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visualization + design
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Watch the video at Numberphile about my art.
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1,000,000 digits of π, φ, e and ASN.
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All the artwork can be purchased from Fine Art America. Most of the pieces were created by myself, and some by Cristian Ilies Vasile.
Round art of π, φ and e
Numerology is bogus, but art based on numbers has a beautiful random quality.
For other examples of numerical art, see my inessiness project. Nixie clock lovers should investigate the accidental similarity number (ASN), which I render in a ASN Nixie poster.
Circos art of π — digit transition paths
It's fitting to use Circos to visualize the digits of π. After all, what is more round than Circos?
Cristian Ilies Vasile had the idea of representing the digits of π as a path traced by links between successive digits. Each digit is assigned a segment around the circle and a link between segment i and j corresponds to the appearance of ij in π. For example, the "14" in "3.14..." is drawn as a link between segment 1 and segment 4.
The position of the link on a digit's segment is associated with the position of the digit π. For example, the "14" link associated with the 2nd digit (1) and the 3rd digit (4) is drawn from position 2 on the 1 segment to position 3 on the 4 segment.
As more digits are added to the path, the image becomes a weaving mandala.
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circos art of π, φ and e — transition paths and bubbles
I added to Cristian's representation by showing the number of transitions between digits in a series of concentric circles placed outside the links. This summary representation counts the number of transition links within a region and addresses the question of what kind of digits appear immediately before or after a given digit in π. The approach is diagrammed below.
The original images were generated using the 10-color Brewer paired qualitative palette, which was later modified as shown below.
The bubbles that count the number of links quickly draw attention to regions where specific digit pairs are frequent. In the image for π below, which shows transitions for the first 1,000 digits, the large bubble on the 9 segment is due to the sequence "999999" sequence at decimal place 762.
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This sequence of 6 9's occurs significantly earlier than expected by chance. Because the distribution and sequence of digits of π is, as far as we know, uniformly random, we can calculate how frequently we should expect a series of 6 identical digits.
For a given digit, the chance that the next 5 digits are the same is 0.00001 (0.1 that the next digit is the same * 0.1 that the second-nex digit is the same * ...). Therefore the chance that a given position the next 5 digits are not the same is 1 - 1/0.00001 = 0.99999. From this, the chance that k consecutive digits don't initiate a 6-digit sequence is therefore 0.99999k.
If I ask what is k for which this value is 0.5, I need to solve 0.99999k, which gives k = 69,314. Thus, chances are 50-50 that in a 69,000 digit random sequence we'll see a run of 6 idendical digits. This calculation is an approximation.
It's fun to look for words in π. For example, love appears at 13,099,586th digit.
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The transition probabilities for each 10 digit bin for the first 2,000 digits of π, φ and e are shown in the image below.
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A tangent into randomness
The digits of π are, as far as we know, randomly distributed. Art based on its digits therefore as a quality that is influenced by this random distribution. To provide a reference of what such a random pattern looks like, below are 16 random numbers represented in the same way. They're all different, yet strangely the same.
Circos art of π — heaps of bubbles
Below are more images by Cristian Ilies Vasile, where dots are used to represent the adjacency between digits. As in the image above, each digit 0-9 is represented by a colored segment. For each digit sequence ij, a dot is placed on ith's segment at the position of i colored by j.
For example, for π the dot coordinates for the first 7 digits are (segment:position:label) 3:0:1 → 1:1:4 → 4:2:1 → 1:3:5 → 5:4:9 ...
segment position colored_by 3 0 1 1 1 4 4 2 1 1 3 5 5 4 9 9 5 2 2 6 6
Because there is a large number of digits, the dots stack up near their position to avoid overlapping. The layout of the dots is automated by Circos' text track layout.
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When the digits of π, e and φ are aligned, positions at which the three numbers have the same digit yield the accidental similarity number (ASN). Below is a dot plot of the transition of the ASN.
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spiral art of π
By mapping the digits onto a red-yellow-blue Brewer palette (0 
9) and placing them as circles on an Archimedean spiral a dense and pleasant layout can be obtained.
Why the Archimedean spiral? This spiral is defined as r = a + bθ and has the interesting property that a ray from the origin will intersect the spiral every 2πb. Thus, each spiral can accomodate inscribed circles of radius πb.
Why the Brewer palette? These color schemes have some very useful perceptual properties and are commonly used to encode quantitative and categorical data.
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!