Perhaps the most striking idea in fractal geometry is its peculiar view of dimension. Since Euclid’s day, an imaginary mathematical point has had no dimension, a line has had one, a plane, two, and the familiar space we live in, three. Einstein added a fourth, time. Mathematics can generalize the idea, and imagine higher dimensions – purely fictitious, but useful for solving a problem in engineering, economics, or physics. Topology, the mathematical study of surfaces, adds some interesting new twists. From a topological point of view, a cucumber is the same as an orange because on can be remolded into the same shape as the other without having to cut the surface. And the circumference of a circle has the same dimension, one, as a jagged coastline on a shipping map. They are both continuous lines; one can be transformed into the other just by bending, folding and stretching – without cutting.
But is that all there is to dimension? Look at a ball of thread, and think about it first from the idealized viewpoint of Euclid. Assume it is five inches in diameter, made of fiber a fraction of an inch thick. From a long distance away, you can barely see the ball; it is, effectively, a point – of no dimension, according to classical geometry. Hold it in your hand, and it resolves to a normal, three-dimensional ball. Bring it up closer: You see it is a tangle of one-dimensional fibers. Closer still, and the fibers are clearly three-dimensional strands. Keep going until the atoms resolve in an electron microscope: Back to zero-dimensional points again. So what is this ball of thread, anyway? Zero, one, or three dimensions? It depends on your point of view. For a complex natural shape, dimension is relative. It varies with the observer. The same object can have more than one dimension, depending on how you measure it and what you want to do with it. And dimension need not be a whole number; it can be fractional. Now an ancient concept dimension, becomes thoroughly modern. (129)
Think of dimension, not as an inherent property, but as a tool of measurement. So how do you actually measure something? If you want to measure a straight line, you get a ruler. If you want to measure a curved line, you could use a smaller ruler, inching it along the curve and counting how many times you moved it. You could get a more accurate, if tedious, measure by using a still-smaller ruler; its measurement will be a bit longer than the first, crude one. Eventually, as the ruler keeps shirking, the measurement settles down to one number that you call the curve’s length. But what if the curve is jagged and irregular? What if it is the coast of Scotland? You can start off with a surveyor’s glass – a big ruler – and measure from promontory to promontory. Then a long tape might measure point to point. Then a yardstick, then calipers, then a microscope. But this is useless: Unlike the smooth curve, the rocky coastline never provides just one “best” estimate of length. It depends on the scale of the map you want to draw – or your political motives. One researcher, Lewis Fry Richardson, who investigated this paradox nearly a century ago, looked in official references for the surveyed length of political borders between countries. Spanish authorities reckoned their border with Portugal to be 987 kilometers long, whereas the plucky Portuguese counted 1,214 kilometers. The Netherlands measured its border with smaller, poorer Belgium at 380 kilometers, whereas the Belgians counted 449 kilometers.
So how long is it? A useless question, as we have seen. But one way around the problem is to plot on graph paper the measurement you get for each size ruler you use. Of course, the measurements increase as the rulers shrink. But – happy surprise – they often do so at a near-steady rate. Start with a trivial example, a straight line. Say the first ruler you use happens to be exactly the length of the line. Now try a smaller ruler, half as big: it measures the line as two of its lengths. Another ruler, half again as big as the last; the line is four of its lengths. You get the picture. But now try measuring that jagged coastline mentioned earlier. Something unusual develops as you use ever-smaller rulers: The length you measure is growing faster than the rulers are shrinking. (130) And that phenomenon is measured by a quantity called fractal dimension. Begin simply. For a straight line, the fractal dimension is 1. And one dimension is exactly what we expect a straight line to have. But the British coastline, it turns out, has a fractal dimension of about 1.25. Does that make sense? Certainly. A rugged coast is more intricate than a one-dimensional straight line; but however numerous its crags and bays, its outline would not be so intensely convoluted as to fill a two dimensional square.
That is not all. The Australian coastline, less rugged than the Cornish, turns out to have a fractal dimension of 1.13. By contrast, the smooth South African shore has a dimension 1.02, only slightly rougher than a straight line. Another example: rivers. A U.S. Geological Survey study of the course of large American rivers found they have a typical fractal dimension of 1.3 in the East; but in the wilder West, it is 1.4. Again, the measurement fits our intuition of the difference between the rugged Colorado and the placid Charles. Other examples: If you measure the immensely intricate surface area inside the lungs, through which a network of branching bronchia stretch, you find that the total area is vast – something like that of a tennis court. But the fractal dimension is very close to 3. The lining is so convoluted and folded in upon itself that it partakes something of a three-dimensional nature.
What have we here? A new tool to measure, not how long, heavy, hot, or loud something is, but how convoluted and irregular it is. It provides science with its first yardstick for roughness. (131)
Mandelbrot, Benoit B., & Richard L. Hudson. The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004.
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