1,000,000 digits of π, φ, e and ASN.
All the artwork can be purchased from Fine Art America. Most of the pieces were created by myself, and some by Cristian Ilies Vasile.
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.
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)=ex 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).
A concept created for this visualization, the iness of a number measures how close each of its digits is to a given number, i.
The iness 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 iness, for i=4.
Thanks to Lance Bailey for suggesting how to measure iness.
In the sequence of Pi (π)
3.1415 the neighbours of the 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 iness posters, the 4ness is mapped onto a color and the standard deviation of the differences onto a size.
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.
We look at what happens how uncertainty of two variables combines and how this impacts the increased uncertainty when two samples are compared and highlight the differences between the two-sample and paired t-tests.
When performing any statistical test, it's important to understand and satisfy its requirements. The t-test is very robust with respect to some of its assumptions, but not others. We explore which.
Krzywinski, M. & Altman, N. (2014) Points of Significance: Comparing Samples — Part I Nature Methods 11:215-216.
Krzywinski, M. & Altman, N. (2013) Points of Significance: Significance, P values and t-tests Nature Methods 10:1041-1042.
Beautiful Science explores how our understanding of ourselves and our planet has evolved alongside our ability to represent, graph and map the mass data of the time. The exhibit runs 20 February — 26 May 2014 and is free to the public. There is a good Nature blog writeup about it, a piece in The Guardian, and a great video that explains the the exhibit narrated by Johanna Kieniewicz, the curator.
I am privileged to contribute an information graphic to the exhibit in the Tree of Life section. The piece shows how sequence similarity varies across species as a function of evolutionary distance. The installation is a set of 6 30x30 cm backlit panels. They look terrific.
Quick, name three chart types. Line, bar and scatter come to mind. Perhaps you said pie too—tsk tsk. Nobody ever thinks of the box plot.
Box plots reveal details about data without overloading a figure with a full frequency distribution histogram. They're easy to compare and now easy to make with BoxPlotR (try it). In our fifth Points of Significance column, we take a break from the theory to explain this plot type and—I hope— convince you that they're worth thinking about.
The February issue of Nature Methods kicks the bar chart two more times: Dan Evanko's Kick the Bar Chart Habit editorial and a Points of View: Bar charts and box plots column by Mark Streit and Nils Gehlenborg.
Krzywinski, M. & Altman, N. (2014) Points of Significance: Visualizing samples with box plots Nature Methods 11:119-120.
For specialists, visualizations should expose detail to allow for exploration and inspiration. For enthusiasts, they should provide context, integrate facts and inform. For the layperson, they should capture the essence of the topic, narrate a story and deligt.
Wired's Brandon Keim wrote up a short article about me and some of my work—Circle of Life: The Beautiful New Way to Visualize Biological Data.
Experimental designs that lack power cannot reliably detect real effects. Power of statistical tests is largely unappreciated and many underpowered studies continue to be published.
This month, Naomi and I explain what power is, how it relates to Type I and Type II errors and sample size. By understanding the relationship between these quantities you can design a study that has both low false positive rate and high power.
Krzywinski, M. & Altman, N. (2013) Points of Significance: Power and Sample Size Nature Methods 10:1139-1140.
20 Tips for Interpreting Scientific Claims is a wonderful comment in Nature warning us about the limits of evidence.
Sutherland WJ, Spiegelhalter D & Burgman M (2013) Policy: Twenty tips for interpreting scientific claims. Nature 503:335–337.