Chaos - James Gleick [45]
In fact, he argued, any coastline is—in a sense—infinitely long. In another sense, the answer depends on the length of your ruler. Consider one plausible method of measuring. A surveyor takes a set of dividers, opens them to a length of one yard, and walks them along the coastline. The resulting number of yards is just an approximation of the true length, because the dividers skip over twists and turns smaller than one yard, but the surveyor writes the number down anyway. Then he sets the dividers to a smaller length—say, one foot—and repeats the process. He arrives at a somewhat greater length, because the dividers will capture more of the detail and it will take more than three one-foot steps to cover the distance previously covered by a one-yard step. He writes this new number down, sets the dividers at four inches, and starts again. This mental experiment, using imaginary dividers, is a way of quantifying the effect of observing an object from different distances, at different scales. An observer trying to estimate the length of England’s coastline from a satellite will make a smaller guess than an observer trying to walk its coves and beaches, who will make a smaller guess in turn than a snail negotiating every pebble.
A FRACTAL SHORE. A computer-generated coastline: the details are random, but the fractal dimension is constant, so the degree of roughness or irregularity looks the same no matter how much the image is magnified.
Common sense suggests that, although these estimates will continue to get larger, they will approach some particular final value, the true length of the coastline. The measurements should converge, in other words. And in fact, if a coastline were some Euclidean shape, such as a circle, this method of summing finer and finer straight-line distances would indeed converge. But Mandelbrot found that as the scale of measurement becomes smaller, the measured length of a coastline rises without limit, bays and peninsulas revealing ever-smaller subbays and subpeninsulas—at least down to atomic scales, where the process does finally come to an end. Perhaps.
SINCE EUCLIDEAN MEASUREMENTS—length, depth, thickness—failed to capture the essence of irregular shapes, Mandelbrot turned to a different idea, the idea of dimension. Dimension is a quality with a much richer life for scientists than for non-scientists. We live in a three-dimensional world, meaning that we need three numbers to specify a point: for example, longitude, latitude, and altitude. The three dimensions are imagined as directions at right angles to one another. This is still the legacy of Euclidean geometry, where space has three dimensions, a plane has two, a line has one, and a point has zero.
The process of abstraction that allowed Euclid to conceive of one– or two-dimensional objects spills over easily into our use of everyday objects. A road map, for all practical purposes, is a quintessentially two-dimensional thing, a piece of a plane. It uses its two dimensions to carry information of a precisely two-dimensional kind. In reality, of course, road maps are as three-dimensional as everything else, but their thickness is so slight (and so irrelevant to their purpose) that it can be forgotten. Effectively, a road map remains two-dimensional, even when it is folded up. In the same way, a thread is effectively one-dimensional and a particle has effectively no dimension at all.
Then what is the dimension of a ball of twine? Mandelbrot answered, It depends on your point of view. From a great distance, the ball is no more than a point, with zero dimensions. From closer, the ball is seen to fill spherical space, taking up three dimensions. From closer still, the twine comes into view, and the object becomes effectively one-dimensional, though the one dimension is certainly tangled up around itself in a way that makes use of three-dimensional space. The notion of how many numbers it takes to specify a point remains useful. From far away, it takes none—the point is all there is. From closer,