to describe what seem like irregularities in the geometry of natural forms. The more a viewer zooms into “rough” natural forms, the clearer it becomes that they actually consist of repeating patterns. Not only are fractals beautiful, they have immense practical value in fields as diverse as art to medicine.

Their hike sparked a theory that reveals a new class of fractals, one that dispensed with the problem of infinity. 'It’s as though we no longer needed to see all the stars in the universe, because the pattern that keeps repeating forever can be seen on a three-mile walk to Tallulah Falls,' Ono says.

"This new class of fractals" no doubt have value to mathematicians reading this story. And tomorrow's technologists will almost certainly put them to use. But I was struck, as well, by the importance of simple observation to discovery. Cultivating the patience and openness needed to see what we're looking at is a skill that benefits scientists and gallery goers alike.

Related, Ono and Kent visualized what it would be like to walk through partition numbers. Einstein is said to have made his relativistic breakthroughs in part because he visualized what it would be like to catch up to a beam of light, and the expression E=MC2 is not only an elegant and powerful statment about the relationship between mass and energy, but has become a cultural touchstone as well.

Do numbers have an independent existence, something "that it is to be like a number?" Probably not. But as Ono and Kent discovered, they are where we find them.