A fine mess

© Jeff Thompson

 

You might think this image is a mess.

It is. It’s a beautiful, fascinating, perfect mess. The platonic ideal of mess.

Starting at the grey dot, the scribble is a line composed out of the decimal digits in pi, or the number 3.14159…(click on the image to see it in more detail).

Each digit is represented as a line in one of 10 directions. For each successive digit, you continue the line from where you ended up.

This way of portaying randomness – and the digits in pi pass every test for randomness that we know – is what’s called a “random walk”, or a “drunkard’s walk”. It’s like the path of a drunk staggering in a random direction from a lamppost, who collapses after a metre, wakes up and the staggers for a metre in another random direction.

The mathematician who came up with the idea of a random walk was John Venn in the 19th century. He also used the digits in pi, but he disregarded the digits 8 and 9, and used the following digits to represent the following directions:

This gave him the following image, from his 1888 book The Logic of Chance, of the first 707 digits:

In his excellent blog, Jeff Thompson has recreated this experiment but using all ten digits – separated by 36˚ – and carried on for 1120 decimal places, giving the image above.

It’s very cool and its interesting how bits of it dont actually look that random – we like to project some kind of order on there, even if no order exists. The path of the pi drunk almost bumps into the starting lamppost…it will do so full on, eventally, if the digits in pi are perfectly random.

Pi is the ratio of a circle’s circumference to its diameter. Split the circle open like an egg and you make this wonderful mess.

Jeff also did the same thing in three dimensions. Worth a look.