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▲ 2024 π DAY | 768 digits of `\pi` as a garden at night. Explore the gardens (BUY ARTWORK)

`\pi` Day 2016 Art Posters

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

▲ 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.

On March 14th celebrate `\pi` Day. Hug `\pi`—find a way to do it.

For those who favour `\tau=2\pi` will have to postpone celebrations until July 26th. That's what you get for thinking that `\pi` is wrong. I sympathize with this position and have `\tau` day art too!

If you're not into details, you may opt to party on July 22nd, which is `\pi` approximation day (`\pi` ≈ 22/7). It's 20% more accurate that the official `\pi` day!

Finally, if you believe that `\pi = 3`, you should read why `\pi` is not equal to 3.

support my work ·

Most of the art is available for purchase as framed prints and, yes, even pillows. Sleep's never been more important — I take custom requests.

This year's `\pi` day art collection celebrates not only the digit but also one of the fundamental forces in nature: gravity.

In February of 2016, for the first time, gravitational waves were detected at the Laser Interferometer Gravitational-Wave Observatory (LIGO).

The signal in the detector was sonified—a process by which any data can be encoded into sound to provide hints at patterns and structure that we might otherwise miss—and we finally heard what two black holes sound like. A buzz and chirp.

The art is featured in the Gravity of Pi article on the Scientific American SA Visual blog.

this year's theme music

All the art was processed while listening to Roses by Coeur de Pirate, a brilliant female French-Canadian songwriter, who sounds like a mix of Patricia Kaas and Lhasa. The lyrics Oublie-moi (Forget me) are fitting with this year's theme of gravity.

Mais laisse-moi tomber, laisse-nous tomber
Laisse la nuit trembler en moi
Laisse-moi tomber, laisse nous tomber
Cette fois

But let me fall, let us fall
Let the night tremble in me
Let me fall, let us fall
This time

The art is generated by running a simulation of gravity in which digits of `\pi` are each assigned a mass and allowed to collide eand orbit each other.

The mathematical details of the simulation can be found in the code section.

exploring force of gravity in `\pi`

A simulation starts with taking `n` digits of `\pi` and arranging them uniformly around a circle. The mass of each digit, `d_i` (e.g. 3), is given by `(1+d)^k` where `k` is a mass power parameter between 0.01 and 1. For example, if `k=0.42` then the mass of 3 is `(1+3)^{0.42} = 1.79`.

collapsing three digits—3.14 collide

The figure below shows the evolution of a simulation with `n=3` digits and `k=1`. The digits 3 and 4 collide to form the digit `3+4 = 7` and immediately collides with 1 to form `7+1=8`. With only one mass left in the system, the simulation stops.

▲ The evolution of a simulation of gravity using `n=3` digits of `\pi` and the mass power `k=1`. The masses are initialized with zero velocity. (zoom)

adding initial velocity to each mass

When masses have initial velocities, the patterns quickly start to get interesting. In the figure above, the masses are initalized with zero velocity. As soon as the simulation, each mass immediately starts to move directly towards the center of mass of the other two masses.

When the initial velocity is non-zero, such as in the figure below, the masses don't immediately collapse towards one another. The masses first travel with their initial velocity but immediately the gravitational force imparts acceleration that alters this velocity. In the examples below, only those simulations in which the masses collapsed within a time cutoff are shown.

▲ The evolution of a simulation of gravity using `n=3` digits of `\pi` and the mass power `k=1` in which all masses collapsed. The masses are initialized with a random velocity. (zoom)

▲ The evolution of 16 simulations of gravity using `n=3` digits of `\pi` and the mass power `k=1` in which all masses collapsed. The masses are initialized with a random velocity. (zoom)

▲ The evolution of 49 simulations of gravity using `n=3` digits of `\pi` and the mass power `k=1` in which all masses collapsed. The masses are initialized with a random velocity. (zoom)

allowing the simulation to evolve

Depending on the initial velocities, some systems collapse very quickly, which doesn't make for interesting patterns.

For example, the simulations above evolved over 100,000 steps and in some cases the masses collapsed within 10,000 steps. In the figure below, I require that the system evolves for at least 15,000 steps before collapsing. Lovely doddles, don't you think?

