Rescaled range analysis and the fractal dimension of pi

P. Vanouplines

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

Conclusion and recommendations for further research

With the help of the rescaled range analysis it is possible to compute a fractal dimension for the first 1,25 million digits of pi-3. This fractal dimension of pi is not 1.5 as expected, but 1.45. This means that the decimal expansion of pi-3 is maybe not entirely random. Remarkable is that for longer ranges, the computed fractal dimension is not longer smaller than 1.5, but bigger. This would mean that there is some persistency in the digits.

Over a large range of digits, the rescaled range analysis cannot distinguish pi from the Knuth-Press random number generator. This means that pi is a 'good' random sequence, as already stated in other papers by various authors.

Borland's TurboPascal 6 random number generator gives differences compared to the Press-Knuth random number generator, when the digits are analysed with the rescaled range method.

It is clear that more than the present 1.25 million digits of pi should be studied. This would bring clarity in the behaviour of pi (and random number generators) at large ranges.

Other 'classical' constants (like e, Champerowne's number, ...) should also be studied with R/S. It would be interesting as well to study series of random numbers generated with other 'good' random number generators in order to check their behaviour at larger ranges. Such investigations can bring clarity in the 'odd' fractal dimension for pi of 1.45 and explain what happens at the larger ranges.

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