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Distractions and amusements, with a sandwich and coffee.

Here we are now at the middle of the fourth large part of this talk.
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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`.

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.

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

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

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

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?

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

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.

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.

In this redesign of a pie chart figure from a Nature Medicine article [1], I look at how to organize and present a large number of categories.

I first discuss some of the benefits of a pie chart—there are few and specific—and its shortcomings—there are few but fundamental.

I then walk through the redesign process by showing how the tumor categories can be shown more clearly if they are first aggregated into a small number groups.

(bottom left) Figure 2b from Zehir et al. Mutational landscape of metastatic cancer revealed from prospective clinical sequencing of 10,000 patients. (2017) Nature Medicine doi:10.1038/nm.4333

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.

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.

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.

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.

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.

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.

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.

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.

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

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.