P. Vanouplines
University Library, Free University Brussels,Read all about the reasons for writing this paper ...
The purpose of this paper is to determine the so-called 'global fractal dimension' of the digits of pi-3, and to study the variability of the fractal dimension over different ranges. It is seen that the digits of pi have a fractal dimension of nearly 1.5, as could be expected for a random sequence. This result is related to the fractal Brownian movement which has the same fractal dimension. The value of 1.5 is not exactly reached by pi, certainly not at longer ranges. A comparison is made with digits calculated from two popular random number generators. This indicates that over the longer ranges, in the order of thousands of digits, the digits of pi are more random than those obtained with the random number generators.
I sincerely want to acknowledge Prof. Dr. Frank Plastria, Director of the Dept. of Industrial Locatin of the Vrije Universiteit Brussel (VUB), for his wounderful ideas that contributed to this paper in a substantial way, not only by his enthousiasm, but also by his distinguished ideas which he contributed to me during several discussions.
Secondly I wish to acknowledge Prof. Dr. Jan Mariën, Director of the Center for Statistics and Operational Research of the same university, who kindly accepted the task of refereeing this paper. Prof. Mariën did much more than that: also his ideas are markedly incorporated in this paper now, ideas that where mostly formulated during our train travels to and from the university. No need to say that this paper "would have been worse than it is now", without their input ...
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An (only slightly) outdated PostScript version of the complete paper is available (beware: the file contains almost 4 million bytes).
A word of advice
If it indeed takes a very long time for downloading this complete file, realise that if you were to download all digits of the decimal expansion of pi computed up to now, it would take at least a thousand times longer!
It is strongly advised to
make this hyperlink
both before and after reading the full paper.
Last updated on Wednesday 7 February 1996.
©Patrick Vanouplines.