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▲ 2025 π DAY | TTCAGT: A sequence of digits. 768 digits of $\pi$ as a Sanger sequencing trace of 1,536 peaks. Decode the sequence (BUY ARTWORK)

The art of Pi ($\pi$), Phi ($\varphi$) and $e$

▲ 2024 π DAY | Explore the garden of digits.

▲ 2023 π DAY | Repeated sequence

▲ 2022 π DAY | three one four: a number of digits

▲ 2021 π DAY | Good things grow for those who wait.' edition.

▲ 2020 π DAY | The piku.

▲ 2019 π DAY | Hundreds of digits, hundreds of languages and a special kids' edition.

▲ 2018 π DAY | Street maps to new destinations.

▲ 2017 π DAY | Imagine the sky in a new way.

▲ 2016 π APPROXIMATION DAY | What would happen if about right was right.

▲ 2016 π DAY | These digits really fall for each other.

▲ 2015 π DAY | A transcendental experience.

▲ 2014 π APPROXIMATION DAY | Spirals into roughness.

▲ 2014 π DAY | Hypnotizes you into looking.

▲ 2014 π DAY | Come into the fold.

▲ 2013 π DAY | Where it started.

▲ CIRCULAR π ART | And other distractions.

This section contains various art work based on $\pi$, $\varphi$ and $e$ that I created over the years.

Some of the numerical art reveals interesting and unexpected observations. For example, the sequence 999999 in π at digit 762 called the Feynman Point. Or that if you calculate π to 13,099,586 digits you will find love.

$\pi$ day art and $\pi$ approximation day art is kept separate.

Count along ·

Like music with numbers? Try Angels at My Door (Una), Pt vs Ys (Yoshinori Sunahara), 2wicky (Hooverphonic), One (Aimee Mann), Straight to Number One (Touch and Go), 99 luftbaloons (Nena), and Aquarius (Boards of Canada)

Cristian Ilies Vasile had the idea of representing the digits of $\pi$ 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 $\pi$. 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 $\pi$. 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.

circos art of $\pi$, $\varphi$ 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 $\pi$. 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 $\pi$ below, which shows transitions for the first 1,000 digits, the large bubble on the 9 segment is due to the "999999" sequence at decimal place 762. This is the Feynman point, which I describe below.

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

The image below shows how this representation of $\pi$ compares to that of $\varphi$ and $e$.

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

The transition probabilities for each 10 digit bin for the first 2,000 digits of $\pi$, $\varphi$ and $e$ are shown in the image below.

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

Feynman point

This sequence of 6 9's occurs significantly earlier than expected by chance. Because the distribution and sequence of digits of $\pi$ is thought to be normal, 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 even (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 $\pi$. For example, love appears at 13,099,586th digit.

A tangent into randomness

The digits of $\pi$ 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 $\pi$—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 the $i$th 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 $\pi$ 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 $\pi$. Created with Circos. (PNG, BUY ARTWORK)

spiral art of $\pi$

▲ The Archimedean spiral embodies $\pi$.

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 $r=a+b\theta$ and has the interesting property that a ray from the origin will intersect the spiral every $2pib$. Thus, each spiral can accomodate inscribed circles of radius $\pi b$.

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.

▲ Calculating (x,y) coordinates for each digit along the 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)

I have use the Archimedean spiral to make art for $\pi$ approximation day

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▲ Pi Approximation Day Art Poster | July 22nd is Pi Approximation Day. Celebrate with this post-modern poster. (PNG, BUY ARTWORK)