Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov

and the (k – j) rows up (we “lost” (k – j) IE).

      In the second step, we take the point Cj for the new beginning of coordinates. In this case, the second step consists in enclosing the j IE to some points D₁, D₂, …, Dj of the new uncertainty interval CjC_j+₁. The coordinates of the points D1,D2, …,Dj relative to the new beginning of coordinates (the point Cj) are at the(n – 1) th columns of the arithmetic square above the binomial coefficient F(n – 1, j), that is,

      This process ends when either all the steps of the measurement algorithm or all the indicatory elements will be exhausted.

       1.10.4. An example of the optimal (n, k, 1)-algorithm

      As an example, let’s consider the operation of the optimal (3, 3, 1)-algorithm on the initial segment [0, 20] (Fig. 1.11).

Fig. 1.11. The optimal (3, 3, 1)-algorithm.

      The optimal (3, 3, 1)-algorithm is realized in the three steps and uses three IE in this case. Note that the number F(3, 3) = 20, which is located at the intersection of the third column and the third row of the arithmetic square (Table 1.1), is equal to the length of the initial uncertainty interval for this algorithm (Table 1.1). The coordinates of the applications of IE in the first step of (3, 3, 1)-algorithm are in the third column of the arithmetic square (Table 1.1) above the number 20 (in bold).

      The first step of the (3, 3, 1)-algorithm on the initial segment [0, 20] consists in the enclosing of the three IE to points 1, 4, 10 (Fig. 1.10). After the first step, based on the indications of the IE, four situations may arise (Figs. 1.10(a)–(d)).

      The second step: Situation (a): For this situation, the measurement process ends because all the IE are to the right of point X.

      Situation (b): For this situation, we have only one IE, which is enclosed to point 2.

      Situation (c): For this situation, we have two IE, which are enclosed to points 5 and 7, respectively.

      Situation (d): For this situation, we have three IE. In this case, point 10 is the new beginning of the coordinates and three IE are enclosed to points 1, 3, 6 relative to the new beginning of coordinates (point 10). By summing up the numbers 1, 3, 6 to the number 10, we get the coordinates of the enclosed points of three IE at the next step: 11, 13, 16.

      After the second step, the following situations may arise: [1, 2], [2, 4], [4, 5], [5, 7], [7, 10], [10, 11], [11, 13], [13, 16], [16, 20]. Note that for situations [1, 2], [4, 5], [10, 11], the measurement process ends at the second step.

      The third step: For the situations [2, 4], [5, 7] and [11, 13], the third step is the enclosing of the one IE to points 3, 6, 12, respectively.

      For the situations [7, 10] and [13, 16], the third step consists in the enclosing of the two IE to points 8, 9 or 14, 15, respectively.

      For the situation [16, 20], the third step consists in the enclosing of the three IE to points 17, 18, 19.

       1.10.5. The extreme particular cases of the optimal (n, k, 1)-algorithm

      Now, we will consider the extreme special cases of the optimal (n, k, 1)-algorithm, that is, when n = 1 or k = 1. For the case n = 1, the expression (1.36) takes the following form:

      It is easy to verify that in the case of n = 1, the optimal (1, k, 1)-algorithm reduces to the readout algorithm discussed above (Fig. 1.7). For the case k = 1, the expression (1.36) takes the following form:

      It is easy to verify that in the case of k = 1, the optimal (n, 1, 1)-algorithm reduces to the counting algorithm discussed above (Fig. 1.5).

       1.10.6. The importance of the binomial algorithms for mathematics and computer science

      Thus, the main mathematical result, obtained in Ref. [16] at the synthesis of the optimal (n, k, 1)-algorithm, is the fact that these studies led us to the arithmetic square and binomial coefficients. We emphasize that this result is much unexpected because by starting the synthesis of this algorithm, we did not assume its connection with the binomial coefficients. This gives us a reason to call the (n, k, 1)-algorithms the binomial algorithms of measurement.

      What is the significance of the binomial measurement algorithms for mathematics? Here, it is important to emphasize that the optimal (n, k, 1)-algorithm is a generalization of the counting algorithm (Fig. 1.5), which, as mentioned above, historically underlies the elementary number theory, the fundamentals of which are outlined in Euclid’s Elements. Therefore, we can assume that the above-considered binomial measurement algorithms can be used for further development of the elementary number theory created by the ancient Greeks.

      There is another idea arising from the interpretation of the optimal (n, k, S)-algorithms as new positional representations of natural numbers. With this approach, the binomial algorithms can be of practical importance for modern computer science. And maybe some of our readers, who can be fascinated by such unusual numeral system, will design a new (binomial) computer. Those readers, who are interested in the binomial computers can refer to the book [136] of the Ukrainian scientist Alexey Borisenko (Sumy University).