The art of π, φ and e

Numerology is bogus, but art based on numbers has a beautiful random quality.

For other examples of numerical art, see my inessiness project. Nixie clock lovers should investigate the accidental similarity number.

art of π

Cristian Ilies Vasile had the idea of representing the digits of π as links that connected segments associated with successive digits. The image is composed of links (segment:position) 3:0 → 1:1 → 4:2 → 1:3 → 5:4 …

▲ Progression of the first 10,000 digits of π By Cristian Ilies Vasile. Created with Circos. (PNG)

art of π, φ and e

I added to Cristian's representation by showing the transition probabilities for each digit across bins of 10 digits. For a given digit segment, d, the inner bubble plot shows the number of times a digit i was found immediately before d. The outer bubble plot shows the number of times i followed d.

▲ Progression and transition for the first 1,000 digits of π. Created with Circos. (PNG, PDF)

▲ Progression and transition for the first 1,000 digits of π, φ and e. Created with Circos. (PNG)

▲ Progression and transition for the first 1,000 digits of φ. Created with Circos. (PNG, PDF)

▲ Progression and transition for the first 1,000 digits of e. Created with Circos. (PNG, PDF)

The transition probabilities for each 10 digit bin for the first 2,000 digits of π, φ and e are shown in the image below. The large bubble on the 9 segment for π is due to the unusual "999999" sequence at decimal place 762, which occurs significantly earlier than expected.

▲ Progression and transition for the first 2,000 digits of e. Created with Circos. (PNG, PDF)

Below are more images by Cristian Ilies Vasile, where dots are used to represent the adjacency between digits. Each digit 0-9 is represented by a colored segment. Dots assigned to a segment represent digits that follow the digit represented by the segment. The position of the dot is roughly the position within the number that the digit appears.

For example, for π the dot coordinates for the first 7 digits are The image is composed of dots using third coordinates (segment:position:label) 3:0:1 >>> 1:1:4 >>> 4:2:1 >>> 1:3:5 >>> 5:4:9 ....

segment position label

digit3  0        1
digit1  1        4
digit4  2        1
digit1  3        5
digit5  4        9
digit9  5        2
digit2  6        6 

▲ Progression and transition for the first 1,000 digits of π. Created with Circos. (PNG)

▲ Progression and transition for the first 1,000 digits of φ. Created with Circos. (PNG)

▲ Progression and transition for the first 1,000 digits of e. Created with Circos. (PNG)

When the digits of π, e and φ are aligned, positions at which the three numbers have the same digit yield the accidental similarity number (ASN). Below is a dot plot of the transition of the ASN.

▲ Progression and transition for the first 1,000 digits of the accidental similarity number. Created with Circos. (PNG)

art of π

It's fitting to use Circos to visualize the digits of π. After all, what is more round than Circos?

By mapping the digits onto a red-yellow-blue Brewer palette (0 9) and placing them as circles on an Archimedean spiral a dense and pleasant layout can be obtained.

▲ Distribution of the first 13,689 digits of π. (PNG, SVG)

▲ Distribution of the first 3,422, 13,689 and 123,201 digits of π. (PNG)

▲ Distribution of the first 3,422 digits of π. (PNG, SVG)

▲ Distribution of the first 123,201 digits of π. (PNG, SVG)