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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%.

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 0.16443654876822 7 3.142857142857 0.000402499435 improved23 7 3.285714285714 0.045875340318 24 7 3.428571428571 0.091348181202 ... 354 113 3.132743362832 0.002816816734355 113 3.141592920354 0.000000084914 improved356 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.

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.

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.

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 1133.1415929203 0.00000008491 52163 166043.1415923873 0.00000008474

After 52,162 there is a slew improvements to the approximation.

104348 332153.1415926539 0.000000000106 208341 663173.1415926534 0.0000000000389 312689 995323.1415926536 0.00000000000927 833719 2653813.141592653581 0.00000000000277 1146408 3649133.14159265359 0.000000000000513 3126535 9952073.141592653588 0.000000000000364 4272943 13601203.1415926535893 0.000000000000129 5419351 17250333.1415926535898 0.00000000000000705 42208400 134353513.1415926535897 0.00000000000000669 47627751 151603843.14159265358977 0.00000000000000512 53047102 168854173.14159265358978 0.00000000000000388 58466453 186104503.14159265358978 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`).

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*.

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.

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.

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

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.

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.