Much of the fun of an amusement park ride results from its stomach-churning, mind-jangling unpredictability. The Tilt-A-Whirl, for example, spins its passengers in one direction, then another, sometimes hesitating between forays and sometimes swinging abruptly from one motion to another. A rider never knows exactly what to expect next.
Yet these complicated, surprising movements arise from a remarkably simple geometry. A passenger rides in one of seven cars, each mounted near the edge of its own circular platform but free to pivot about the center. The platforms, in turn, move at a constant speed along an undulating circular track that consists of thr