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Distractions and amusements, with a sandwich and coffee.

listen; there's a hell of a good universe next door: let's go.
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Numbers are a lot of fun. They can start conversations—the interesting number paradox is a party favourite: every number must be interesting because the first number that wasn't would be very interesting! Of course, in the wrong company they can just as easily end conversations.

The art here is my attempt at transforming famous numbers in mathematics into pretty visual forms, start some of these conversations and awaken emotions for mathematics—other than dislike and confusion

Numerology is bogus, but art based on numbers can be beautiful. Proclus got it right when he said (as quoted by M. Kline in *Mathematical Thought from Ancient to Modern Times*)

Wherever there is number, there is beauty.

—Proclus Diadochus

The consequence of the interesting number paradox is that all numbers are interesting. But some are more interesting than others—how Orwellian!

All animals are equal, but some animals are more equal than others.

—George Orwell (Animal Farm)

Numbers such as `\pi` (or `\tau` if you're a revolutionary), `\phi`, `e`, `i = \sqrt{-1}`, and `0` have captivated imagination. Chances are at least one of them appears in the next physics equation you come across.

π φ e

= 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 ... = 1.61803 39887 49894 84820 45868 34365 63811 77203 09179 ... = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 ...

Of these three transcendental numbers, `\pi` (3.14159265...) is the most well known. It is the ratio of a circle's circumference to its diameter (`d = \pi r`) and appears in the formula for the area of the circle (`a = \pi r^2`).

The Golden Ratio (`\phi`, 1.61803398...) is the attractive proportion of values `a > b` that satisfy `{a+b}/2 = a/b`, which solves to `a/b = {1 + \sqrt{5}}/2`.

The last of the three numbers, `e` (2.71828182...) is Euler's number and also known as the base of the natural logarithm. It, too, can be defined geometrically—it is the unique real number, `e`, for which the function `f(x) = e^x` has a tangent of slope 1 at `x=0`. Like `\pi`, `e` appears throughout mathematics. For example, `e` is central in the expression for the normal distribution as well as the definition of entropy. And if you've ever heard of someone talking about log plots ... well, there's `e` again!

Two of these numbers can be seen together in mathematics' most beautiful equation, the Euler identity: `e^{i\pi} = -1`. The tau-oists would argue that this is even prettier: `e^{i\tau} = 1`.

Did you notice how the 13th digit of all three numbers is the same (9)? This accidental similarity generates its own number—the Accidental Similarity Number (ASN).

Today is the day and it's hardly an approximation. In fact, `22/7` is 20% more accurate of a representation of `\pi` than `3.14`!

Time to celebrate, graphically. This year I do so with perfect packing of circles that embody the approximation.

By warping the circle by 8% along one axis, we can create a shape whose ratio of circumference to diameter, taken as twice the average radius, is 22/7.

If you prefer something more accurate, check out art from previous `\pi` days: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, and 2016 `\pi` Day.

*Regression can be used on categorical responses to estimate probabilities and to classify.*

The next column in our series on regression deals with how to classify categorical data.

We show how linear regression can be used for classification and demonstrate that it can be unreliable in the presence of outliers. Using a logistic regression, which fits a linear model to the log odds ratio, improves robustness.

Logistic regression is solved numerically and in most cases, the maximum-likelihood estimates are unique and optimal. However, when the classes are perfectly separable, the numerical approach fails because there is an infinite number of solutions.

Altman, N. & Krzywinski, M. (2016) Points of Significance: Logistic regression. *Nature Methods* **13**:541-542.

Altman, N. & Krzywinski, M. (2016) Points of Significance: Regression diagnostics? *Nature Methods* **13**:385-386.

Altman, N. & Krzywinski, M. (2015) Points of Significance: Multiple Linear Regression *Nature Methods* **12**:1103-1104.

Altman, N. & Krzywinski, M. (2015) Points of significance: Simple Linear Regression *Nature Methods* **12**:999-1000.

Genomic instability is one of the defining characteristics of cancer and within a tumor, which is an ever-evolving population of cells, there are many genomes. Mutations accumulate and propagate to create subpopulations and these groups of cells, called clones, may respond differently to treatment.

It is now possible to sequence individual cells within a tumor to create a profile of genomes. This profile changes with time, both in the kinds of mutation that are found and in their proportion in the overall population.

Clone evolution diagrams visualize these data. These diagrams can be qualitative, showing only trends, or quantitative, showing temporal and population changes to scale. In this Molecular Cell forum article I provide guidelines for drawing these diagrams, focusing with how to use color and navigational elements, such as grids, to clarify the relationships between clones.

I'd like to thank Maia Smith and Cydney Nielsen for assistance in preparing some of the figures in the paper.

Krzywinski, M. (2016) Visualizing Clonal Evolution in Cancer. Mol Cell 62:652-656.

*Limitations in print resolution and visual acuity impose limits on data density and detail.*

Your printer can print at 1,200 or 2,400 dots per inch. At reading distance, your reader can resolve about 200–300 lines per inch. This large gap—how finely we can print and how well we can see—can create problems when we don't take visual acuity into account.

The column provides some guidelines—particularly relevant when showing whole-genome data, where the scale of elements of interest such as genes is below the visual acuity limit—for binning data so that they are represented by elements that can be comfortably discerned.

Krzywinski, M. (2016) Points of view: Binning high-resolution data. Nature Methods 13:463.

*Residual plots can be used to validate assumptions about the regression model.*

Continuing with our series on regression, we look at how you can identify issues in your regression model.

The difference between the observed value and the model's predicted value is the residual, `r = y_i - \hat{y}_i`, a very useful quantity to identify the effects of outliers and trends in the data that might suggest your model is inadequate.

We also discuss normal probability plots (or Q-Q plots) and show how these can be used to check that the residuals are normally distributed, which is one of the assumptions of regression (constant variance being another).

Altman, N. & Krzywinski, M. (2016) Points of Significance: Analyzing outliers: Influential or nuisance? *Nature Methods* **13**:281-282.

Altman, N. & Krzywinski, M. (2015) Points of Significance: Multiple Linear Regression *Nature Methods* **12**:1103-1104.

Altman, N. & Krzywinski, M. (2015) Points of significance: Simple Linear Regression *Nature Methods* **12**:999-1000.

*Some outliers influence the regression fit more than others.*

This month our column addresses the effect that outliers have on linear regression.

You may be surprised, but not all outliers have the same influence on the fit (e.g. regression slope) or inference (e.g. confidence or prediction intervals). Outliers with large leverage—points that are far from the sample average—can have a very large effect. On the other hand, if the outlier is close to the sample average, it may not influence the regression slope at all.

Quantities such as Cook's distance and the so-called hat matrix, which defines leverage, are useful in assessing the effect of outliers.

Altman, N. & Krzywinski, M. (2015) Points of Significance: Multiple Linear Regression *Nature Methods* **12**:1103-1104.

Altman, N. & Krzywinski, M. (2015) Points of significance: Simple Linear Regression *Nature Methods* **12**:999-1000.