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The art of Pi (`\pi`), Phi (`\phi`) and `e`
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.
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.
Ishihara's Tests for Colour Deficiency
Authentic and accurate images of Ishihara's test plates photographed (and lovingly color-corrected) from the 38-plate Ishihara's Tests for Colour Deficiency.
Symmetric alternatives to the ordinary least squares regression
What immortal hand or eye, could frame thy fearful symmetry? — William Blake, "The Tyger"
This month, we look at symmetric regression, which, unlike simple linear regression, it is reversible — remaining unaltered when the variables are swapped.
Simple linear regression can summarize the linear relationship between two variables `X` and `Y` — for example, when `Y` is considered the response (dependent) and `X` the predictor (independent) variable.
However, there are times when we are not interested (or able) to distinguish between dependent and independent variables — either because they have the same importance or the same role. This is where symmetric regression can help.
Luca Greco, George Luta, Martin Krzywinski & Naomi Altman (2025) Points of significance: Symmetric alternatives to the ordinary least squares regression. Nat. Methods 22:1610–1612.
Beyond Belief Campaign BRCA Art
Fuelled by philanthropy, findings into the workings of BRCA1 and BRCA2 genes have led to groundbreaking research and lifesaving innovations to care for families facing cancer.
This set of 100 one-of-a-kind prints explore the structure of these genes. Each artwork is unique — if you put them all together, you get the full sequence of the BRCA1 and BRCA2 proteins.
Propensity score weighting
The needs of the many outweigh the needs of the few. —Mr. Spock (Star Trek II)
This month, we explore a related and powerful technique to address bias: propensity score weighting (PSW), which applies weights to each subject instead of matching (or discarding) them.
Kurz, C.F., Krzywinski, M. & Altman, N. (2025) Points of significance: Propensity score weighting. Nat. Methods 22:638–640.
Happy 2025 π Day—
TTCAGT: a sequence of digits
Celebrate π Day (March 14th) and sequence digits like its 1999. Let's call some peaks.
Crafting 10 Years of Statistics Explanations: Points of Significance
I don’t have good luck in the match points. —Rafael Nadal, Spanish tennis player
Points of Significance is an ongoing series of short articles about statistics in Nature Methods that started in 2013. Its aim is to provide clear explanations of essential concepts in statistics for a nonspecialist audience. The articles favor heuristic explanations and make extensive use of simulated examples and graphical explanations, while maintaining mathematical rigor.
Topics range from basic, but often misunderstood, such as uncertainty and P-values, to relatively advanced, but often neglected, such as the error-in-variables problem and the curse of dimensionality. More recent articles have focused on timely topics such as modeling of epidemics, machine learning, and neural networks.
In this article, we discuss the evolution of topics and details behind some of the story arcs, our approach to crafting statistical explanations and narratives, and our use of figures and numerical simulations as props for building understanding.
Altman, N. & Krzywinski, M. (2025) Crafting 10 Years of Statistics Explanations: Points of Significance. Annual Review of Statistics and Its Application 12:69–87.