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Trance opera—Spente le Stellebe dramaticmore quotes

pi: fun


EMBO Practical Course: Bioinformatics and Genome Analysis, 5–17 June 2017.


visualization + design

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The 2017 Pi Day art imagines the digits of Pi as a star catalogue with constellations of extinct animals and plants. The work is featured in the article Pi in the Sky at the Scientific American SA Visual blog.

`\pi` Approximation Day Art Posters


Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2017 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` approximation day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2015 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` approx day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2013 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Circular `\pi` art

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`.

There are two kinds of `\pi` Approximation Day posters.

The first uses the Archimedean spiral for its design, which I've used before for other numerical art. The second packs warped circles, whose ratio of circumference to average diameter is `22/7` into what I call `\pi`-approximate circular packing.

As you probably know, the ratio of the circumference of a circle to its diameter is `\pi`. $$ C / d = \pi $$

For `\pi` approximation day, let's ask what would happen if $$ C / d = 22/7 $$

where now `C` is the circumference of some shape other than a circle. What could this shape be?

A good place to start is to think about an ellipse. I've done this before in the 22/7 Universe article, in which I considered an ellipse with a major axis of `r+\delta` and a minor axis of `r` and solved for `\delta` such that the circumference of the ellipse divided by `2 r` would be `22/7`. Doing so means numerically solving the equation $$ \frac{C(r,r+\delta)}{2r} = 22/7 $$

where `r + \delta` is the major axis, `r` is the minor axis and `C(r,r+\delta)` is the circumference of the ellipse. Substituting the expression for the circumference, $$ 4(r+\delta) \int_0^{\pi/2} \sqrt { 1 - \left(1-\frac{r}{(r+\delta)^2}\right)\sin^2 \theta } d \theta = 2 r \frac{22}{7}$$

If we set `r=1` and solve it turns out that only a very minor deformation is required and `\delta = 0.0008`. You can verify this at Wolfram Alpha.

I wanted to make some art based on the shape of the this ellipse, but a deformation of 0.08% is not perceptible. So I came up with a slightly different approach to how I define the original circumference-to-diameter ratio.

Instead of treating the diameter as `r` and using `r + \delta` as the major axis, I now define the diameter as twice the average radius, or `2r + \delta`. This means that the equation to solve is $$ \frac{C(r,r+\delta)}{2r+\delta} = 22/7 $$

As before, setting `r=1` and substituting the expression for the circumference of an ellipse, we get $$ 4(1+\delta) \int_0^{\pi/2} \sqrt { 1 - \left(1-\frac{1}{(1+\delta)^2}\right)\sin^2 \theta } d \theta = (2+\delta) \frac{22}{7}$$

and solving this for `\delta` find $$ \delta = 0.083599769... $$

You can verify this at Wolfram Alpha.

This is a more useable approach since an 8% warping of a circle can be easily perceived.

Pi Approximation Day Art Poster / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The ratio of the circumference of a circle, `C(r)`, to its dimameter, `2r`, is `\pi`. If we warp the circle by 8%, the corresponding ratio, if we use twice the average radius as the diameter, is 22/7. This deformation can be easily identified.

Below is matrix of perfect circles along side the 8% deformed circles.

Pi Approximation Day Art Poster / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A matrix of perfect circles and ones which have been stretched by 8% along one axis and then randomly rotated. The deformed circles embody the `\pi` approximation of 22/7.

The art posters are based on a packing of these deformed circles.

Pi Approximation Day Art Poster / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Warped circles, packed.
Pi Approximation Day Art Poster / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Even more warped circles, packed.

By superimposing perfect circles on the warped circles, fun patterns appear.

Pi Approximation Day Art Poster / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Superposition of perfect and warped circles, packed.

perfect vs approximate packing

If you pack perfect circles perfectly, the area occupied by the circles is `\pi/4 = 78.5%`.

What is the area occupied by perfect packing of warped and randomly rotated (like in the posters) circles?

color scheme

To motivate choice of colors, I chose images with a 1970's feel.

Pi Approximation Day Art Poster / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Images used for color schemes. The colors of each image were grouped into clusters—8 for the first two images and 6 for the third—to obtain proportions of representative colors.

Using my color summarizer, I analyzed each image for its representative colors. Using these colors and their proportions, I colored the perfect and warped circles.

Pi Approximation Day Art Poster / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Packed warped circles colored in proportion to color schemes derived from the images above.

