Fractals, Chaos and the Mandelbrot Set Workshop - Pt. 2

The following article is an excellent, concise explanation of one of the key ideas in chaos theory, "sensitive dependence on initial conditions." It appeared in the Globe & Mail on Feb. 24, 1990.

The Real Reason for
Chaotic Behaviour of the Cue Ball

By Derek York

Many years ago as a teen-ager in northern England, I used to sneak into a dingy billiard hall with a friend and play snooker for an hour. Later we graduated to more respectable places to play, but I never became very good at snooker or billiards. Now thanks to science, I know why. Well almost. It is all to do with chaos and an electron at the edge of our galaxy. How was I to know its gravitational field was important as I chalked my cue tip and judged the angles?

Of course, I am greatly exaggerating. And yet, that little electron and the cue ball I used to strike were curiously linked by gravitational fields. Some theorists of chaos, one of the hottest fields in mathematics and physics right now, like to stun their listeners with a comment such as: "If we neglect the effect on the cue ball of the gravitational field of an electron at the edge of our galaxy, our calculations of where the balls will be roughly one or two minutes after the cue ball is struck (neglecting friction, so the balls keep moving for more than a minute) will be hopelessly wrong."

Now, those of you who never had time for billiards because of homework will immediately say: "This is absurd." After all, the edge of our galaxy is said to be about 100,000 light-years away. Not only that, an electron is unimaginably small - about one thousand billion, billion, billionth of a kilogram. In fact, celebrated mathematician Michael Berry of England calculates that the angle through which the cue-ball would typically be deflected by the distant electron's gravity would be so small that it would be represented very roughly as a fraction of a degree by a decimal point followed by 100 zeros, then a one. Such a minuscule deflection is impossible to measure. Yet, if snooker balls kept rolling for more than a minute after being struck, its effect would become profound.

And this is the nub of chaos theory. Little things mean a lot. Incredibly minute uncertainties in the initial state of a system lead to total uncertainty in our best conceivable predictions of the futures of those systems. For instance, if you knew more or less a point in space that marked the edge of the solar system at its birth, it would be impossible, using that slightly inexact information to plot where the edge is now. It would therefore never be possible to predict the weather accurately beyond a few weeks, or the behaviour of the planets more than a few hundred million years from now (that may seem like a long time to humans, but it is a trivial amount of time astronomically and geologically). The theorists tell us it is because minute effects may grow exponentially in such non-linear systems. While this is difficult to explain in a simple way for the weather or the solar system, the snooker illustration is far simpler to grasp. It is all because of the way snooker balls bounce off each other. Because of the balls' curved ivory-like surfaces, the incredibly tiny effect of the electron at the edge of the galaxy is multiplied by 10 every time the balls collide, according to Dr. Berry. So after 101 collisions, the decimal point I referred to earlier will have marched 101 positions to the right and the effect of the distant electron will have grown to 1 degree. One more collision means a 10- degree change in direction and two more gives 1,000 degrees, an enormous effect in a short time. Snooker balls, of course, don't run for a couple of minutes and the effect is therefore hypothetical.

However, there is an essentially frictionless form of snooker taking place all around us in the air we breathe. Imagine dancing oxygen molecules to be minute snooker balls, unceasingly bouncing off each other, forever moving in new directions. Such imaginary molecular snooker balls would have incredibly tightly curved surfaces, leading to an error amplification factor of roughly 600, instead of 10 for real snooker, per molecular collision, according to Dr. Berry. He therefore calculated that after about 50 collisions, which occur in less than a blink of an eye for molecules, not taking into account the gravitational pull of one electron at the edge of the galaxy would lead to enormous error in estimating what directions an oxygen molecule went. So even assuming a purely classical Newtonian world, as Dr. Berry did to make a point, we see that there is an extraordinary and unavoidable unpredictability all around us. This is the meaning of chaos. When the limits to predictability of the smallest particles of matter are added to this, it is remarkable that on a large scale there is so much order.

Dr. York is a geophysicist and professor at the University of Toronto. His article is reproduced here with his kind permission.