Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - contact me Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca on Twitter Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - Lumondo Photography Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - Pi Art Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - Hilbertonians - Creatures on the Hilbert CurveMartin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - Pi Day 2020 - Piku
Sun is on my face ...a beautiful day without you.Royskoppbe apartmore quotes

numbers: curious


The Outbreak Poems — artistic emissions in a pandemic


visualization + design

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A `\pi` day music video!: Transcendental Tree Map premieres on 2020 Pi Day from Max Cooper's Yearning for the Infinite. Animation by Nick Cobby and myself. Watch live from Barbican Centre.
Music video of the “Transcendental Tree Map” Max Cooper's Yearning for the Infinite album. This video premiered on 2020 Pi Day. Music by Max Cooper. Animation by Nick Cobby and myself.
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The 2020 Pi Day art celebrates digits of `\pi` with piku (パイク) —poetry inspired by haiku.
They serve as the form for The Outbreak Poems.
Tau Day tree map animation of 8,909 digits of `\tau = 2 \pi` created with 40,015 lines. The video is 6:28 minutes long.

`\pi` Approximation Day Art Posters


Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2019 `\pi` has hundreds of digits, hundreds of languages and a special kids' edition.

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2018 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2017 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` approximation day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2015 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` approx day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2013 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Circular `\pi` art

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`.

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

The SEIRS model for infectious disease dynamics

Thu 18-06-2020

Realistic models of epidemics account for latency, loss of immunity, births and deaths.

We continue with our discussion about epidemic models and show how births, deaths and loss of immunity can create epidemic waves—a periodic fluctuation in the fraction of population that is infected.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: The SEIRS model for infectious disease dynamics. (read)

This column has an interactive supplemental component (download code) that allows you to explore epidemic waves and introduces the idea of the phase plane, a compact way to understand the evolution of an epidemic over its entire course.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: The SEIRS model for infectious disease dynamics. (Interactive supplemental materials)

Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: The SEIRS model for infectious disease dynamics. Nature Methods 17:557–558.

Background reading

Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: Modeling infectious epidemics. Nature Methods 17:455–456.

Gene Machines

Fri 05-06-2020

Shifting soundscapes, textures and rhythmic loops produced by laboratory machines.

In commemoration of the 20th anniversary of Canada's Michael Smith Genome Sciences Centre, Segue was commissioned to create an original composition based on audio recordings from the GSC's laboratory equipment, robots and computers—to make “music” from the noise they produce.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Gene Machines by Segue. Now available on vinyl.

Virus Mutations Reveal How COVID-19 Really Spread

Mon 01-06-2020

Genetic sequences of the coronavirus tell story of when the virus arrived in each country and where it came from.

Our graphic in Scientific American's Graphic Science section in the June 2020 issue shows a phylogenetic tree based on a snapshot of the data model from Nextstrain as of 31 March 2020.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Virus Mutations Reveal How COVID-19 Really Spread. Text by Mark Fischetti (Senior Editor), art direction by Jen Christiansen (Senior Graphics Editor), source: Nextstrain (enabled by data from GISAID).

Cover of Nature Cancer April 2020

Mon 27-04-2020

Our design on the cover of Nature Cancer's April 2020 issue shows mutation spectra of patients from the POG570 cohort of 570 individuals with advanced metastatic cancer.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Each ellipse system represents the mutation spectrum of an individual patient. Individual ellipses in the system correspond to the number of base changes in a given class and are layered by mutation count. Ellipse angle is controlled by the proportion of mutations in a class within the sample and its size is determined by a sigmoid mapping of mutation count scaled within the layer. The opacity of each system represents the duration since the diagnosis of advanced disease. (read more)

The cover design accompanies our report in the issue Pleasance, E., Titmuss, E., Williamson, L. et al. (2020) Pan-cancer analysis of advanced patient tumors reveals interactions between therapy and genomic landscapes. Nat Cancer 1:452–468.

Modeling infectious epidemics

Tue 16-06-2020

Every day sadder and sadder news of its increase. In the City died this week 7496; and of them, 6102 of the plague. But it is feared that the true number of the dead this week is near 10,000 ....
—Samuel Pepys, 1665

This month, we begin a series of columns on epidemiological models. We start with the basic SIR model, which models the spread of an infection between three groups in a population: susceptible, infected and recovered.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Modeling infectious epidemics. (read)

We discuss conditions under which an outbreak occurs, estimates of spread characteristics and the effects that mitigation can play on disease trajectories. We show the trends that arise when "flattenting the curve" by decreasing `R_0`.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Modeling infectious epidemics. (read)

This column has an interactive supplemental component (download code) that allows you to explore how the model curves change with parameters such as infectious period, basic reproduction number and vaccination level.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Modeling infectious epidemics. (Interactive supplemental materials)

Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: Modeling infectious epidemics. Nature Methods 17:455–456.

The Outbreak Poems

Sat 04-04-2020

I'm writing poetry daily to put my feelings into words more often during the COVID-19 outbreak.

Ideas,
light
in a shadow.
Falling Moon
caught
by horizon.
Distant shout
comes 
in a whisper.

Read the poems and learn what a piku is.