This love loves love. It's a strange love, strange love.find a way to love

# similarity: beautiful

Malofiej 23 Bronze award for 1% Difference information graphic.

# visualization + design

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.

# The art of Pi (pi), Phi (phi) and e

Support Ellie Balk's Kickstarter community math mural project in which Brooklyn students learn math and art to visualize pi.
2013 pi day
2014 pi day
2015 pi day
2014 pi approx day
Circular pi art

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.

2,258 digits of phi, 3,855 digits of e and 3,628 digits of pi in 6 level treemaps. Uniform line thickness. Bauhaus prime colors in Piet Mondrian style. (2015 pi day posters)

## the numbers π, φ and e

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 numbers \pi, \phi and e nearly form a right-angled triangle.

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 see something special?

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

Like music with numbers? Try Angels at My Door (Una), Pt vs Ys (Yoshinori Sunahara), 2wicky (Hooverphonic), One (Aimee Mann), Straight to Number One (Touch and Go), 99 luftbaloons (Nena).

## is π normal?

It is not yet known whether the digits of π are normal—determining this is an important problem in mathematics. In other words, is the distribution of digit frequencies in π uniform? Do each of the digits 0–9 appear exactly 1/10th of the time, does every two-digit string appear exactly 1/100th of the time and so on for every finite-length string1?

1 One interesting finite-length string is the 6-digit Fenyman Point (...999999...) which appears at digit 762 in π. The Feynman Point was the subject of 2014 pi Day art.

This question can be posed for different representations of π—in different bases. The distribution frequencies of 1/10, 1/100, and so on above refer to the representation of π in base 10. This is the way we're used to seeing numbers. However, if π is encoded as binary (base 2), would all the digits in 11.00100100001111... be normal? The table below shows the first several digits of π in each base from 2 to 16, as well as the natural logarithm base, e.

 base, b pi_b base, b pi_b 2 11.00100100001111 10 3.14159265358979 3 10.01021101222201 11 3.16150702865A48 4 3.02100333122220 12 3.184809493B9186 5 3.03232214303343 13 3.1AC1049052A2C7 6 3.05033005141512 14 3.1DA75CDA813752 7 3.06636514320361 15 3.21CD1DC46C2B7A 8 3.11037552421026 16 3.243F6A8885A300 e 10.10100202000211 source: virtuescience.com

Because the digits in the numbers are essentially random (this is a conjecture), the essence of the art is based on randomness.

A vexing consequence of π being normal is that, because it is non-terminating, π would contain all patterns. Any word you might think of, encoded into numbers in any way, would appear infinitely many times. The entire works of Shakespeare, too. As well, all his plays in which each sentence is reversed, or has one spelling mistake, or two! In fact, you would eventually find π within π, but only if you have infinite patience.

This is why any attempts to use the digits of pi to infer meaning about anything is ridiculous. The exact opposite of what you find is also in pi.

#### Stoneham's constant

A number can be normal in one base, but another. For example, Stoneham's constant,

\alpha_{2,3} = 1/2 + 1/(2^{3^1} 3^1) + 1/(2^{3^2} 3^2) + 1/(2^{3^3} 3^3) + ... + 1/(2^{3^k} 3^k) + ...

is 0.54188368083150298507... in base 10 and 0.100010101011100011100011100... in base 2.

Stoneham's constant is provably normal in base 2. In some other bases, such 6, Stoneham's constant is provably not normal.

All art posters are available for purchase.
I take custom requests.

## patterns in the art

Some of the numerical art reveals interesting and unexpected observations. For example, the sequence 999999 in π at digit 762 called the Feynman Point. Or that if you calculate π to 13,099,586 digits you will find love.

# Bayes' Theorem

Mon 06-04-2015

In our first column on Bayesian statistics, we introduce conditional probabilities and Bayes' theorem

P(B|A) = P(A|B) × P(B) / P(A)

This relationship between conditional probabilities P(B|A) and P(A|B) is central in Bayesian statistics. We illustrate how Bayes' theorem can be used to quickly calculate useful probabilities that are more difficult to conceptualize within a frequentist framework.

Nature Methods Points of Significance column: Bayes' Theorem. (read)

Using Bayes' theorem, we can incorporate our beliefs and prior experience about a system and update it when data are collected.

Altman, N. & Krzywinski, M. (2015) Points of Significance: Bayes' Theorem Nature Methods 12:277-278.

Oldford, R.W. & Cherry, W.H. Picturing probability: the poverty of Venn diagrams, the richness of eikosograms. (University of Waterloo, 2006)

# Happy 2015 Pi Day—can you see pi through the treemap?

Sat 14-03-2015

Celebrate pi Day (March 14th) with splitting its digit endlessly. This year I use a treemap approach to encode the digits in the style of Piet Mondrian.

Digits of pi, phi and e. (details)

The art has been featured in Ana Swanson's Wonkblog article at the Washington Post—10 Stunning Images Show The Beauty Hidden in pi.

I also have art from 2013 pi Day and 2014 pi Day.

# Split Plot Design

Tue 03-03-2015

The split plot design originated in agriculture, where applying some factors on a small scale is more difficult than others. For example, it's harder to cost-effectively irrigate a small piece of land than a large one. These differences are also present in biological experiments. For example, temperature and housing conditions are easier to vary for groups of animals than for individuals.

Nature Methods Points of Significance column: Split plot design. (read)

The split plot design is an expansion on the concept of blocking—all split plot designs include at least one randomized complete block design. The split plot design is also useful for cases where one wants to increase the sensitivity in one factor (sub-plot) more than another (whole plot).

Altman, N. & Krzywinski, M. (2015) Points of Significance: Split Plot Design Nature Methods 12:165-166.

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

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

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

# Color palettes for color blindness

Tue 03-03-2015

In an audience of 8 men and 8 women, chances are 50% that at least one has some degree of color blindness1. When encoding information or designing content, use colors that is color-blind safe.

A 12-color palette safe for color blindness

# Points of Significance Column Now Open Access

Tue 10-02-2015

Nature Methods has announced the launch of a new statistics collection for biologists.

Nature Methods Points of Significance column is now open access. (column archive)

As part of that collection, announced that the entire Points of Significance collection is now open access.

This is great news for educators—the column can now be freely distributed in classrooms.

# Before and After—Designing Tiny Figures for Nature Methods

Tue 13-01-2015

I've posted a writeup about the design and redesign process behind the figures in our Nature Methods Points of Significance column.

I have selected several figures from our past columns and show how they evolved from their draft to published versions.

Fig 2 from Points of Significance: Nested designs. (Krzywinski, M. & Altman, N. (2014) Nature Methods 11:977-978.) (...more)

Clarity, concision and space constraints—we have only 3.4" of horizontal space— all have to be balanced for a figure to be effective.

Fig 2c (excerpt) from Points of Significance: Designing comparative experiments. (Krzywinski, M. & Altman, N. (2014) Nature Methods 11:597-598.) (...more)

It's nearly impossible to find case studies of scientific articles (or figures) through the editing and review process. Nobody wants to show their drafts. With this writeup I hope to add to this space and encourage others to reveal their process. Students love this. See whether you agree with my decisions!