One dial, then chaos

In 1838 a mathematician in Brussels named Pierre Francois Verhulst read Malthus and did what Malthus had not. Malthus had said populations grow by multiplying, each generation a fixed fraction larger than the last, and that food does not, so people must outrun their bread. Verhulst asked what happens to the growth when the room runs out, and wrote the correction as one term: the rate at which a population grows falls off in proportion to how full the place already is. The curve that comes out starts slow, rises steeply, and bends over to a ceiling it approaches and never crosses, an S lying on its side. He called it the logistic curve in a second paper in 1845 and fitted it to the census returns of Belgium, France and England. The fits were good. The ceilings were not; more on that at the end. The curve was forgotten for seventy-five years, rediscovered in 1920 by two men at Johns Hopkins who fitted it to the American census and thought they had invented it, and it has been the first equation in every population biology course since. Yeast in a flask follow it. Sheep in Tasmania followed it. Bacteria on a dish follow it until the dish is full.

The dial

Written for one year at a time, the equation is short enough to say aloud. Next year's population is this year's population, times the breeding rate, times the fraction of the room still empty. Call the population a fraction of the most the field can hold, so it runs from nought to one. If the field is nearly empty the last term is nearly one and the rabbits multiply freely. If the field is nearly full the last term is nearly nought and almost nobody is born. In between it balances, and for a breeding rate of two or two and a half the rabbits climb to a level and stay there, year after year, exactly as Verhulst's curve says.

In 1976 Robert May, an Australian physicist who had moved into biology at Princeton, published a paper in Nature about what happens when you keep turning the breeding rate up. Past three, the rabbits stop settling. They overshoot the level one year and undershoot it the next, a big year then a small year, for ever: the population has a period of two. Turn the dial a little further and the big years split into a bigger and a less big, the small years likewise, and the cycle has a period of four. Then eight. Then sixteen. The doublings come faster and faster, and at a breeding rate of about 3.57 they have all happened, and beyond it the yearly count never repeats. It is not random, since every year follows from the last by the same one line of arithmetic, and it is not predictable either, because two fields that start one rabbit apart are on different paths within a few years. May called it chaos, and his paper ended with a plea that every student be shown the equation early, so that they would know that simple rules do not promise simple behaviour.

Doubling

Mitchell Feigenbaum, at Los Alamos in 1975, took the places on the dial where the doublings happen and measured the gaps between them on a programmable pocket calculator, which was slow enough that he had time to notice something while it worked. Each gap was smaller than the one before by the same factor: 4.669, and then more decimals, the same number every time. He then tried a completely different equation, a sine curve instead of a parabola, and got 4.669 again. The number did not belong to the rabbits. It belonged to the act of doubling itself, and any system that goes to chaos by doubling its period, whatever it is made of, approaches the edge at that one rate. The claim was so strange that the journals turned the paper down for two years.

Then the number turned up in the world. Albert Libchaber in Paris heated a cell of liquid helium a few millimetres across from below and watched the rolls of convection begin to wobble, then wobble with a period of two, then four, with the gaps between the doublings shrinking by Feigenbaum's factor; he did it again in mercury. A graduate student in Santa Cruz named Robert Shaw put a microphone under a dripping tap and timed the drops: evenly spaced at a slow flow, then in pairs, then in fours, then with no pattern at all, the same road to chaos in plain water from a kitchen fitting. Hearts going into fibrillation, lasers, electrical circuits, the drive on a loudspeaker: the doublings and the number were in all of them. A quantity that was found on a calculator in a study of imaginary rabbits is a constant of nature, in the same sense that pi is.

The set

There is one more thing in the diagram, and it is the reason this page sits on the shelf next to the coastline of everything. Draw the tree of doublings with the dial along the bottom and the yearly populations stacked above each setting, so that a settled population is one line, a period of two is two lines, and chaos is a band of dust. Inside the dust there are clear windows where the branches come back, and the widest of them, near a breeding rate of 3.83, has exactly three lines in it. Now take the Mandelbrot set, the figure from the coastline page, and lay it on its side along the same axis. The big cardioid covers exactly the range of the dial where the rabbits settle. The round bulb attached to it covers the range where they alternate between two values. The next bulb is the period of four, and the needle that runs out to the point of the set is the chaos, with a small copy of the whole set sitting at the exact place where the window of three opens. The two figures are the same object. The Mandelbrot set is made by the same rule as the rabbits, multiply and add, carried out on numbers with two parts instead of one, and the rabbit equation is the slice of it that runs along the real line. Benoit Mandelbrot did not know this when he drew the set in 1980, and neither did May when he turned the dial in 1976. The pictures were put together afterward.

The instrument

One dial, the breeding rate, from one half to four. The dial mode shows sixty years of rabbits at the setting you choose, the tree mode shows every setting at once with your setting marked, and the set mode lays the Mandelbrot set under the tree so you can see the bulbs line up with the forks. Drag the dial slowly through three, through 3.45, through 3.57, and stop at 3.83 for the window of three. Tap the disc in the tree or the set to jump the dial to that place.