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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!
The `22/7` approximation of `\pi` is more accurate than using the first three digits `3.14`. In light of this, it is curious to point out that `\pi` Approximation Day depicts `\pi` 20% more accurately than the official `\pi` Day! The approximation is accurate within 0.04% while 3.14 is accurate to 0.05%.
first 10,000 approximations to `\pi`
For each `m=1...10000` I found `n` such that `m/n` was the best approximation of `\pi`. You can download the entire list, which looks like this
m n m/n relative_error best_seen?
1 1 1.000000000000 0.681690113816 improved
2 1 2.000000000000 0.363380227632 improved
3 1 3.000000000000 0.045070341449 improved
4 1 4.000000000000 0.273239544735
5 2 2.500000000000 0.204225284541
7 2 3.500000000000 0.114084601643
8 3 2.666666666667 0.151173636843
9 4 2.250000000000 0.283802756086
10 3 3.333333333333 0.061032953946
11 4 2.750000000000 0.124647812995
12 5 2.400000000000 0.236056273159
13 4 3.250000000000 0.034507130097 improved
14 5 2.800000000000 0.108732318685
16 5 3.200000000000 0.018591635788 improved
17 5 3.400000000000 0.082253613025
18 5 3.600000000000 0.145915590262
19 6 3.166666666667 0.007981306249 improved
20 7 2.857142857143 0.090543182332
21 8 2.625000000000 0.164436548768
22 7 3.142857142857 0.000402499435 improved
23 7 3.285714285714 0.045875340318
24 7 3.428571428571 0.091348181202
...
354 113 3.132743362832 0.002816816734
355 113 3.141592920354 0.000000084914 improved
356 113 3.150442477876 0.002816986561
...
9998 3183 3.141061891298 0.000168946885
9999 3182 3.142363293526 0.000245302310
10000 3183 3.141690229343 0.000031059327
As the value of `m` is increased, better approximations are possible. For example, each of `13/4`, `16/5`, `19/6` and `22/7` are in turn better approximations of `\pi`. The line includes the improved flag if the approximation is better than others found thus far.
next best after 22/7
After `22/7`, the next better approximation is at `179/57`.
Out of all the 10,000 approximations, the best one is `355/113`, which is good to 7 digits (6 decimal places).
pi = 3.1415926
355/113 = 3.1415929
I've scanned to beyond `m=1000000` and `355/113` still remains as the only approximation that returns more correct digits than required to remember it.
increasingly accurate approximations
Here is a sequence of approximations that improve on all previous ones.
1 1 1.000000000000 0.681690113816 improved
2 1 2.000000000000 0.363380227632 improved
3 1 3.000000000000 0.045070341449 improved
13 4 3.250000000000 0.034507130097 improved
16 5 3.200000000000 0.018591635788 improved
19 6 3.166666666667 0.007981306249 improved
22 7 3.142857142857 0.000402499435 improved
179 57 3.140350877193 0.000395269704 improved
201 64 3.140625000000 0.000308013704 improved
223 71 3.140845070423 0.000237963113 improved
245 78 3.141025641026 0.000180485705 improved
267 85 3.141176470588 0.000132475164 improved
289 92 3.141304347826 0.000091770575 improved
311 99 3.141414141414 0.000056822190 improved
333 106 3.141509433962 0.000026489630 improved
355 113 3.141592920354 0.000000084914 improved
For all except one, these approximations aren't all good value for your digits.
For example, `179/57` requires you to remember 5 digits but only gets you 3 digits of `\pi` correct (3.14).
Only `355/113` gets you more digits than you need to remember—you need to memorize 6 but get 7 (3.141592) out of the approximation!
You could argue that `22/7` and `355/113` are the only approximations worth remembering. In fact, go ahead and do so.
approximations for large `m` and `n`
It's remarkable that there is no better `m/n` approximation after `355/113` for all `m \le 10000`.
What do we find for `m > 10000`?
Well, we have to move down the values of `m` all the way to 52,163 to find `52163/16604`. But for all this searching, our improvement in accuracy is miniscule—0.2%!
pi 3.141592653589793238
m n m/n relative_error
355 113 3.1415929203 0.00000008491
52163 16604 3.1415923873 0.00000008474
After 52,162 there is a slew improvements to the approximation.
104348 33215 3.1415926539 0.000000000106 208341 66317 3.1415926534 0.0000000000389 312689 99532 3.1415926536 0.00000000000927 833719 265381 3.141592653581 0.00000000000277 1146408 364913 3.14159265359 0.000000000000513 3126535 995207 3.141592653588 0.000000000000364 4272943 1360120 3.1415926535893 0.000000000000129 5419351 1725033 3.1415926535898 0.00000000000000705 42208400 13435351 3.1415926535897 0.00000000000000669 47627751 15160384 3.14159265358977 0.00000000000000512 53047102 16885417 3.14159265358978 0.00000000000000388 58466453 18610450 3.14159265358978 0.00000000000000287
I stopped looking after `m=58,466,453`.
Despite their accuracy, all these approximations require that you remember more or equal the number of digits than they return. The last one above requires you to memorize 17 (9+8) digits and returns only 14 digits of `\pi`.
The only exception to this is `355/113`, which returns 7 digits for its 6.
You can download the first 175 increasingly accurate approximations, calculated to extended precision (up to `58,466,453/18,610,450`).
Propensity score weighting
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.
Symmetric alternatives to the ordinary least squares regression
What immortal hand or eye, could frame thy fearful symmetry? — William Blake, "The Tyger"
This month, we look at symmetric regression, which, unlike simple linear regression, it is reversible — remaining unaltered when the variables are swapped.
Simple linear regression can summarize the linear relationship between two variables `X` and `Y` — for example, when `Y` is considered the response (dependent) and `X` the predictor (independent) variable.
However, there are times when we are not interested (or able) to distinguish between dependent and independent variables — either because they have the same importance or the same role. This is where symmetric regression can help.
Luca Greco, George Luta, Martin Krzywinski & Naomi Altman (2025) Points of significance: Symmetric alternatives to the ordinary least squares regression. Nat. Methods 22:1610–1612.
Beyond Belief Campaign BRCA Art
Fuelled by philanthropy, findings into the workings of BRCA1 and BRCA2 genes have led to groundbreaking research and lifesaving innovations to care for families facing cancer.
This set of 100 one-of-a-kind prints explore the structure of these genes. Each artwork is unique — if you put them all together, you get the full sequence of the BRCA1 and BRCA2 proteins.
