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lunch break
Distractions and amusements, with a sandwich and coffee.
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`).
news + thoughts
Oryza longistaminata genome cake
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.
Optimal experimental design
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.
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
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.
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
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.
McDonald, T. (2018) Why we can't give up this odd way of typing, BBC, 25 May 2018.
Molecular Case Studies Cover
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.
Happy 2018 `\tau` Day—Art for everyone
