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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
(2⁶) + 32 (2⁵) + 4 (2²). 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 × 2⁶ + 1 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 1 ×

105

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