Real Number Walks versus Algorithmic Random Walks

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This Demonstration compares an irrational number walk (based on its digital expansion) with algorithmic random walks. The irrational number walks are mathematical constants , , (the golden ratio), (Euler–Mascheroni constant), , log(2) and , where is a prime; they seem to be indistinguishable from algorithmic random walks.

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Walks based on the digits of Liouville's constant and the like, which omit some digits entirely, clearly cannot be considered random at all.

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Contributed by: Khoa Tran and Laila Zhexembay  (January 2017)
(Illinois Mathematics Summer REU Program 2016)
Open content licensed under CC BY-NC-SA


Snapshots


Details

This Demonstration is based on [1], where walks are constructed based on real numbers as follows: the step of the walk is a unit step in direction , where is the digit of the expansion in base of the number. For example, in the case of base 2, the random walk is a one-dimensional walk that moves by for the digit 1 and by for the digit 0. Such real number walks can help visualize the randomness in the digits of famous irrational constants such as .

In the 3D case, the base is six, to match the six directions of the 3D axes.

Reference

[1] F. J. A. Artacho, D. H. Bailey, J. M. Borwein and P. B. Borwein, "Walking on Real Numbers," The Mathematical Intelligencer, 35(1), 2013 pp. 42–60. doi:10.1007/s00283-012-9340-x.



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