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
And she looks like the moon. So close and yet, so far.Future Islandsaim highmore quotes

pi day: beautiful


In Silico Flurries: Computing a world of snow. Scientific American. 23 December 2017


visualization + design

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The 2018 Pi Day art celebrates the 30th anniversary of `\pi` day and connects friends stitching road maps from around the world. Pack a sandwich and let's go!

`\pi` Approximation Day Art Posters


Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2018 `\pi` day shrinks the world and celebrates road trips by stitching streets from around the world together. In this version, we look at the boonies, burbs and boutique of `\pi` by drawing progressively denser patches of streets. Let's go places.

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

Oryza longistaminata genome cake

Mon 24-09-2018

Data visualization should be informative and, where possible, tasty.

Stefan Reuscher from Bioscience and Biotechnology Center at Nagoya University celebrates a publication with a Circos cake.

The cake shows an overview of a de-novo assembled genome of a wild rice species Oryza longistaminata.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Circos cake celebrating Reuscher et al. 2018 publication of the Oryza longistaminata genome.

Optimal experimental design

Tue 31-07-2018
Customize the experiment for the setting instead of adjusting the setting to fit a classical design.

The presence of constraints in experiments, such as sample size restrictions, awkward blocking or disallowed treatment combinations may make using classical designs very difficult or impossible.

Optimal design is a powerful, general purpose alternative for high quality, statistically grounded designs under nonstandard conditions.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Optimal experimental design. (read)

We discuss two types of optimal designs (D-optimal and I-optimal) and show how it can be applied to a scenario with sample size and blocking constraints.

Smucker, B., Krzywinski, M. & Altman, N. (2018) Points of significance: Optimal experimental design Nature Methods 15:599–600.

Background reading

Krzywinski, M., Altman, N. (2014) Points of significance: Two factor designs. Nature Methods 11:1187–1188.

Krzywinski, M. & Altman, N. (2014) Points of significance: Analysis of variance (ANOVA) and blocking. Nature Methods 11:699–700.

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

The Whole Earth Cataloguer

Mon 30-07-2018
All the living things.

An illustration of the Tree of Life, showing some of the key branches.

The tree is drawn as a DNA double helix, with bases colored to encode ribosomal RNA genes from various organisms on the tree.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The circle of life. (read, zoom)

All living things on earth descended from a single organism called LUCA (last universal common ancestor) and inherited LUCA’s genetic code for basic biological functions, such as translating DNA and creating proteins. Constant genetic mutations shuffled and altered this inheritance and added new genetic material—a process that created the diversity of life we see today. The “tree of life” organizes all organisms based on the extent of shuffling and alteration between them. The full tree has millions of branches and every living organism has its own place at one of the leaves in the tree. The simplified tree shown here depicts all three kingdoms of life: bacteria, archaebacteria and eukaryota. For some organisms a grey bar shows when they first appeared in the tree in millions of years (Ma). The double helix winding around the tree encodes highly conserved ribosomal RNA genes from various organisms.

Johnson, H.L. (2018) The Whole Earth Cataloguer, Sactown, Jun/Jul, p. 89

Why we can't give up this odd way of typing

Mon 30-07-2018
All fingers report to home row.

An article about keyboard layouts and the history and persistence of QWERTY.

My Carpalx keyboard optimization software is mentioned along with my World's Most Difficult Layout: TNWMLC. True typing hell.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
TNWMLC requires seriously flexible digits. It’s 87% more difficult than using a standard Qwerty keyboard, according to Martin Krzywinski, who created it (Credit: Ben Nelms). (read)

McDonald, T. (2018) Why we can't give up this odd way of typing, BBC, 25 May 2018.

Molecular Case Studies Cover

Fri 06-07-2018

The theme of the April issue of Molecular Case Studies is precision oncogenomics. We have three papers in the issue based on work done in our Personalized Oncogenomics Program (POG).

The covers of Molecular Case Studies typically show microscopy images, with some shown in a more abstract fashion. There's also the occasional Circos plot.

I've previously taken a more fine-art approach to cover design, such for those of Nature, Genome Research and Trends in Genetics. I've used microscopy images to create a cover for PNAS—the one that made biology look like astrophysics—and thought that this is kind of material I'd start with for the MCS cover.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Cover design for Apr 2018 issue of Molecular Case Studies. (details)

Happy 2018 `\tau` Day—Art for everyone

Wed 27-06-2018
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
You know what day it is. (details)