Chaos - James Gleick [38]
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* For convenience, in this highly abstract model, “population” is expressed as a fraction between zero and one, zero representing extinction, one representing the greatest conceivable population of the pond.
So begin: Choose an arbitrary value for r, say, 2.7, and a starting population of .02. One minus .02 is .98. Multiply by 0.02 and you get .0196. Multiply that by 2.7 and you get .0529. The very small starting population has more than doubled. Repeat the process, using the new population as the seed, and you get .1353. With a cheap programmable calculator, this iteration is just a matter of pushing one button over and over again. The population rises to .3159, then .5835, then .6562—the rate of increase is slowing. Then, as starvation overtakes reproduction, .6092. Then .6428, then .6199, then .6362, then .6249. The numbers seem to be bouncing back and forth, but closing in on a fixed number: .6328, .6273, .6312, .6285, .6304, .6291, .6300, .6294, .6299, .6295, .6297, .6296, .6297, .6296, .6296, .6296, .6296, .6296, .6296, .6296. Success!
In the days of pencil-and–paper arithmetic, and in the days of mechanical adding machines with hand cranks, numerical exploration never went much further.
* With a parameter of 3.5, say, and a starting value of .4, he would see a string of numbers like this:
.4000, .8400, .4704, .8719,
.3908, .8332, .4862, .8743,
.3846, .8284, .4976, .8750,
.3829, .8270, .4976, .8750,
.3829, .8270, .5008, .8750,
.3828, .8269, .5009, .8750,
.3828, .8269, .5009, .8750, etc.
A Geometry
of Nature
And yet relation appears,
A small relation expanding like the shade
Of a cloud on sand, a shape on the side of a hill.
—WALLACE STEVENS
“Connoisseur of Chaos”
A PICTURE OF REALITY built up over the years in Benoit Mandelbrot’s mind. In 1960, it was a ghost of an idea, a faint, unfocused image. But Mandelbrot recognized it when he saw it, and there it was on the blackboard in Hendrik Houthakker’s office.
Mandelbrot was a mathematical jack-of–all-trades who had been adopted and sheltered by the pure research wing of the International Business Machines Corporation. He had been dabbling in economics, studying the distribution of large and small incomes in an economy. Houthakker, a Harvard economics professor, had invited Mandelbrot to give a talk, and when the young mathematician arrived at Littauer Center, the stately economics building just north of Harvard Yard, he was startled to see his findings already charted on the older man’s blackboard. Mandelbrot made a querulous joke—how should my diagram have materialized ahead of my lecture?—but Houthakker didn’t know what Mandelbrot was talking about. The diagram had nothing to do with income distribution; it represented eight years of cotton prices.
From Houthakker’s point of view, too, there was something strange about this chart. Economists generally assumed that the price of a commodity like cotton danced to two different beats, one orderly and one random. Over the long term, prices would be driven steadily by real forces in the economy—the rise and fall of the New England textile industry, or the opening of international trade routes. Over the short term, prices would bounce around more or less randomly. Unfortunately, Houthakker’s data failed to match his expectations. There were too many large jumps. Most price changes were small, of course, but the ratio of small changes to large was not as high as he had expected. The distribution did not fall off quickly enough. It had a long tail.
The standard model for plotting variation was and is the bell-shaped curve. In the middle, where the hump of the bell rises, most data cluster around the average. On the sides, the low and high extremes fall off rapidly. A statistician uses a bell-shaped curve the way an internist uses a stethoscope, as the instrument of first resort. It represents the standard, so-called Gaussian distribution of things—or, simply, the normal distribution. It makes a statement about the nature of randomness. The point is that when things vary, they