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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`.

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

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.

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.

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.

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.

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.

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)!

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.

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.

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`.

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.

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.

My cover design on the 11 April 2022 Cancer Cell issue depicts depicts cellular heterogeneity as a kaleidoscope generated from immunofluorescence staining of the glial and neuronal markers MBP and NeuN (respectively) in a GBM patient-derived explant.

LeBlanc VG *et al.* Single-cell landscapes of primary glioblastomas and matched explants and cell lines show variable retention of inter- and intratumor heterogeneity (2022) *Cancer Cell* **40**:379–392.E9.

Browse my gallery of cover designs.

My cover design on the 4 April 2022 Nature Biotechnology issue is an impression of a phylogenetic tree of over 200 million sequences.

Konno N *et al.* Deep distributed computing to reconstruct extremely large lineage trees (2022) *Nature Biotechnology* **40**:566–575.

Browse my gallery of cover designs.

My cover design on the 17 March 2022 Nature issue depicts the evolutionary properties of sequences at the extremes of the evolvability spectrum.

Vaishnav ED *et al.* The evolution, evolvability and engineering of gene regulatory DNA (2022) *Nature* **603**:455–463.

Browse my gallery of cover designs.

three one four: a number of notes

Celebrate `\pi` Day (March 14th) and finally hear what you've been missing.

“three one four: a number of notes” is a musical exploration of how we think about mathematics and how we feel about mathematics. It tells stories from the very beginning (314…) to the very (known) end of π (...264) as well as math (Wallis Product) and math jokes (Feynman Point), repetition (nn) and zeroes (null).

The album is scored for solo piano in the style of 20th century classical music – each piece has a distinct personality, drawn from styles of Boulez, Feldman, Glass, Ligeti, Monk, and Satie.

Each piece is accompanied by a piku (or πku), a poem whose syllable count is determined by a specific sequence of digits from π.

Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, 2016 `\pi` Day, 2017 `\pi` Day, 2018 `\pi` Day, 2019 `\pi` Day, 2020 `\pi` Day and 2021 `\pi` Day.

My design appears on the 25 January 2022 PNAS issue.

The cover shows a view of Earth that captures the vision of the Earth BioGenome Project — understanding and conserving genetic diversity on a global scale. Continents from the Authagraph projection, which preserves areas and shapes, are represented as a double helix of 32,111 bases. Short sequences of 806 unique species, sequenced as part of EBP-affiliated projects, are mapped onto the double helix of the continent (or ocean) where the species is commonly found. The length of the sequence is the same for each species on a continent (or ocean) and the sequences are separated by short gaps. Individual bases of the sequence are colored by dots. Species appear along the path in alphabetical order (by Latin name) and the first base of the first species is identified by a small black triangle.

Lewin HA et al. The Earth BioGenome Project 2020: Starting the clock. (2022) PNAS 119(4) e2115635118.

As part of the COVID Charts series, I fix a muddled and storyless graphic tweeted by Adrian Dix, Canada's Health Minister.

I show you how to fix color schemes to make them colorblind-accessible and effective in revealing patters, how to reduce redundancy in labels (a key but overlooked part of many visualizations) and how to extract a story out of a table to frame the narrative.