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Let me tell you about something.
lunch break
Distractions and amusements, with a sandwich and coffee.
visualization + design
download
numbers.tgz
1,000,000 digits of
π
,
φ
,
e
and ASN.
find your own path
The source code is freely available. Read how you can compute your own π path!
watch video
Watch the video at Numberphile about my art.
2013 Pi Day art
Explore Pi Day art for 2013.
buy artwork
All the artwork can be purchased from Fine Art America.
The art of Pi (π), Phi (φ) and e
the art
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 numbers
Of these three transcendental numbers, π is the most well known. It is the ratio of a circle's circumference to its diameter (d = πr).
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...
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 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).
methods
perl, SVG, Illustrator
Happy Pi Day!
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).
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The 2013 posters were inspired by the beautiful AIDS posters by Elena Miska.
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4ness of Pi (π)
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.
example
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.
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accidental similarity number
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).
example
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 …
Circos numerical art
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.
news + thoughts
Before and After—Designing Tiny Figures for Nature Methods
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.
Clarity, concision and space constraints—we have only 3.4" of horizontal space— all have to be balanced for a figure to be effective.
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!
Sources of Variation
Past columns have described experimental designs that mitigate the effect of variation: random assignment, blocking and replication.
The goal of these designs is to observe a reproducible effect that can be due only to the treatment, avoiding confounding and bias. Simultaneously, to sample enough variability to estimate how much we expect the effect to differ if the measurements are repeated with similar but not identical samples (replicates).
We need to distinguish between sources of variation that are nuisance factors in our goal to measure mean biological effects from those that are required to assess how much effects vary in the population.
Altman, N. & Krzywinski, M. (2014) Points of Significance: Two Factor Designs Nature Methods 11:5-6.
Background reading
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.
Two Factor Designs
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.
Background reading
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.
Nested Designs—Assessing Sources of Noise
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.
Background reading
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.
Replication—Quality over Quantity
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.
Background reading
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.
Monkeys on a Hilbert Curve—Scientific American Graphic
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!