A piece of pi
The number pi or π, (approximately 3.14159265) is one of the most important quantities in mathematics. It is perhaps best known in the context of circles: a circle of diameter 1 has circumference equal to π.
This picture, created by Cristian Ilies Vasile, is a graphical representation of the first 10,000 digits of π. Each segment represents a digit from 0 to 9, and within each segment, there are 10,000 positions, one for each digit of π being represented. We write “m:n” as shorthand for “position n of segment m”.
Each of the coloured strands represents a link between two successive digits, so the first two digits of π (3 and 1) are represented by a strand from 3:0 to 1:1; in other words, from position 0 of segment 3 to position 1 of segment 1. The sequence of coloured strands continues according to the sequence 3:0 → 1:1 → 4:2 → 1:3 → 5:4 …
The sequence of digits of π never terminates and never goes into an endlessly repeating loop. This is because π is an irrational number, which means that it cannot be expressed exactly as a fraction. Sometimes people say that π is equal to 22/7, but this is merely a convenient approximation.
In some sense, π is “more irrational” than numbers such as the golden ratio (approximately 1.618): although neither number is equal to a fraction, the golden ratio is a root of the polynomial x^2 – x – 1, whereas π is not a root of any polynomial with integer coefficients. Mathematicians express this by saying that the golden ratio is an algebraic number, whereas π is a transcendental number.
The upshot of this is that the digits of π are more or less random. However, there is a sequence of six consecutive 9s, called the Feynman point after physicist Richard Feynman, which appears after only 762 decimal places. Feynman once stated during a lecture that he would like to memorise the digits of π until that point, so he could recite them and quip “nine nine nine nine nine nine and so on”, thus implying that π was rational. (“Surely you’re joking, Mr Feynman.”)
There is a lot more fascinating π artwork on Martin Krzywinski’s web site (http://mkweb.bcgsc.ca/pi/art/). Thanks to Skip Jimroo for telling me about this picture!