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visualization + design
`\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`).
VIEW ALL
news + thoughts
Cancer Cell cover
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.
Nature Biotechnology cover
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.
Nature cover — Gene Genie
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.
Happy 2022 `\pi` Day—
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.
PNAS Cover — Earth BioGenome Project
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.
The COVID charts — hospitalization rates
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.
