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detail, but rather than being geometrically similar, it is dynamically similar. Call this dynamical self-similarity. Still, there’s a clear parallel to standard statistical self-similarity: fractal dimension for normal physical objects roughly quantifies how much interesting spatial detail persists between magnification operations, and how much magnification one must do move from one level of detail to another. Similarly, dynamical complexity roughly quantifies how much interesting detail[1] there is in the patterns present in the behavior of the system (rather than in the shape of the system itself), and how much coarse-graining (and what sort) can be done while still preserving this self-similar of detail. This allows us to recover and greatly expand some of the conceptual underpinnings of fractal dimensionality as a measure of complexity—indeed, it ends up being one of the more accurate measures we discussed.

2.2 Effective Complexity: The Mathematical Foundation of Dynamical Complexity

Finally, what of Shannon entropy? First, notice that this account of dynamical complexity also gives us a neat way of formalizing the state of a system as a sort of message so that its Shannon entropy can be judged: the state of a system is represented by its position in configuration space, and facts about how the system changes over time are represented as patterns in how that system moves through configuration space. All these facts can easily be expressed numerically. The deeper conceptual problem with Shannon entropy remains, though: if the correlation condition fails (which it surely still does), how can we account for the fact that there does seem to be some relationship between Shannon entropy and dynamical complexity? That is, how do we explain the fact that where there is no strict, linear correlation between changes in dynamical complexity and changes in Shannon entropy, there does indeed seem to be

  1. We will consider how this quantification works in just a moment. There is a mathematical formalism behind all of this with the potential to make things far more precise.