Bernard - 2014-04-05

CloisterWalk

Introduction

Thousands of years ago, geometers discovered that the diagonal/edge ratio in a square is not a fraction of natural numbers. This early "irrational number" is now calculated as square root of 2. Other famous examples of irrational numbers include the length/diameter ratio in a circle, or Pi (3.1415…). An interesting property of irrational numbers is that their decimal expansions never repeat and seem "random". This randomness can be examined using graphs known as "walks" [1]. For example, millions of decimals of Pi were used to produce a huge "random" walk using powerful computers [2,3]. Graphical methods used to represent numbers were reviewed in a recent article [4] by Jean-Paul Delahaye [5]. One of these, devised by Benoit Cloitre (cloister in French) is intriguing because it appears to reveal unexpected fractal patterns in irrational numbers [6, 7] and does not require much computer power. However, the reason orderly graphs emerge from the simple procedure implemented in CloisterWalk remains to be fully elucidated.

Method / Algorithm

Walks take place on a grid and start with a single step to the "north" from the origin: (0,0) to (0,1). For each successive step (1, 2, 3…), the number under examination is multiplied by the step number. Depending on whether the result is even or odd, the next step is taken to the left or to the right. The process is repeated many times to produce a graph. The grid can be either "square" (90 degree left and right turns) or "hexagonal" (120 degrees to the left and 60 degrees to the right). Graphs are more compact on the hexagonal grid because each left step tends to backtrack.

Using the software

Select a number that you would like to test, using the popup menu at the top left of the window, or choose "custom" to enter your favorite number in the adjacent field. Known irrationals are at the top of the list, and numbers not yet proven to be irrational are found further down [8, 9, 10]. A few rational approximations are also included at the bottom of the list for comparison. Note that 1 is a trivial example producing alternatively right and left turns and a staircase plot on a square grid. Use the popup menu near the top right of the window to switch between square and hexagonal grids. At the bottom of the window, enter the desired number of steps to walk. Preferably start with a small number of steps, before trying larger numbers (scientific notation, such as 1.23e2 for 123 is allowed). The forward/backward popup menu allows you to repeat a portion of the walk using fewer steps at a time. To improve speed at high step counts, CloisterWalk will display a limited number of data points by default. Checking the "fine plot" box before clicking the Go button will take longer and draw all steps. The current step count is displayed near the bottom left of the window. Press the Reset button at the bottom left, or change either the grid type or number under examination at the top of the window to start from scratch.

Limitations

CloisterWalk slows down noticeably when step counts increase, especially when the "fine plot" box is checked. Rounding errors may occur by combining a huge step count with either a small or a large test number. All data are stored in memory, which may be limiting when performing long calculations, depending on your configuration.

License / Disclaimer

This software is in the public domain and offered without any warranty.

Programming

Xojo (formerly Real Basic)
http://xojo.com/

DataPlot by Roger Meier
http://opensource.the-meiers.org/

References

[1] "Irrational Digits Walk" from the Wolfram Demonstrations Project
http://demonstrations.wolfram.com/IrrationalDigitsWalk/

[2] Francisco J. Aragon Artacho, David H. Bailey, Jonathan M. Borwein and Peter B. Borwein. "Walking on real numbers", Mathematical Intelligencer, 2013, 35:42-60.
http://www.carma.newcastle.edu.au/jon/numtools.pdf
http://www.davidhbailey.com/dhbtalks/dhb-walking.pdf

[3] website of David H. Bailey
http://davidhbailey.com/

[4] Jean-Paul Delahaye. "Figuration de nombres" (review article in French), Pour la Science, February 2014, 436, page 78.
http://www2.lifl.fr/~delahaye/pls/2014/243.pdf
http://www.pourlascience.fr/ewb_pages/a/article-figurations-de-nombres-32547.php

[5] website of Jean-Paul Delahaye
http://www.lifl.fr/~delahaye/

[6] Benoit Cloitre. "Fractal walks and irrationality of numbers".
https://dl.dropboxusercontent.com/u/46675017/FractalWalks16oct.pdf

[7] website of Benoit Cloitre
http://bcmathematics.monsite-orange.fr/

[8] Irrational Number article in Wikipedia
http://en.wikipedia.org/wiki/Irrational_number#Open_questions

[9] Irrational Number article in Wolfram MathWorld
http://mathworld.wolfram.com/IrrationalNumber.html

[10] Transcendental Number article in Wolfram MathWorld
http://mathworld.wolfram.com/TranscendentalNumber.html

 

Last edit: Bernard 2014-04-29