division between top and bottom and do so in a consistent way), then we can keep

division between top and bottom and do so in a consistent way), then we can keep score with a random looking binary series such as 11001001010.... that describes the sequence of rotations. The advantage that accrues by doing so is that this coin flip counting eliminates details in favor of a computable over all measure and supports several forms of entropy calculations for its use in deciding if this system is behaving like that system, an equivalence relation. One can imagine a series of coin flips with 1 being heads and 0 being tails such that the statistics of a characteristic series is determined by the fairness of the coin. As noted above, Donald Ornstein’s famous theorem says that the entropy of these kinds of hardware and software systems is the only general basis for finding correspondence between characterizations of two such irregularly behaving systems. The important idea here is that a series of 1’s and 0’s may not be identical but the two systems can be isomorphically equivalent with respect to their entropy. Notice again that the physical process of hard spheres bouncing off each other on a flat surface has been captured by an abstract representation in binary numbers that, like a series of coin flips, can be quantified as entropies (which would be maximal for an ideal, fair coin). After describing the process of real number representation by the binary code, we will show how entropies can be computed for these binary series. We remind ourselves that we are struggling to obtain some kind of knowing in a representative system manifesting the tension and mystery between emptiness and form. We can translate all finite real numbers into this language, making them accessible to standard entropy computations. The following discussion of the process of transforming numbers into binary series is in the spirit of the famous number theory theorem that every natural number (the positive integers such as 1, 2, 3, 4...) can be expressed as the sum of at most four squared numbers. Encoding any number by a series of 0’s or 1’s in what is called a binary transformation, begins with its separation, called partition, into a sum of powers of 2, for example, 100 = 64 (2°) + 32 (2°) + 4 (2). A short hand description of this sum begins with a form indicating the presence or absence of each successive power by a 1 or 0 coming before the relevant power of two; i.e. 100 = 1 x 2°+1 x 2°+0 x 24+0 x 2°+1 x 105 HOUSE_OVERSIGHT_013605