Sunday, December 19, 2010

Some coastlines are more infinite than others

In October, the mathematician Benoît B. Mandelbrot died in Cambridge, Massachusetts. His obituary in the Times reminded me of two things from the schools of S.A.D. 6: first, the mysterious and beautiful Mandelbrot Set, the famous fractal, and second, the erroneous trivia (repeated even to this day on the tourist office's website) that Maine has the longest coastline of any state save Alaska.

The reasoning behind the latter goes like this: California and Florida's coasts may look longer on a map, but look closely at Maine's coast and you'll find thousands of islands, peninsulae, and inlets - features that those other states don't have. But take this reasoning even further and see where it gets you: you'd need to count the waterline on every mangrove root in the Everglades, and the outline of every grain of sand on California's beaches.

Every coastline infinitely long. Not only that, but every state gains and loses an infinitely long length of coastline every time the tide goes out, or every time a wave washes ashore.

I thought of this problematic state trivia because Mandelbrot's first academic paper, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension," tried to tackle this paradox. It observed that how long a coastline looks depends on how much detail you care to look for - whether you're measuring on a classroom map of the contiguous states, or on individual grains of sand under a microscope. For instance, if we use a 50-mile yardstick, Maine's coast looks about 230 miles long:

But try using a 5-mile yardstick, and you'll find that there are at least 200 miles of coastline in Casco Bay alone:

And we're still missing so much detail! Who wants to try measuring Casco Bay with a mile-long ruler? Or an actual yardstick?*

If Maine's coastline were smooth, like Florida's, the 40-odd 5-mile rulers in the bottom map would have covered roughly as much territory as four 50-mile rulers.

Mandelbrot's insight was this: you could measure the "crookedness" of a coastline by calculating the relationship between how carefully you measure something (the scale of your ruler) and the total length you get.

This relationship is similar to the fractal dimension, a mathematical concept useful for calculating how some infinite sets are "more infinite" than others. It's also a way of describing things that are neither one-dimensional lines, nor two-dimensional planes, but fuzzy and in between - like the infinite coastlines that emerge and sink away under every one of the ocean's infinite ripples.

For what it's worth, these guys say that the coast of Maine's fractal dimension is 1.27 - slightly more infinite than the coast of Britain (at 1.25), but not nearly as infinite as the coast of Norway (1.52).

* Please don't.


awoodill said...

I ran into this idea in a book about logging in British Columbia (Canada). It claimed that BC's coast was equivalent to the entire coastline from a spot in Norway east all the way to a spot in Greece. It's interesting that BC's complex array of islands and inlets could be compared to an entire continent... but only the coastline between two of the most complex coastal countries in Europe.

turboglacier said...

I would argue that, for human purposes, the yardstick for this should be-- a yardstick. That is, about the length of a human pace. I imagine it would be pretty easy for a computer + digital satellite images to spit out a number.

But would you measure with the tide in, or out?

I once tried to estimate how much area (two-dimension) Maine gains and loses with the tide. There were a lot of unknowns (chiefly, the average slope of the intertidal region) but I came up with something like 0.2% of the state's land area.