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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.

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.

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.

Altman, N. & Krzywinski, M. (2016) Points of significance: Analyzing outliers: Influential or nuisance. Nature Methods 13:281–282.

Two-level factorial experiments, in which all combinations of multiple factor levels are used, efficiently estimate factor effects and detect interactionsâ€”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.

Krzywinski, M. & Altman, N. (2014) Points of significance: Designing comparative experiments.. Nature Methods 11:597–598.

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.

One moment you're `:)`

and the next you're `:-.`

Make sense of it all with my Tree of Emotional life—a hierarchical account of how we feel.

One of my color tools, the `colorsnap`

application snaps colors in an image to a set of reference colors and reports their proportion.

Below is Times Square rendered using the colors of the MTA subway lines.