Re-Pairing Out-of-Phase: Chaos Theory and the Dripping Faucet

From Trialogues at the Edge of the West, by R. Abraham, T. McKenna, R. Sheldrake, pp. 24-5.

For example, a good laboratory for the study of chaotic dynamics is the dripping faucet. The dripping faucet was discovered as an ideal demonstrator for chaos theory because lectures are usually given in a physics lecture hall and they always seem to have sinks and faucets in the front. When you crack the tap a little bit, the water drips out very regularly. If you crack the tap a little more, the drips speed up, but they're still regular. When you crack it a little more, they sound irregular, like rain dripping off a roof. If you measure the time between drops, and make a list of these numbers, you have the paradigmatic example of a chaotic time series.

Somebody decided to study this dripping faucet seriously after seeing it in physics lectures. This person, Rob Shaw, is now one of the leading people in chaos theory. He did a very fine study by placing a microphone in the sink where the drop would hit it, getting an electronic beep, putting the time intervals into his computer, and analyzing the results. These portions of his study all belonged to level one, the lowest layer of figure 3, the physical world. Then he made a model for the dripping faucet on level three. In this mathematical model, the water drop gets bigger and bigger, and when its mass reaches a critical value the drop falls off the faucet. From this model on level three, Shaw wrote a computer program to simulate it, which was a model on level two. He ran the simulator and produced data that was almost exactly like the experimental data from the actual faucet on level one. Opening the model's tap eventually changed the simulated data from periodic to chaotic through a bifurcation.

This is an example of modeling in the three-level context. The point of this kind of modeling is to gain understanding as part of a hermeneutic circle. You look at the data, try to build a model, and you fail. You then observe in a different way, which helps you to build a better model, and, as the circle turns, the level of your understanding grows. This is what Rob Shaw did with the dripping faucet. The different way of observing data that came to him from the model was a method now known as chaoscopy. In this method, you take the sequence of numbers the time between drops-and visualize a vertical column of numbers. Then you make a copy of this column of numbers over to the right. You whack one number off the top of this second column and move the entire column up one number. Now you have a column of pairs of numbers. Then you plot these pairs in a plane figure as a series of points.

There's a film available from Aerial Press that shows a machine actually doing this. From totally chaotic data viewed in this particular way, through chaoscopy, you geta set of points in a plane. If the data were really random, the dots would be all over the plane. Instead they lie along a smooth curve! This indicates a chaotic attractor.

A hidden order in chaos is revealed by a new way of looking. From the observation of the data in this way, the smoothness of the curve suggests, to a chaos theorist, a model that you can actually take off the shelf and apply to other data.