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ignorance, the emptiness and its mystery. Computations of the entropy of systems in motion convert questions and answers concerning the detailed workings of the leg’s neuromuscular machinery to global statistical descriptions of more abstract thematic motifs, forms, expressed in the dance. Patterns of behavior of these properties can suggest intuitive ideas and imagery about global mechanisms, approach/avoid, smooth/discrete, wild/tame, as well as correlated and objective physical observables.

To learn more about this abstract, topology tinged (none numeric) style of model building, we can go to school on a long studied physical example. It connects a simple and well understood real world observable with abstract statistical patterns resulting from motions using the one-to-one correspondence (the equivalence relation called isomorphism) between their entropies. As we have discussed, the Stanford mathematician and Field’s Medal Winner, Donald Ornstein, proved that in statistical studies of even point-to-point unpredictable, chaotic systems, entropy is the only isomorphism. The hardware of this physical example is what the statistical physicists call a dilute gas of some fixed number, n, of uniform hard spheres, moving scatterers, that, absent of dissipative friction, wander continuously around, changing their directions when bumping into each other. In a two dimensional bounded arena of randomly rolling balls, this game has been called Sinai’s billiards. It was named for previously mentioned Ya Sinai, an eminent Russian mathematician He is now at Princeton and was previously a student of Andrei Nikolaevic Kolmogorov, the Russian guru of many of the Twentieth Century’s world-class Russian mathematicians. Kolmogorov axiomatized the field of probability and, more relevantly, initiated the theory of statistical descriptions, the ergodic theory, of nonlinear dynamical systems. In the language of statistical physics, we will see that the same system produced by high number of elements executing Newton’s deterministic laws can be generated by a so-called random system such as that resulting from flipping a suitably biased coin. Our example can also serve as a metaphor, used extensively in the mathematical biology of the late Professor Art Winfree, for the temporal features of life on a topological circle: the natural irregularities of the recurrent beat of the heart, the in and out breathing of lungs, the

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