`\pi` Approximation Day Art Posters

▲ 2021 `\pi` reminds us that good things grow for those who wait.' edition.

▲ 2019 `\pi` has hundreds of digits, hundreds of languages and a special kids' edition.

▲ 2018 `\pi` day stitches street maps into new destinations.

▲ 2017 `\pi` day imagines the sky in a new way.

▲ 2016 `\pi` approximation day wonders what would happen if about right was right.

▲ 2016 `\pi` day sees digits really fall for each other.

▲ 2015 `\pi` day maps transcendentally.

▲ 2014 `\pi` approx day spirals into roughness.

▲ 2014 `\pi` day hypnotizes you into looking.

▲ 2014 `\pi` day

▲ 2013 `\pi` day is where it started

▲ Circular `\pi` art and other distractions

The never-repeating digits of `\pi` can be approximated by 22/7 = 3.142857 to within 0.04%. These pages artistically and mathematically explore rational approximations to `\pi`. This 22/7 ratio is celebrated each year on July 22nd. If you like hand waving or back-of-envelope mathematics, this day is for you: `\pi` approximation day!

Want more math + art? Discover the Accidental Similarity Number. Find humor in my poster of the first 2,000 4s of `\pi`.

Vague mathematics and alternative explanations to reality ahead.
Equip your imagination.

What would circles look like if `\pi`=22/7?

Tiny loop, Folded dimensions and solidland

Imagine that the circle had a tiny loop at one of its points. The circumference of this loop would be added to the circumference of the circle, but the loop would be so small that we would never notice it.

▲ In a universe where Pi=22/7, circles might have a single point at which another dimension is curled up, contributing to the additional component of circumference.

This is reminiscent of how string theories describe higher dimensions—as tiny loops at each point in space, except in my example the loop is only at one point.

▲ A 3rd dimension explained on a 2d plane. The 3rd dimension is represented by a circle of a very small radius.

This idea originated with Klein, who explained the fourth dimension as a curled up circle of a very small radius. Another way in which this curling-up is used is to say that the fifth dimension is a curled up Planck length, as explained in this Imagining 10 Dimensions video.

flatlanders and solidlanders

If this idea is difficult to wrap your head around, you're not alone. We cannot think of additional dimensions in the regular spatial sence since we have no means of experiencing such phenomena. We can however imagine how flatlanders might explain the 3rd dimension, since we can perceive it. They would draw the curled up circles in their plane because they would not have the experience of drawing with perspective mimicking our 3rd dimension.

▲ How creatures that live in a 2-dimensional world, so-called flatlanders, might explain the 3rd dimension (left) and how we ourselves might visualize their explanation (right).

We would draw their explanation as shown on the right in the figure above, borrowing from our concept of the 3rd spatial dimension. Now imagine showing our explanation to a flatlander. They would not see the same thing as you—the circles would not intuitively imply the higher dimension to them.

This is analogous to why we cannot draw folded up dimensions. We are merely solidlanders—flatlanders in 3d space. Creatures that can perceive more spatial dimensions would use us as examples of diminished perceptual ability.

Did you notice the fallacy in the term solidlander? We refer to solids as objects that occupy the maximum number of spatial dimensions. There's no reason to think that creatures that perceive more dimensions wouldn't use this word the same way we do. We're solidlanders from our perspective and they're solidlanders from theirs.

relativistic speeds, frames of reference and length contraction

Another way to imagine how a circle might look is a little more realistic. The theory of special relativity tells us that when we travel at speed relative to another object the dimensions of that object appear contracted to us in the direction of motion.

This contraction is always present, but essentially imperceptible unless we're travelling fast enough. For example, in order for a 1 meter object to appear contracted by the length of a hydrogen molecule (0.3 nm) we would have to be travelling at 7.3 km/s (Wolfram Alpha calculation)!

▲ If we travel at a speed of 0.04c and use the radius length along our direction of motion, the circumference of the circle will appear to be `2 \times 22/7 \times r`.

How fast would we have to be going to compress the circle sufficiently so that its circumference and radius ratio embody the `22/7` approximation of `\pi`? Pretty fast, it turns out. If we travel at just over 12,000 km/sec (0.04 times the speed of light, Wolfram Alpha calculation), the circle will compress as shown in the figure above, and the ratio of its circumference to the radius along direction of motion will make `\pi` appear to be `22/7`.

This compression in length would be barely perceptible to us. Below are both circles, shown overlapping, with `delta` being the extra length in radius required.

▲ Deformation of the circle required to change the ratio of its circumference to original radius from \pi to 22/7.

The value of `\delta`, which is 0.0008049179155 (if `r = 1`), can be calculated by considering the perimeter of an ellipse. The fact that `\delta` is small shouldn't be surprising since `22/7` is an excellent approximation of `\pi`, good to 0.04%.

Calculating the parameter of an ellipse is more complicated than calculating it for a circle because it uses something called an elliptic integral. This integral has no analytical solution and requires numerical approximation. Luckily, we have computers.

▲ If a circle compressed slightly in one direction (e.g. vertically) then we can make the ratio of its circumference to the new radius be 2 × 22/7.

We can use the expression shown above for the perimeter of the ellipse to determine how much the circle needs to be deformed. Let's write `a = r + \delta` (original radius with slight deformation `\delta`) and `b=r`. Since `22/7 > \pi` we know that `\delta > 0`.

It remains to solve the equation below for a value of `\delta` that will yield a ratio of circumference to `r` of `2 \times 22/7`.

▲ The equation we need to solve to determine how much of a stretch the circle needs.

To make things simpler, let set `r=1`. Solving the equation numerically, I find $$\delta = 0.0008049179155$$

You can verify this solution at Wolfram Alpha.

the meaning of full-circle

After all this, we come full-circle to the meaning of full-circle.

You might ask why I didn't change the definition of `\pi` to `22/7` in the upper limit of the integral. After all, why not make the approximation exercise more faithful to the approximation?

It turns out that if I did that I would get `\delta=0`, which brings us back to the original circle. How is this possible?

Technically, this is because the integral returns the upper limit as its answer if the eccentricity is zero (i.e., `E(x,0)=x`).

Intuitively, this is because changing the upper limit of the integral actually redefines the angle of a full revolution. Now, full-circle isn't `2 \pi` radians, but `2 \times 22/7`. Given that the ratio of the circumference of a circle to its radius is exactly the size, in radians, of a full revolution, we don't need to change the shape of the circle if we're willing to change what a full revolution means.