▲ The evolution of 36 simulations of gravity using `n=3` digits of `\pi` and the mass power `k=1` in which all masses collapsed after a minimum amount of time. The masses are initialized with a random velocity. (zoom)

exploring ensembles

When a simulation is repeated with different initial conditions, the set of outcomes is called an ensemble.

Below, I repeat the simulation 100 times with `n=3` and `k=0.2`, each time with slightly different initial velocity. The velocities have their `x`- and `y`-components normally distributed with zero mean and a fixed variance. Each of the four ensembles has its simulations evolve over progressively more time steps: 5,000, 7,500, 10,000, and 20,000.

You can see that with 5,000 steps the masses don't yet have a chance to collide. After 7,500, there have been collisions in a small number of systems. The blue mass corresponds to the 3 colliding with 4 and the green mass to 1 colliding with 4. After 10,000, even more collisions are seen and in 3 cases we see total collapse (all three digits collided). After 20,000,

▲ The evolution of 100 simulations of gravity over total time `t` using `n=3` digits of `\pi` and the mass power `k=0.2`. Within each ensemble, the masses are initialized with a different random velocity in each instance. (zoom)

varying masses

The value of `k` greatly impacts the outcome of the simulation. When `k` is very small, all the digits have essentially the same mass. For example, when `k=0.01` the 0 has a mass of 1 and 9 has a mass of 1.02.

When `k` is large, the difference in masses is much greater. For example, for `k=2` the lightest mass is `(1+0)^2=1` and the heaviest `(1+9)^2=10`. Because the acceleration of a mass is proportional to the mass that is attracting it, in a pair of masses the light mass will accelerate faster.

▲ Larger values of `k` create greater diversity among the masses. Shown are simulations of 36 digits with `k` values varying from 0.1 to 3. The total mass of the system, `\Sigma m`, is also shown.`. (zoom)

increasing number of masses

As the number of digits is increased, the pattern of collapse doesn't qualitatively change.

▲ Simulations for `n = 50, 100, 250` and `500` masses with `k = 0.5`. (zoom)

gravity makes beautiful doodles

I ran a large number of simulations. For various values of `n` and `k`, I repeated the simulation several times to sample different intial velocities.

▲ Thumbnails of `\pi` digit orbital simulations for various values of `n` and `k`. (zoom)

▲ Gravitational attraction paths of the first 100 digits of `\pi` for `k = 0.3`, `0.6` and `0.8` with initial velocities randomly set. Three instances of the simulation are shown, each with different intital velocities. (zoom)

▲ Gravitational attraction paths of the first 60 digits of `\pi` for `k = 1`. After 100,000 time steps, some masses are still orbiting within the canvas (e.g. green mass at bottom right). The numbers next to the masses correspond to the digits (those around the circle are the first 50 digits of `\pi` and others are the sum (mod 10) of digits that collided). Also shown next to the numbers is their mass, index and indices of masses that formed them. (zoom)

▲ Gravitational attraction paths of the first 50 digits of `\pi` for `k = 0.4`. The numbers next to the masses correspond to the digits (those around the circle are the first 50 digits of `\pi` and others are the sum (mod 10) of digits that collided). (zoom)

Below is a great example of how a stable orbital pattern of a pair of masses can be disrupted by the presence of another mass. You can see on the left that once the light red mass moves away from the orange/green pair, they settle into a stable pattern.

▲ Gravitational attraction paths of the first 50 digits of `\pi` for `k = 0.9`. The numbers next to the masses correspond to the digits (those around the circle are the first 50 digits of `\pi` and others are the sum (mod 10) of digits that collided). (zoom)

The figure below shows one of my favourite patterns. As the digits collide, three masses remain, which leave the system. They remain under each other's gravitational influence, but are moving too quickly to return to the canvas within the time of the simulation.

▲ Gravitational attraction paths of the first 90 digits of `\pi` for `k = 0.8`. The digits collide, leaving three rapidly-moving masses, which leave the canvas. (zoom)

how the idea developed

interactive gravity simulator

Use this fun inteactive gravity simulator if you want to drop your own masses and watch them orbit.