Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - contact me Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca on Twitter Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Lumondo Photography Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Pi Art Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Hilbertonians - Creatures on the Hilbert Curve
In your hiding, you're alone. Kept your treasures with my bones.Coeur de Piratecrawl somewhere bettermore quotes

DNA on 10th — street art, wayfinding and font


visualization + design

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The 2019 Pi Day art celebrates digits of `\pi` with hundreds of languages and alphabets. If you're a kid at heart—rejoice—there's a special edition for you!

`\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

Analyzing outliers: Robust methods to the rescue

Sat 30-03-2019
Robust regression generates more reliable estimates by detecting and downweighting outliers.

Outliers can degrade the fit of linear regression models when the estimation is performed using the ordinary least squares. The impact of outliers can be mitigated with methods that provide robust inference and greater reliability in the presence of anomalous values.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Analyzing outliers: Robust methods to the rescue. (read)

We discuss MM-estimation and show how it can be used to keep your fitting sane and reliable.

Greco, L., Luta, G., Krzywinski, M. & Altman, N. (2019) Points of significance: Analyzing outliers: Robust methods to the rescue. Nature Methods 16:275–276.

Background reading

Altman, N. & Krzywinski, M. (2016) Points of significance: Analyzing outliers: Influential or nuisance. Nature Methods 13:281–282.

Two-level factorial experiments

Fri 22-03-2019
To find which experimental factors have an effect, simultaneously examine the difference between the high and low levels of each.

Two-level factorial experiments, in which all combinations of multiple factor levels are used, efficiently estimate factor effects and detect interactions—desirable statistical qualities that can provide deep insight into a system.

They offer two benefits over the widely used one-factor-at-a-time (OFAT) experiments: efficiency and ability to detect interactions.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Two-level factorial experiments. (read)

Since the number of factor combinations can quickly increase, one approach is to model only some of the factorial effects using empirically-validated assumptions of effect sparsity and effect hierarchy. Effect sparsity tells us that in factorial experiments most of the factorial terms are likely to be unimportant. Effect hierarchy tells us that low-order terms (e.g. main effects) tend to be larger than higher-order terms (e.g. two-factor or three-factor interactions).

Smucker, B., Krzywinski, M. & Altman, N. (2019) Points of significance: Two-level factorial experiments Nature Methods 16:211–212.

Background reading

Krzywinski, M. & Altman, N. (2014) Points of significance: Designing comparative experiments.. Nature Methods 11:597–598.

Happy 2019 `\pi` Day—
Digits, internationally

Tue 12-03-2019

Celebrate `\pi` Day (March 14th) and set out on an exploration explore accents unknown (to you)!

This year is purely typographical, with something for everyone. Hundreds of digits and hundreds of languages.

A special kids' edition merges math with color and fat fonts.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
116 digits in 64 languages. (details)
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
223 digits in 102 languages. (details)

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

Tree of Emotional Life

Sun 17-02-2019

One moment you're :) and the next you're :-.

Make sense of it all with my Tree of Emotional life—a hierarchical account of how we feel.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A section of the Tree of Emotional Life.

Find and snap to colors in an image

Sat 29-12-2018

One of my color tools, the colorsnap application snaps colors in an image to a set of reference colors and reports their proportion.

Below is Times Square rendered using the colors of the MTA subway lines.


Colors used by the New York MTA subway lines.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Times Square in New York City.
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Times Square in New York City rendered using colors of the MTA subway lines.
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Granger rainbow snapped to subway lines colors from four cities. (zoom)

Take your medicine ... now

Wed 19-12-2018

Drugs could be more effective if taken when the genetic proteins they target are most active.

Design tip: rediscover CMYK primaries.

More of my American Scientific Graphic Science designs

Ruben et al. A database of tissue-specific rhythmically expressed human genes has potential applications in circadian medicine Science Translational Medicine 10 Issue 458, eaat8806.