Rescaled range analysis and the fractal dimension of pi

P. Vanouplines

Processing the digits of pi-3

Hurst studied the water levels in a reservoir. The measured water levels are a result of both inflow and outflow of water. These water levels can be presented in a graph, with time in the abscissa and the levels in the ordinate. Consider now the digits of pi-3. The digits can be handled as if they were observations of some quantity over a time span, just like the in- and outflow of water in a reservoir. Every digit adds to the general look of a profile-type curve. The n-th digit of the decimal expansion of the mathematical constant, determines together with all the digits, from the first one to the digit number n-1, the value of some kind of cumulative sum at instant n. The cumulative value at instant n+1 is determined by the values of the digits 1 through n+1.

The most simple way for creating a profile-type curve on the basis of the digits of pi-3 would be to make the cumulative sum of all digits. That results in the following numerical sequence:

The cumulative sum always increases, which is difficult to handle. The next cumulative sum can be used, in order to obtain a curve which resembles the profile-like curve in a better way:

This means that for a given digit number p, the cumulative sum pip is made of all pii-digits (from the first to the p-th digit of pi-3), which are each multiplied by two and from which nine is subtracted. The graphical presentation of these partial cumulative sums for the hundred first digits of pi-3 is given in figure 3.

Figure 3 Graphical representation of the modified cumulative sum of the hundred first digits of pi-3 (by multiplying each digit with two, and by subtracting nine from each).

Another method is to look at the distances between two successive digits along a circle (Plastria, 1993). This straightforward method is illustrated for a binary system in figure 4. A number represented in the binary system consists of a series of zeroes and ones. If a zero is followed by a one, then the difference, interpreted as a distance, is equal to minus one. In the reverse case, the difference is plus one. The difference of two digits of the same value is zero.

Figure 4 Circular representation of a binary system.

Table 2 gives in the first column all possible sequences of the two digits (zero and one) and in binary system. The second column presents the circular difference value.

Next, consider a ternary system, in which the possible digits are zero, one and two. Take the example of the sequence '01'. The circular difference is then 0-1=-1. That means that when 'moving' clockwise, the circular difference is negative, and vice versa. This illustrated in figure 5.

Figure 5 Circular representation of a ternary system.

Table 3 gives the circular difference values in a ternary system. Note the absolute values of the differences are one. Indeed, as we can see from figure 5, the shortest distance between zero and two is not two. When going counter clockwise, one finds the shorter distance +1.

Let us finally look at the decimal system. Analogous with what was said about the ternary system, the absolute value of the biggest distance is only five. Note also that the sum of all circular differences is always zero, for any system (binary, ternary, or decimal). That means that, for a long sequence of random numbers, the sum of the circular should be zero. If that is not the case, then the computed sum should be smaller than the variance of the circular differences. Table 4 gives the values of the circular difference values in a decimal system.

Figure 6 gives the graphical representation of the modified cumulative sum for the hundred first digits of pi-3, computed with the circular difference method for the decimal system. Compare this with figure 3. The last method produces smaller deviations from the zero-axis, since the biggest difference between two digits is smaller.

Figure 6 Graphical representation of the modified cumulative sum for the hundred first digits of pi-3 (by converting each two digits to their decimal circular difference, see text).

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Last updated on Sunday 19 November 1995.
©Patrick Vanouplines.