Let me tell you about something.

Distractions and amusements, with a sandwich and coffee.

Twenty — minutes — maybe — more.
•
• choose four words

Typography geek? If you like the geometry and mathematics of these posters, you may enjoy something more lettered. Visions of type: Type Peep Show: The Private Curves of Letters posters.

numbers.tgz

1,000,000 digits of
π
,
φ
,
e
and ASN.

The source code is freely available. Read how you can compute your own π path!

Watch the video at Numberphile about my art.

Explore Pi Day art for 2013.

All the artwork can be purchased from Fine Art America.

Numbers are a lot of fun. They can start conversations—the interesting number paradox is a party favourite. Of course, in the wrong company they can just as easily end conversations.

The art here represents my attempt at transforming famous numbers in mathematics into pretty visual forms. This work is 99% art and 1% data visualization. Because the digits in the numbers are essentially random (as far as we know), the essence of the art is based on randomness.

In a few cases, the art reveals an interesting and unexpected observation. For example, the sequence 999999 in π at digit 762 appears significantly earlier than expected by chance. Or that if you calculate π to 13,099,586 digits you will find love, as encoded by 1114214 in the scheme a=0, b=1, c=2...

Keep in mind that because the digits are random and never terminating, they have the property that they contain all observations about numbers within them. In fact, because the digits go on forever, you'll eventually find π within π.

Of these three transcendental numbers, π is the most well known. It is the ratio of a circle's circumference to its diameter (*d* = π*r*).

The Golden Ratio (φ) is the attractive proportion of values *a* and *b* (*a* > *b*) that satisfy (*a*+*b*)/*a* = *a*/*b*, which solves to *a*/*b* = (1+√5)/2.

The last of the three numbers, e 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 π, 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!

π φ e

= 3.141592653589793238462643... = 1.618033988749894848204586... = 2.718281828459045235360287...

These three numbers have the curious property that they are almost Pythagorean. In other words, if they are made into sides of a triangle, the triangle is nearly a right-angled triangle (89.1°).

Did you notice how in the 12th decimal point all three numbers have the same digit—9? This accidental similarity generates its own number—the Accidental Similarity Number (ASN).

perl, SVG, Illustrator

Hug π on March 14th and celebrate Pi Day. Those who favour τ will have to postpone celebrations until July 26th (τ = 2 π). If you're not into details, you may opt to party on July 22nd, which is π approximation day (π ≈ 22/7).

The 2013 posters were inspired by the beautiful AIDS posters by Elena Miska.

A concept created for this visualization, the *i*ness of a number measures how close each of its digits is to a given number, *i*.

The *i*ness is calculated for each digit from the average of the relative difference between *i* and the digit's neighbours.

The 4ness of Pi (π) is a specific case of an *i*ness, for *i*=4.

Thanks to Lance Bailey for suggesting how to measure *i*ness.

In the sequence of Pi (π) `3.1`

the neighbours of the __4__15__4__ are 3, 1, 1 and 5. The relative distances to 4 are -1, -3, -1 and 1. The average, which is the 4ness, of this digit (which is also a 4, coincidentally) is -1.5. The 4ness of each of the other digits is computed identically.

In the *i*ness posters, the 4ness is mapped onto a color and the standard deviation of the differences onto a size.

The accidental similarity number is a kind of overlap between numbers. I came up with this concept after creating typographical art about the 4ness of Pi (π).

To construct this number for Pi (π), Phi (φ) and e we first write the numbers on top of each other and then identify positions for which the numbers have the same digit.

3.141 … 3589793 … 7067982 … 7019385 … 1.618 … 8749894 … 1137484 … 5959395 … 2.718 … 8459045 … 6427427 … 6279434 …

These digits are then used to create the accidental similarity number. In thise case,

asn(π,φ,e) = 0.979 …

Numerology is bogus, but art based on numbers is pretty, in a random non-metaphysical way.

These depictions were generated using my Circos software by Cristian Ilies Vasile and myself.

We've previously written about how to analyze the impact of one variable in our ANOVA column. Complex biological systems are rarely so obliging—multiple experimental factors interact and producing effects.

ANOVA is a natural way to analyze multiple factors. It can incorporate the possibility that the factors interact—the effect of one factor depends on the level of another factor. For example, the potency of a drug may depend on the subject's diet.

We can increase the power of the analysis by allowing for interaction, as well as by blocking.

Krzywinski, M., Altman, (2014) Points of Significance: Two Factor Designs *Nature Methods* **11**:1187-1188.

Blainey, P., Krzywinski, M. & Altman, N. (2014) Points of Significance: Replication *Nature Methods* **11**:879-880.

Krzywinski, M. & Altman, N. (2014) Points of Significance: Analysis of variance (ANOVA) and blocking *Nature Methods* **11**:699-700.

