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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
Markov Chains
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 state—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.
1-bit zoomable gigapixel maps of Moon, Solar System and Sky
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
Quantile regression robustly estimates the typical and extreme values of a response.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.
Background reading
Altman, N. & Krzywinski, M. (2015) Points of significance: Simple linear regression. Nature Methods 12:999–1000.
Analyzing outliers: Robust methods to the rescue
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.
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
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.
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
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.
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.