Rescaled range analysis and the fractal dimension of pi

P. Vanouplines

University Library, Free University Brussels,
Pleinlaan 2, 1050 Brussels, Belgium

Computation of the fractal dimension of a profile-like curve

One popular way to compute the fractal dimension of a set is the block-count algorithm. It is popular since it can both be used for computations by hand, and also be programmed easily.

Reassemble the map of the Norwegian coastline. Collect some five and ten sheets of transparent paper. On the first sheet you draw a grid of squares measuring precisely one by one millimetre. On the second sheet you draw a grid with two by two millimetre squares. On the third you use four by four millimetres. Then eight by eight millimetres.

Put the first sheet on top of the map. Count through how many squares the coastline crosses. Repeat this for all the sheets you have. Notice that it is even advised to use different origins for the different sheets. Write the results in a table and add some computed values. The following table is an aid to process your data. The headings in this table are:

  1. l, size of the squares (in millimetres);
  2. n(l), number of squares (of size l) crossed;
  3. N(l), length of the coastline (in millimetres, simply the multiplication of the two previous values);
  4. log(l), logarithm of the size of the squares;
  5. log(N(l)), logarithm of the length of the coastline;
  6. D, fractal dimension (see later).

The values in the following table are fictive.

ln(l)N(l)log(l)log(N(l))D
1699869980.0003.845
1.39
2267953580.3013.729
1.35
4105442160.6023.625
1.32
842433920.9033.530
1.31
1617127361.2043.437
1.33
326821761.5053.338
1.33
642717281.8063.238
1.44
1281012802.1073.107

Notice that the coastline (column 3) becomes apparently longer when it is measured with a finer grid. This should be understood as the coastline being measured more precisely with the smaller grid. Is there any 'best' length for the coastline? The answer is negative for any physical data. By taking a finer and finer grid, one finally arrives at the size of the smallest grains. An even smaller grid size is senseless.

The data in columns 4 and 5 can be represented in a graph, as follows in figure 1.

Regression line coastline data

Figure 1 Graphical representation of the fictive coastline data. The regression line fits the data as Log(coast length) = 3.838 - 0.339Log(grid size), with a regression coefficient of 0.999.

Column 6 displays the fractal dimension, each value computed from columns 4 and 5 as the slope of the line between two observed points (Log(grid size),Log(coast length)). Indeed, the fractal dimension for the box count algorithm is given as D = 1-m, where m is the slope of the regression line.

One can easily find a similar method in order to compute the fractal dimension for the digits of pi-3. The first one hundred digits of pi-3 are:

     14159265358979323846264338327950288419716939937510
     58209749445923078164062862089986280348253421170679

These digits are presented in a curve in figure 2. The shape of this curve looks like a profile of (very) mountainous area. We therefore call this type of curve not a 'coast-line'-like curve, but rather a profile-like curve.

profile-like curve

Figure 2 Graphical representation (as a 'profile-like curve') of the first hundred digits of pi-3.

However, there is one severe objection for applying the box count algorithm in this case. The dimension of the x- and the y-coordinate are the same for the cartographic representation of the coastline: both are expressed in units of length. This is not the case in figure 2. An adequate method for this type of data is the rescaled range analysis.


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