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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
Analysis of Variance (ANOVA) and Blocking—Accounting for Variability in Multi-factor Experiments
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, t2 = 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.
Background reading
Krzywinski, M. & Altman, N. (2014) Points of Significance: Designing Comparative Experiments 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.
Designing Experiments—Coping with Biological and Experimental Variation
This month, Points of Significance begins a series of articles about experimental design. We start by returning to the two-sample and paired t-tests for a discussion of biological and experimental variability.
We introduce the concept of blocking using the paired t-test as an example and show how biological and experimental variability can be related using the correlation coefficient, ρ, and how its value imapacts the relative performance of the paired and two-sample t-tests.
We also emphasize that when reporting data analyzed with the paired t-test, differences in sample means (and their associated 95% CI error bars) should be shown—not the original samples—because the correlation in the samples (and its benefits) cannot be gleaned directly from the sample data.
Krzywinski, M. & Altman, N. (2014) Points of Significance: Designing Comparative Experiments Nature Methods 11:597-598.
Background reading
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.
Have skew, will test
Our May Points of Significance Nature Methods column jumps straight into dealing with skewed data with Non Parametric Tests.
We introduce non-parametric tests and simulate data scenarios to compare their performance to the t-test. You might be surprised—the t-test is extraordinarily robust to distribution shape, as we've discussed before. When data is highly skewed, non-parametric tests perform better and with higher power. However, if sample sizes are small they are limited to a small number of possible P values, of which none may be less than 0.05!
Krzywinski, M. & Altman, N. (2014) Points of Significance: Non Parametric Testing Nature Methods 11:467-468.
Background reading
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.
Mind your p's and q's
In the April Points of Significance Nature Methods column, we continue our and consider what happens when we run a large number of tests.
Observing statistically rare test outcomes is expected if we run enough tests. These are statistically, not biologically, significant. For example, if we run N tests, the smallest P value that we have a 50% chance of observing is 1–exp(–ln2/N). For N = 10k this P value is Pk=10–kln2 (e.g. for 104=10,000 tests, P4=6.9×10–5).
We discuss common correction schemes such as Bonferroni, Holm, Benjamini & Hochberg and Storey's q and show how they impact the false positive rate (FPR), false discovery rate (FDR) and power of a batch of tests.
Krzywinski, M. & Altman, N. (2014) Points of Significance: Comparing Samples — Part II — Multiple Testing Nature Methods 11:215-216. 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.
Happy Pi Day— go to planet π
Celebrate Pi Day (March 14th) with the art of folding numbers. This year I take the number up to the Feynman Point and apply a protein folding algorithm to render it as a path.
For those of you who liked the minimalist and colorful digit grid, I've expanded on the concept to show stacked ring plots of frequency distributions.
And if spirals are your thing...
Have data, will compare
In the March Points of Significance Nature Methods column, we continue our discussion of t-tests from November (Significance, P values and t-tests).
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.