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visualizations
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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
Statistics vs Machine Learning
We conclude our series on Machine Learning with a comparison of two approaches: classical statistical inference and machine learning. The boundary between them is subject to debate, but important generalizations can be made.Inference creates a mathematical model of the datageneration process to formalize understanding or test a hypothesis about how the system behaves. Prediction aims at forecasting unobserved outcomes or future behavior. Typically we want to do both and know how biological processes work and what will happen next. Inference and ML are complementary in pointing us to biologically meaningful conclusions.
Statistics asks us to choose a model that incorporates our knowledge of the system, and ML requires us to choose a predictive algorithm by relying on its empirical capabilities. Justification for an inference model typically rests on whether we feel it adequately captures the essence of the system. The choice of pattern-learning algorithms often depends on measures of past performance in similar scenarios.
Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Statistics vs machine learning. Nature Methods 15:233–234.
Background reading
Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.
Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: supervised methods. Nature Methods 15:5–6.
Happy 2018 `\pi` Day—Boonies, burbs and boutiques of `\pi`
Celebrate `\pi` Day (March 14th) and go to brand new places. Together with Jake Lever, this year we shrink the world and play with road maps.
Streets are seamlessly streets from across the world. Finally, a halva shop on the same block!
Intriguing and personal patterns of urban development for each city appear in the Boonies, Burbs and Boutiques series.
No color—just lines. Lines from Marrakesh, Prague, Istanbul, Nice and other destinations for the mind and the heart.
The art is featured in the Pi City on the Scientific American SA Visual blog.
Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, 2016 `\pi` Day and 2017 `\pi` Day.
Machine learning: supervised methods (SVM & kNN)
Supervised learning algorithms extract general principles from observed examples guided by a specific prediction objective.We examine two very common supervised machine learning methods: linear support vector machines (SVM) and k-nearest neighbors (kNN).
SVM is often less computationally demanding than kNN and is easier to interpret, but it can identify only a limited set of patterns. On the other hand, kNN can find very complex patterns, but its output is more challenging to interpret.
We illustrate SVM using a data set in which points fall into two categories, which are separated in SVM by a straight line "margin". SVM can be tuned using a parameter that influences the width and location of the margin, permitting points to fall within the margin or on the wrong side of the margin. We then show how kNN relaxes explicit boundary definitions, such as the straight line in SVM, and how kNN too can be tuned to create more robust classification.
Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Machine learning: a primer. Nature Methods 15:5–6.
Background reading
Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.
Human Versus Machine
Balancing subjective design with objective optimization.In a Nature graphics blog article, I present my process behind designing the stark black-and-white Nature 10 cover.
Nature 10, 18 December 2017
Machine learning: a primer
Machine learning extracts patterns from data without explicit instructions.In this primer, we focus on essential ML principles— a modeling strategy to let the data speak for themselves, to the extent possible.
The benefits of ML arise from its use of a large number of tuning parameters or weights, which control the algorithm’s complexity and are estimated from the data using numerical optimization. Often ML algorithms are motivated by heuristics such as models of interacting neurons or natural evolution—even if the underlying mechanism of the biological system being studied is substantially different. The utility of ML algorithms is typically assessed empirically by how well extracted patterns generalize to new observations.
We present a data scenario in which we fit to a model with 5 predictors using polynomials and show what to expect from ML when noise and sample size vary. We also demonstrate the consequences of excluding an important predictor or including a spurious one.
Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.
Snowflake simulation
Symmetric, beautiful and unique.Just in time for the season, I've simulated a snow-pile of snowflakes based on the Gravner-Griffeath model.
The work is described as a wintertime tale in In Silico Flurries: Computing a world of snow and co-authored with Jake Lever in the Scientific American SA Blog.
Gravner, J. & Griffeath, D. (2007) Modeling Snow Crystal Growth II: A mesoscopic lattice map with plausible dynamics.