The present bloggers, together with Francisco Aragon Artacho (University of Newcastle, Australia) and Peter Borwein (Simon Fraser University, Canada, and Jonathan Borwein’s brother), have just completed the paper Tools for visualizing real numbers: Planar number walks.
This manuscript describes analysis of the digits of pi and many other real numbers and quantifies various techniques of modern computer visualization. In most of these analyses, the authors address a real number (represented in base-4 digits, i.e., 0, 1, 2, 3) as a “random walk,” typically by moving one unit east, north, west or south, depending on whether the digit at a given position is 0, 1, 2 or 3. The color of the dot indicates its position in the sequence.
The authors show that this type of graphical representation can be used to distinguish numbers whose digits appear to be truly “random” (such as pi, which does not exhibit any patterns in digits analyzed so far), from others, such as Champernowne’s number 0.12345678910111213141516… (a concatenation of successive base-10 integers) or its equivalents in other number bases, which typically exhibit very different digit patterns.
Some other graphical representations, such as a conventional 2-dimensional “matrix-box” image, are also useful in spotting irregularities.
The manuscript has been featured in a column on the Wired magazine website, authored by Samuel Arbesman. He shows some of the graphics from the article, including one depicting a “random” walk on the digits of pi. The Aperiodical also describes some of the same work.