For each poster of these color schemes, two poster versions are available. In one, the perfect cirlces are shown with warped circles as a clip mask. In the other, warped circles are shown, clipped by perfect circles.

VIEW ALL

news + thoughts

Tabular Data

Tue 11-04-2017
Tabulating the number of objects in categories of interest dates back to the earliest records of commerce and population censuses.

After 30 columns, this is our first one without a single figure. Sometimes a table is all you need.

In this column, we discuss nominal categorical data, in which data points are assigned to categories in which there is no implied order. We introduce one-way and two-way tables and the `\chi^2` and Fisher's exact tests.

Altman, N. & Krzywinski, M. (2017) Points of Significance: Tabular data. Nature Methods 14:329–330.

...more about the Points of Significance column

Happy 2017 `\pi` Day—Star Charts, Creatures Once Living and a Poem

Tue 14-03-2017


on a brim of echo,

capsized chamber
drawn into our constellation, and cooling.
—Paolo Marcazzan

Celebrate `\pi` Day (March 14th) with star chart of the digits. The charts draw 40,000 stars generated from the first 12 million digits.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
12,000,000 digits of `\pi` interpreted as a star catalogue. (details)

The 80 constellations are extinct animals and plants. Here you'll find old friends and new stories. Read about how Desmodus is always trying to escape or how Megalodon terrorizes the poor Tecopa! Most constellations have a story.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Find friends and stories among the 80 constellations of extinct animals and plants. Oh look, a Dodo guardings his eggs! (details)

This year I collaborate with Paolo Marcazzan, a Canadian poet, who contributes a poem, Of Black Body, about space and things we might find and lose there.

Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day and and 2016 `\pi` Day.

Data in New Dimensions: convergence of art, genomics and bioinformatics

Tue 07-03-2017

Art is science in love.
— E.F. Weisslitz

A behind-the-scenes look at the making of our stereoscopic images which were at display at the AGBT 2017 Conference in February. The art is a creative collaboration with Becton Dickinson and The Linus Group.

Its creation began with the concept of differences and my writeup of the creative and design process focuses on storytelling and how concept of differences is incorporated into the art.

Oh, and this might be a good time to pick up some red-blue 3D glasses.

BD Genomics 3D art exhibit - AGBT 2017 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A stereoscopic image and its interpretive panel of single-cell transcriptomes of blood cells: diseased versus healthy control.

Interpreting P values

Thu 02-03-2017
A P value measures a sample’s compatibility with a hypothesis, not the truth of the hypothesis.

This month we continue our discussion about `P` values and focus on the fact that `P` value is a probability statement about the observed sample in the context of a hypothesis, not about the hypothesis being tested.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Interpreting P values. (read)

Given that we are always interested in making inferences about hypotheses, we discuss how `P` values can be used to do this by way of the Benjamin-Berger bound, `\bar{B}` on the Bayes factor, `B`.

Heuristics such as these are valuable in helping to interpret `P` values, though we stress that `P` values vary from sample to sample and hence many sources of evidence need to be examined before drawing scientific conclusions.

Altman, N. & Krzywinski, M. (2017) Points of Significance: Interpreting P values. Nature Methods 14:213–214.

Background reading

Krzywinski, M. & Altman, N. (2017) Points of significance: P values and the search for significance. Nature Methods 14:3–4.

Krzywinski, M. & Altman, N. (2013) Points of significance: Significance, P values and t–tests. Nature Methods 10:1041–1042.

...more about the Points of Significance column

Snellen Charts—Typography to Really Look at

Sat 18-02-2017

Another collection of typographical posters. These ones really ask you to look.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Snellen charts designed using physical constants, Braille and elemental abundances in the universe and human body.

The charts show a variety of interesting symbols and operators found in science and math. The design is in the style of a Snellen chart and typset with the Rockwell font.

Essentials of Data Visualization—8-part video series

Fri 17-02-2017
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca

In collaboration with the Phil Poronnik and Kim Bell-Anderson at the University of Sydney, I'm delighted to share with you our 8-part video series project about thinking about drawing data and communicating science.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Essentials of Data Visualization: Thinking about drawing data and communicating science.

We've created 8 videos, each focusing on a different essential idea in data visualization: encoding, shapes, color, uncertainty, design, drawing missing or unobserved data, labels and process.

The videos were designed as teaching materials. Each video comes with a slide deck and exercises.