2? +0 x 2'+0 x 2°(in which the last term, arbitrarily, is 2° = 1, since anything to

2? +0 x 2'+0 x 2°(in which the last term, arbitrarily, is 2° = 1, since anything to the power O = 1). This can be written even more simply as a series of 0’s or 1’s, their presence indicating whether the power represented by each place in the left to right descending sequence of powers of two participates in the sum of the partition. It is in this way that in binary numbers, 100 = 110010. As another example, if we similarly partition the decimal number 729 = 512 (2°) + 128 (2”) + 64 (2°) + 16 (2°) + 8 (2°) + 1(2°), we find that its binary transformation results in 729 = 1011011001, the 0’s representing the descending powers of two that are absent in the powers of two partition. One can compute the binary representations of lower valued numbers immediately; for example, 4 = 1 x 27 +0 x 2'+0 x 2°s0 that there is a 1 in the multiply-the-power-of- two column and 0 the power 1 and power O columns so in binary representation, 4 = 100. Similarly, 6 = 1 x 22+ 1 x 2'+0 x 2° making the binary transformation of 6 = 110. It was the co-inventor (with Isaac Newton) of the calculus, Gottfried Wilhelm Leibniz, in about 1665, who fully developed the binary representation of all decimal numbers. |n a state of wonderment about the simplicity, power and completeness of this 1 and 0 encoding, he is said to have the beliefs that 0 symbolized the emptiness of the universe’s beginnings, 1 represented the complete fullness of God and that this transformation served as metaphoric evidence consistent with God’s creation of the universe out of nothing. The simplicity of binary expressions as in the dynamics of hard spheres or rotations on the circle as well as the transformations such as 729 = 1011011001 make them propitious for exemplifying the methods for computing the entropies of the growth rate of the possible, called the topological entropy, Hr, and the probable, the metric entropy, Hy, which was introduced in a previous chapter called “Sensual In-Between Entropies.” The following exemplify the computations of measures of topological and metric entropies, Hr and Hy, another computable idea called algorithmic complexity, AC and finally, the well known (to statisticians) standard run score, src. Their descriptions have as their purpose a demonstration for the reader that these apparently abstract, perhaps nebulous sounding, words can be transformed into well-defined, concrete, quantitative and computable form of reality. 106 HOUSE_OVERSIGHT_013606