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# Falling Into Chaos

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 bot

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.

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