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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`).

Discover Cantor's transfinite numbers through my music video for the Aleph 2 track of Max Cooper's Yearning for the Infinite (album page, event page).

I discuss the math behind the video and the system I built to create the video.

*Everything we see hides another thing, we always want to see what is hidden by what we see.
—Rene Magritte*

A Hidden Markov Model extends a Markov chain to have hidden states. Hidden states are used to model aspects of the system that cannot be directly observed and themselves form a Markov chain and each state may emit one or more observed values.

Hidden states in HMMs do not have to have meaning—they can be used to account for measurement errors, compress multi-modal observational data, or to detect unobservable events.

In this column, we extend the cell growth model from our Markov Chain column to include two hidden states: normal and sedentary.

We show how to calculate forward probabilities that can predict the most likely path through the HMM given an observed sequence.

Grewal, J., Krzywinski, M. & Altman, N. (2019) Points of significance: Hidden Markov Models. *Nature Methods* **16**:795–796.

Altman, N. & Krzywinski, M. (2019) Points of significance: Markov Chains. *Nature Methods* **16**:663–664.

My cover design for Hola Mundo by Hannah Fry. Published by Blackie Books.

Curious how the design was created? Read the full details.

*You can look back there to explain things,
but the explanation disappears.
You'll never find it there.
Things are not explained by the past.
They're explained by what happens now.
—Alan Watts*

A Markov chain is a probabilistic model that is used to model how a system changes over time as a series of transitions between states. Each transition is assigned a probability that defines the chance of the system changing from one state to another.

Together with the states, these transitions probabilities define a stochastic model with the Markov property: transition probabilities only depend on the current stateâ€”the future is independent of the past if the present is known.

Once the transition probabilities are defined in matrix form, it is easy to predict the distribution of future states of the system. We cover concepts of aperiodicity, irreducibility, limiting and stationary distributions and absorption.

This column is the first part of a series and pairs particularly well with Alan Watts and Blond:ish.

Grewal, J., Krzywinski, M. & Altman, N. (2019) Points of significance: Markov Chains. *Nature Methods* **16**:663–664.

*Places to go and nobody to see.*

Exquisitely detailed maps of places on the Moon, comets and asteroids in the Solar System and stars, deep-sky objects and exoplanets in the northern and southern sky. All maps are zoomable.

Quantile regression explores the effect of one or more predictors on quantiles of the response. It can answer questions such as "What is the weight of 90% of individuals of a given height?"

Unlike in traditional mean regression methods, no assumptions about the distribution of the response are required, which makes it practical, robust and amenable to skewed distributions.

Quantile regression is also very useful when extremes are interesting or when the response variance varies with the predictors.

Das, K., Krzywinski, M. & Altman, N. (2019) Points of significance: Quantile regression. *Nature Methods* **16**:451–452.

Altman, N. & Krzywinski, M. (2015) Points of significance: Simple linear regression. *Nature Methods* **12**:999–1000.