Krzywinski, M. & Altman, N. (2014) Points of Significance: Designing Comparative Experiments *Nature Methods* **11**:597-598.

Sources of noise in experiments can be mitigated and assessed by nested designs. This kind of experimental design naturally models replication, which was the topic of last month's column.

Nested designs are appropriate when we want to use the data derived from experimental subjects to make general statements about populations. In this case, the subjects are *random* factors in the experiment, in contrast to *fixed* factors, such as we've seen previously.

In ANOVA analysis, random factors provide information about the amount of noise contributed by each factor. This is different from inferences made about fixed factors, which typically deal with a change in mean. Using the F-test, we can determine whether each layer of replication (e.g. animal, tissue, cell) contributes additional variation to the overall measurement.

Krzywinski, M., Altman, N. & Blainey, P. (2014) Points of Significance: Nested designs *Nature Methods* **11**:977-978.

Blainey, P., Krzywinski, M. & Altman, N. (2014) Points of Significance: Replication *Nature Methods* **11**:879-880.

Krzywinski, M. & Altman, N. (2014) Points of Significance: Analysis of variance (ANOVA) and blocking *Nature Methods* **11**:699-700.

Krzywinski, M. & Altman, N. (2014) Points of Significance: Designing Comparative Experiments *Nature Methods* **11**:597-598.

It's fitting that the column published just before Labor day weekend is all about how to best allocate labor.

Replication is used to decrease the impact of variability from parts of the experiment that contribute noise. For example, we might measure data from more than one mouse to attempt to generalize over all mice.

It's important to distinguish technical replicates, which attempt to capture the noise in our measuring apparatus, from biological replicates, which capture biological variation. The former give us no information about biological variation and cannot be used to directly make biological inferences. To do so is to commit *pseudoreplication*. Technical replicates are useful to reduce the noise so that we have a better chance to detect a biologically meaningful signal.

Blainey, P., Krzywinski, M. & Altman, N. (2014) Points of Significance: Replication *Nature Methods* **11**:879-880.

Krzywinski, M. & Altman, N. (2014) Points of Significance: Analysis of variance (ANOVA) and blocking *Nature Methods* **11**:699-700.

Krzywinski, M. & Altman, N. (2014) Points of Significance: Designing Comparative Experiments *Nature Methods* **11**:597-598.

I was commissioned by Scientific American to create an information graphic that showed how our genomes are more similar to those of the chimp and bonobo than to the gorilla.

I had about 5 x 5 inches of print space to work with. For 4 genomes? No problem. Bring out the Hilbert curve!

To accompany the piece, I will be posting to the Scientific American blog about the process of creating the figure. And to emphasize that the *genome is not a blueprint*!

As part of this project, I created some Hilbert curve art pieces. And while exploring, found thousands of Hilbertonians!

Celebrate Pi Approximation Day (July 22nd) with the art of arm waving. This year I take the first 10,000 most accurate approximations (*m*/*n*, *m*=1..10,000) and look at their accuracy.

I turned to the spiral again after applying it to stack stacked ring plots of frequency distributions in Pi for the 2014 Pi Day.

Our 10th Points of Significance column! Continuing with our previous discussion about comparative experiments, we introduce ANOVA and blocking. Although this column appears to introduce two new concepts (ANOVA and blocking), you've seen both before, though under a different guise.

If you know the *t*-test you've already applied analysis of variance (ANOVA), though you probably didn't realize it. In ANOVA we ask whether the variation within our samples is compatible with the variation between our samples (sample means). If the samples don't all have the same mean then we expect the latter to be larger. The ANOVA test statistic (*F*) assigns significance to the ratio of these two quantities. When we only have two-samples and apply the *t*-test, *t*^{2} = *F*.

ANOVA naturally incorporates and partitions sources of variation—the effects of variables on the system are determined based on the amount of variation they contribute to the total variation in the data. If this contribution is large, we say that the variation can be "explained" by the variable and infer an effect.

We discuss how data collection can be organized using a randomized complete block design to account for sources of uncertainty in the experiment. This process is called blocking because we are blocking the variation from a known source of uncertainty from interfering with our measurements. You've already seen blocking in the paired *t*-test example, in which the subject (or experimental unit) was the block.

We've worked hard to bring you 20 pages of statistics primers (though it feels more like 200!). The column is taking a month off in August, as we shrink our error bars.

Krzywinski, M. & Altman, N. (2014) Points of Significance: Analysis of Variance (ANOVA) and Blocking *Nature Methods* **11**:699-700.

*Nature Methods* **11**:597-598.

Krzywinski, M. & Altman, N. (2014) Points of Significance: Comparing Samples — Part I — *t*-tests *Nature Methods* **11**:215-216.

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