Repetition doesn’t always have to be humdrum. In mathematics, it is a powerful force, capable of generating bewildering complexity. Even after decades of study, mathematicians find themselves unable to answer questions about the repeated execution of very simple rules — the most basic “dynamical systems.” But in trying to do so, they have uncovered deep connections between those rules and other seemingly distant areas of math. For example, the Mandelbrot set, which I wrote about last month, is a...