Fractals and Measuring "Roughness"


"If you stand back from it, a jagged line will grow smoother," someone once said to me when trying to put the events of a bad day in perspective. "The little bumps are not as plain to see from a distance." 

Is this always true? Not in fractal geometry.

A fractal is a shape that exhibits self-similarity across scales. If you zoom in on a fractal, you'll get more of the same distinction and definition. 

Romanesco broccoli

If you tried to measure the circumference of this cauliflower, for instance, and you cut up the pieces to get a more accurate measurement, you'd be holding the same cauliflower, only smaller.

Fractals have something called "roughness," meaning that because the pattern repeats at all scales, the object is difficult to measure. The closer-in you try to measure a coastline, or a lung, the longer it will be. 

"Roughness is a part of human life," mathematician Benoit Mandelbrot said. Mandelbrot found order in this type of branching complex system for which there was previously no geometry, and discovered that the underlying mathematical rules that created these patterns were short and few. In a fractal pattern, roughness can go on forever and ever--infinite roughness--like finding a map of the coastline printed on a single grain of sand. 

Mandelbrot's formula for fractals

Mandelbrot's formula for fractals

I think "infinite roughness" is a decent metaphor for the mind. We find focus when we sense a pattern; that's what focus is for. Hunting an answer to our mystery, we can and do focus on both the infinitely minute and the macrocosmic, sometimes all at once in unpleasant ways, and sometimes to our benefit. Learning leaves the erudite thirsty for more because now he knows more about what he's missing. 

We often speak of progress in our work, or growing up or becoming a nicer person as "smoothing out the rough edges." But as one gets closer and closer to uncovering some true personality, it eludes us and more roughness is revealed. Ever the onion reveals another layer, until there is no onion.

This is the cosmic punchline, encoded into the very math that unfolds a flower. 

The HumanCurrent podcast is hosted by Angie Cross & Stacy Hale. Subscribe in iTunes or listen at www.human-current.com.

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