Fractals, scale, and measurements.

So, how long is the coast of Great Britain? It depends on the size of your measuring stick.

That's the gist of a paper written by the famous mathematician (aka fractal dude) Benoit Mandelbrot, way back in 1967. His whole idea was that a coastline isn't made up of straight lines; instead it's a craggy, jagged thing that's craggy and jagged no matter what how close you're looking at it. Because of this, the smaller the ruler you use to measure, the larger your final measurement will be, because you can lay it up against more and more of it's curvilinear edge. For example, if you measure the coastline of Great Britain in 200 km lengths, the number comes out around 2400 km, but if you measure it in 50 km lengths it comes out to around 3400. And so forth. In this way, an infinitely small unit of measurement would result in an infinitely large measurement of the perimeter. At least if the object is truly fractal, anyways.

This means that when you're given a measurement for pretty much anything in the real world that isn't strictly linear, the number is meaningless unless you know the scale the measurer used. Most the time the scale used is implicit in the units used to report it, but even so...

I feel like this basic idea can becomes really powerful way to think about systems and tasks once you add one more crucial premise: not only is it important to know what scale you're working with, it's important to use the right one for the job. Measuring out garden plots to the mm is a waste of time, but you had better be at least that accurate if you're milling machine parts. In other words, tasks have a 'best' scale on which to work on.

I realize this has all been pretty straightforward (and quite possibly boring) so far, but please bear with me: Next week I'm going to go out on a limb and try to apply these ideas to sculpture and public zoning laws.

Further reading:

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. (wikipedia)

Chaos, by James Gleick (a really well-written book about the pioneers of chaos science- it's where I first read about the Great Britain example)