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Visualizing Irrationally Beautiful Numbers

From the pages of a textbook, mathematics often gets a bad rap for being either too stagnant or too perplexing. Scientists Cristian Ilies Vasile and Martin Krzywinski have found a way though to spark attention on some special numbers in math by creating colorful interpretations of them.

Using a software designed for circular visualization called Circos, Vasile and Krzywinski have created a kaleidoscope of colors just from the digits of three mathematical constants: π (a ratio for a circle) , φ (the golden ratio), and e (Euler’s number).

First Vasile connected each digit of π (3.141592…) to its successive digit for the first 10,000 digits. He did this by color coding each digit (0-9), and then drawing a line going from the first digit 3 to the second digit 1, then 1 to 4, and so on. This is called a progression. Eventually beautiful patterns and shapes develop to expose the hidden beauty lying behind the plain old numbers.


Vasile then visualizes this in a different way by starting at the color of the first number, drawing a color dot for the next number above it, and then repeating this for each successive number.

Krzywinski then built on Vasile’s first progression plot by surrounding it with a bubble plot of the transition probabilities, shown below.


To understand the above plot of  π, consider the digit 9 (here in light purple). The inner bubble plot shows the number of times a certain (color coded) digit preceded 9, and the outer bubble plot shows the number of times a certain digit follows 9. Notice that in the section for 9, there is a big purple dot both above and below. This happens because hundreds of digits into π, the number is oddly repeated in succession: ‘999999’. Since there is of course  a large likelihood that 9 will precede 9, and that 9 will follow 9, there is a large purple bubble both above and below! Here’s a test: given that 1 is colored in light blue, what number is most likely to follow it? (Answer: 7, found in orange).

Krzywinski finally visualizes π in a continuous, simple way instead: each digit is still color coded, but now he wraps them around in a clockwise spiral that ends in the center.


Check out the other special numbers that these scientists have transformed into intellectually stimulating works of art. Perhaps if more math books used images like these, we wouldn’t be caught snoozing at our desks!

Keara Wright

Aspiring creative author and astrophysicist, with degrees in Physics, Mathematics, and Psychology.

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