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`\pi` Approximation Day Art Posters
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!
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.
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.
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.
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.
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)!
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.
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.
Chromosomes — music, genomes, split-flaps
A short electronic music concert based on structures of chromosomes 21, Y and 1.
Each track uses a microtonal scale derived from the cytogenetic band positions of the chromosome. Rhythms are based on relative lengths of cytogenetic bands.
Consequences of missing data
Either I've been missing something or nothing has been going on. —Karen Elizabeth Gordon
Missing data are everywhere. Subjects may decline to participate in a survey or fail to answer sensitive questions. A cell culture might fail due to contamination. Instrument failures or mishandling of a sample may lead to missing observations. But why are missing data a problem?
This month, we begin a series of articles about practical and statistical aspects of missing data. We'll see that missing data can increase variability and introduce bias and we will ask whether anything can be done to mitigate these consequences. It turns out that, in the case where we know nothing about the missing subjects, no mitigation is possible. We must accept higher variability and, if we have a suspicion that the missing subjects aren’t completely random, possible bias as well.
Tanujit Dey, T., Lipsitz, S.R., Fitzmaurice, G., Krzywinski, M. & Altman, N. (2026) Points of significance: Consequences of missing data. Nat. Methods 23:1661–1663.
Double Robustness
It is not certain that everything is uncertain. —Blaise Pascal
We have already explored how we can mitigate bias caused by confounding variables in observational studies using propensity score (PS) matching (PSM) and propensity score weighting (PSW). However, any statistical model is only as good as its assumptions and, if it is specified incorrectly, it can itself produce biased estimates of the treatment effect.
This month, we explore double robustness, a powerful statistical concept that provides a valuable “safety net” against the risk of an incorrect model. It offers two opportunities, instead of just one, to obtain a valid estimate of the treatment effect — making it possible to draw credible causal inferences from observational data without having to depend on a single set of modeling assumptions.
Kurz, C.F., Krzywinski, M. & Altman, N. (2026) Points of significance: Double Robustness. Nat. Methods 23:868–869.
Nature Biotechnology cover
My cover design on the 7 April 2026 Nature Biotechnology issue shows the dendrogram that represents a cluster of uniquely expressed (or downregulated) genes in human naive stem cells induced from such cells. Within each dendrogram block, the genomic barcode sequence (sampled from Supplementary Table 1) is depicted with a Code 39 barcode. The highlighted barcode is one of those used for cell isolation.
Ishiguro S. et al. A multi-kingdom genetic barcoding system for precise clone isolation (2026) Nature Biotechnology 44:616–629.
Browse my gallery of cover designs.
Happy 2026 π Day—
Art for the 5%
Celebrate π Day (March 14th) and enjoy the art — but only if you're part of the 5%.
Go ahead, see what you can't see.
Ishihara's Tests for Colour Deficiency
Authentic and accurate images of Ishihara's test plates photographed (and lovingly color-corrected) from the 38-plate Ishihara's Tests for Colour Deficiency.
I also provide the position, size, and color of each circle on each test plate.
