When I was a physics student, I spent most of my time studying things that didn't exist--frictionless pulleys, massless springs, gravitational fields that don't change with height. Scientists have recognized for some time that real systems interact with their environment in ways that elude practical measurements, and over time these effects accumulate. So we've known that idealized solutions can describe systems for only so long. What we didn't realize was just how brief this time period could be.

In the 1980s researchers learned that for many real systems the myriad uncooperative complications of nature cause observations to disagree with predictions extremely quickly. You can, for example, take all the temperature, pressure and wind-speed readings you want, but you just won't have enough information to forecast the weather accurately more than seven days ahead. The reason is that the effects of tiny perturbations, even a jet flying over Salt Lake City, can build much more rapidly than scientists had realized, altering the weather in unforeseen ways. Systems that tumble rapidly into unpredictability are aptly termed "chaotic."

Now, thanks to a delightful device developed by Mahlon Kriebel, a professor of neuroscience and physiology at the SUNY Health Science Center at Syracuse, you can explore the subtleties of chaos at your own kitchen table. Kriebel's apparatus is a slow-motion version of another chaos classic--the dripping faucet. By dropping colored water droplets through mineral oil, Kriebel slows them enough that the onset of chaos can be readily observed and studied.

A graduated cylinder with a volume of 1,000 cubic centimeters makes an ideal chimney to hold the mineral oil, but a tall, clear flower vase would also work well. For the reservoir to hold the colored water, you can use a hot-water bottle with tubing attached. Cut the tubing and mate it with a barbed hose adapter to some Tygon tubing. You'll need a hose clamp to control the flow rate through the tubing and nozzle. Almost any hardware store should have the necessary hose adapter, clamps and tubing. To keep the hot-water bottle well above the nozzle, I fastened the bottle to the back of a chair and placed it on top of my kitchen table.

Kriebel fashions his nozzle from the tip of a pipette, but eyedroppers are easier to come by. The opening on mine was too wide, so I filled the tip with candle wax and then used a heated sewing needle to melt a tiny hole in the wax. Hold the tip of the needle against the wax with a pair of pliers and touch a hot soldering iron to the needle right near the dropper. Applying firm and steady pressure will quickly bore the needle through the wax plug. After that's done, insert your nozzle into the Tygon tubing and secure the connection with a liberal dose of aquarium cement.

Next, to keep the tubing in place after you've situated it, tie it to a length of coat hanger. Use sewing thread and not twine, because the latter sheds fibers that will contaminate the oil. Then install the assembly into the cylinder with the nozzle's exit hole centered about 10 centimeters (four inches) below the top of the cylinder.

Lash a second length of Tygon tubing to another coat hanger and install one end at the bottom of the cylinder. Pinch off the other end of the tube with a hose clamp and set it in an empty wine jug. This setup will enable you to siphon some of the colored water that collects at the bottom so that you can periodically return it to the hot-water bottle.

Fill the cylinder with mineral oil from your local drugstore until the liquid just covers the nozzle. Then dye some water with food coloring and pour the solution into the hot-water bottle. Loosen the upper clamp just a bit to allow droplets to form slowly on the nozzle and fall through the oil. You're now ready to begin your excursion into chaos.

The slowest flow rates produce droplets that are all about the same size, and so they fall at nearly the same rate. They reach the bottom one after another, resulting in a pattern that can be recorded as 1, 1, 1, 1,... At some slightly higher flow rate, the drops fluctuate in size. The larger drops fall faster in the mineral oil than the smaller ones do (the former have a higher terminal velocity because of their greater mass per surface area), and so they catch up and push into the smaller drops, thus falling in clumps of two. At this flow rate, the pattern is 2, 2, 2, 2,... If you increase the flow rate a bit more, you may find a setting at which they will tend to fall in groups of three.

At even faster rates, all heck breaks loose, with data that are not predictable but aren't random either. To understand why, consider the rolling of dice, which is similar to the falling of droplets in that both depend on intractable factors. Change the spin rate or the trajectory of a die slightly as it leaves your hand, and you completely change how it rolls. Likewise, the size of a droplet and its rate of descent depend on uncontrollable variables such as its vibration as it pulls away from the water stream and the fluctuating pressures inside the nozzle.

A truly random process, however, is completely unpredictable; whatever you observe one instant has no relation to what came before and no effect on what comes after. In this sense, rolling a die is random, because the odds of getting a "1," for instance, are always one sixth, regardless of how the die has rolled before. But falling drops behave differently, because although one state may not determine the next, it can affect it.

For example, if I continue rolling a die I will eventually roll two consecutive 1s. But at a fast flow rate I never observed two single drops arriving one after another. I thus concluded that creating a single drop always caused the system to enter a state that produced only multiple drops. In other words, the arrival of a single drop guaranteed that a cluster would follow. At least that was some information. And I knew even less about what was coming after that. The fact that each state affected the next but that the outcomes became rapidly less certain is the hallmark of chaos. Chaos inhabits the gap between the perfect predictability of a frictionless pendulum and the pure randomness of rolling dice.

Chaotic systems are easy to spot with the help of a special graph called a return map. For the water-drop data, plot your first point by taking the first number in the series as the x coordinate and the second number as y. For the second point, use the second datum in the series as x and take the third as y. Keep going until you've run out of data. For the steady drip of the low-flow faucet, the points all cluster near a single location. For random data, all pairs are equally likely, so the points would be scattered over the map. A chaotic system, on the other hand, is not random. Because each event affects the one that follows, some combinations are more likely than others. If the connection is strong enough, some areas on the map will contain no points at all.

Return maps can open up a universe filled with chaotic delights. Indeed, countless everyday phenomena are chaotic, such as the time between pedestrians passing by on a busy street, the distance between blossoms on a vine and the spacings between stripes on a cat. Plot these variables to discover how truly chaotic our world is.

For a good primer on chaos theory, read Chaos, by James Gleick. More information about this and other projects can be found on the Society for Amateur Scientists's Web site . You may write to the society at 4735 Clairemont Square, PMB 179, San Diego, CA 92117, or call 1-401-823-7800.

The Society for Amateur Scientists (SAS) is a nonprofit research and educational organization dedicated to helping people enrich their lives by following their passion to take part in scientific adventures of all kinds.