Much of the fun of an amusement ride results from its stomach-churning, mind-jangling unpredictability. The Tilt-A-Whirl for example, spins its passengers from in one direction, then another, sometimes hesitating between forays and sometimes swinging abruptly from one motion to another. The rider never knows exactly what to expect next.

Kautz and Huggard, students at Northern Arizona University in Flagstaff, Arizona, studied the ride's motion in the hope that it might entertain students studying chaos theory, a branch of mathematics that analyzes the behavior of unpredictable systems such as weather patterns. The Tilt-A-Whirl consists of seven cars which travel around an undulating track, spinning in either direction and independently of each other. It is impossible to predict how a car will behave during the ride.

Moreover, what happens to an individual car is highly dependent on the weight of its passengers and where they sit-a characteristic of chaotic systems, in which small changes in the starting conditions produce wildly different results. The two researchers also found that passengers can dramatically affect what their car does by throwing their weight in a certain direction at a key moment. In 100,000 simulated rides, no two are exactly the same.

The ride was invented in 1926 by Herbert Sellner of Minnesota. The Tilt-A-Whirl first operated in 1926 at an amusement park in White Bear Lake, Minnesota. Most likely, the ride's inventor, Herbert W. Sellner discovered its unpredictable dynamics not through mathematical analysis but by building one and trying it out.

The mathematical model indicates that when the platforms travel at very low speeds along the track, the cars complete one backward revolution as their platforms go over each hill. At high speeds, a car swings to its platform's outer edge and stays locked in that position.

The mathematical naivety of the amusement park industry may be a thing of the past. Kautz and Huggard's analysis has caught the attention of manufacturers who are interested in designing more chaotic motion into future models to make them more fun. An engineer with Chance Operations of Wichita, Kansas, says computer programs for designing rides should be able to take account of chaotic motions and introduce extra thrills.