the pendulum’s state space had two dimensions because two variables—the pendulum’s angle and its velocity—were necessary and sufficient to predict its future. They gave us exactly the information we needed to predict its angle and velocity an instant later, and an instant after that, on and on into the future. In that sense, the pendulum is an inherently two-dimensional system. It has a two-dimensional state space. The curse of high dimensions arises when we consider systems more complicated than a pendulum. For example, let’s take the problem that gave Newton a headache, the problem of three
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