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Making Sense of the Numbers

Mathematicians have long searched for a tool that would help verify if strings of numbers are truly random or have some sort of higher order structure. In this quest, one method that has been employed is to explore graphical representations of the number; if there is a repeating pattern or structure you would see it in a plot.

Pi, the ratio of circumference to diameter of a circle, is 3.14159265358979 in a base of 10. The decimals go on infinitely in a seemingly random sequence. If you write Pi using a base-4 numeral system it is 3.02100333122220, a random sequence of digits between 0 and 3.

To help visualize these large mathematical data sets, Francisco Javier Aragón Artacho, research associate at CARMA, School of Mathematical & Physical Sciences, University of Newcastle, Australia, at the suggestion of CARMA Director and collaborator Jonathan Borwein, set about to create a very large image.

Largest Mathematical Picture

Motivated by the desire to show the behavior of the first billions of digits of Pi in just one picture, Francisco used Python to start the image at a pixel and at each step moved the “pen” to one of the four adjacent pixels using the rules (0 – right, 1 – up, 2 – left, 3 – down). With this project, Francisco aimed to represent numbers as planar walks and quantitatively measure their randomness.

This image is a result of continuing this process for the first 100 billion digits of Pi in base-4. The color indicates how the path progressed over time as the walk progresses.

It is unknown whether the digits of Pi are random, but the picture clearly supports this conjecture by showing characteristics of a seemingly ‘random’ walk: it returns to the origin, it is space-filling, and it has a fractal-like structure.

“This is probably the largest mathematical picture ever drawn and surely the largest one made out of the digits of Pi,” Francisco said.

Zoom in on Pi 

After creating the image, Francisco uploaded it to GigaPan Viewer in order to make the image zoomable. With GigaPan Viewer people can zoom in to see extreme detail of the largest mathematical picture ever drawn.

This research has already gone viral in the research community and a corresponding research paper, The Mathematical Intelligencer, was published.

“The viewer makes it extremely easy to navigate through the image. Before I discovered GigaPan, I tried a few other viewers, and all of them were cumbersome and far from satisfactory,” Francisco concluded. “With GigaPan everything was very simple and intuitive.”