← art(π,φ,e)

Round 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 (ASN), which I render in a ASN Nixie poster.

Circos art of π — digit transition paths

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

▲ A path connecting segments traces out the digits of π. Here the transition for the 6 digits is shown. Concept by Cristian Ilies Vasile. Created with Circos.

Cristian Ilies Vasile had the idea of representing the digits of π as a path traced by links between successive digits. Each digit is assigned a segment around the circle and a link between segment i and j corresponds to the appearance of ij in π. For example, the "14" in "3.14..." is drawn as a link between segment 1 and segment 4.

The position of the link on a digit's segment is associated with the position of the digit π. For example, the "14" link associated with the 2nd digit (1) and the 3rd digit (4) is drawn from position 2 on the 1 segment to position 3 on the 4 segment.

As more digits are added to the path, the image becomes a weaving mandala.

▲ Digit transition paths for 10, 100 and 1,000 digits of π. Concept by Cristian Ilies Vasile. Created with Circos.

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▲ Transition paths for the first 10,000 digits of π. Concept by Cristian Ilies Vasile. Created with Circos. (PNG, BUY ARTWORK)

circos art of π, φ and e — transition paths and bubbles

I added to Cristian's representation by showing the number of transitions between digits in a series of concentric circles placed outside the links. This summary representation counts the number of transition links within a region and addresses the question of what kind of digits appear immediately before or after a given digit in π. The approach is diagrammed below.

▲ The number of transitions to and from a given digit within a window of 10 digits is shown by circles. For a given digit segment (here, 9) each circle indicates the presence of a specific digit appearing before (inner track) or after (after track) the digit. Solid circles are used for the digit that appears most often and if all digits appear equally often, the choice is arbitrary. In some images the order of digits in the inner track is outward. (zoom)

The original images were generated using the 10-color Brewer paired qualitative palette, which was later modified as shown below.

▲ For added visual impact, I inverted the color palette and added hue shift and vibrance effects.

The bubbles that count the number of links quickly draw attention to regions where specific digit pairs are frequent. In the image for π below, which shows transitions for the first 1,000 digits, the large bubble on the 9 segment is due to the sequence "999999" sequence at decimal place 762.

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

This sequence of 6 9's occurs significantly earlier than expected by chance. Because the distribution and sequence of digits of π is, as far as we know, uniformly random, we can calculate how frequently we should expect a series of 6 identical digits.

For a given digit, the chance that the next 5 digits are the same is 0.00001 (0.1 that the next digit is the same * 0.1 that the second-nex digit is the same * ...). Therefore the chance that a given position the next 5 digits are not the same is 1 - 1/0.00001 = 0.99999. From this, the chance that k consecutive digits don't initiate a 6-digit sequence is therefore 0.99999^k.

If I ask what is k for which this value is 0.5, I need to solve 0.99999^k, which gives k = 69,314. Thus, chances are 50-50 that in a 69,000 digit random sequence we'll see a run of 6 idendical digits. This calculation is an approximation.

It's fun to look for words in π. For example, love appears at 13,099,586th digit.

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

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

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

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.

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

A tangent into randomness

The digits of π are, as far as we know, randomly distributed. Art based on its digits therefore as a quality that is influenced by this random distribution. To provide a reference of what such a random pattern looks like, below are 16 random numbers represented in the same way. They're all different, yet strangely the same.

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▲ Digit transition paths of sixteen 1,000 digit random numbers. (PNG, BUY ARTWORK)

Circos art of π — heaps of bubbles

Below are more images by Cristian Ilies Vasile, where dots are used to represent the adjacency between digits. As in the image above, each digit 0-9 is represented by a colored segment. For each digit sequence ij, a dot is placed on ith's segment at the position of i colored by j.

▲ In a digit bubble heap, a digit is represented by a bubble and placed on the segment of its previous neighbour at the index position of the neighbour.

For example, for π the dot coordinates for the first 7 digits are (segment:position:label) 3:0:1 → 1:1:4 → 4:2:1 → 1:3:5 → 5:4:9 ...

segment position colored_by

3       0        1
1       1        4
4       2        1
1       3        5
5       4        9
9       5        2
2       6        6 

Because there is a large number of digits, the dots stack up near their position to avoid overlapping. The layout of the dots is automated by Circos' text track layout.

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▲ Progression and transition for the first 10,000 digits of π. Created with Circos. (PNG, BUY ARTWORK)

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▲ Progression and transition for the first 10,000 digits of φ. Created with Circos. (PNG, BUY ARTWORK)

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▲ Progression and transition for the first 10,000 digits of e. Created with Circos. (PNG, BUY ARTWORK)

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.

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▲ Progression and transition for the first 10,000 digits of the accidental similarity number. Created with Circos. (PNG, BUY ARTWORK)

spiral art of π

▲ The Archimedean spiral embodies π.

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.

Why the Archimedean spiral? This spiral is defined as r = a + bθ and has the interesting property that a ray from the origin will intersect the spiral every 2πb. Thus, each spiral can accomodate inscribed circles of radius πb.

Why the Brewer palette? These color schemes have some very useful perceptual properties and are commonly used to encode quantitative and categorical data.

▲ The digits of π assembled along an Archimedean spiral.

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

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

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

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

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