`\pi` Day 2021 Art Posters - A forest of `\pi` (a Lindenmayer system)
On March 14th celebrate `\pi` Day. Hug `\pi`—find a way to do it.
For those who favour `\tau=2\pi` will have to postpone celebrations until July 26th. That's what you get for thinking that `\pi` is wrong. I sympathize with this position and have `\tau` day art too!
If you're not into details, you may opt to party on July 22nd, which is `\pi` approximation day (`\pi` ≈ 22/7). It's 20% more accurate that the official `\pi` day!
Finally, if you believe that `\pi = 3`, you should read why `\pi` is not equal to 3.
The trees along this city street,
Save for the traffic and the trains,
Would make a sound as thin and sweet
As trees in country lanes.
—Edna St. Vincent Millay (City Trees)
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Welcome to this year's celebration of `\pi` and mathematics.
The theme this year is flower and flowers—in contrast to last year's understandable downturn in mood.
This year's `\pi` poem City Trees by Edna St. Vincent Millay.
This year's `\pi` day song is Sway by Laleh.
In past years, I've used the digits to draw a star map, run a gravity simulation draw a star map, draw streets of imagined cities. I even took a stab at waxing poetic.
Play time isn't over. This year, the digits of `\pi` sprout an infinite and irrational forest.
Good things grow for those who wait.
Can you see the digits through the forest?
The digits of `\pi` are shown as a forest. Each tree in the forest represents the digits of `\pi` up to the next 9. The first 10 trees are "grown" from the digit sets 314159, 2653589, 79, 3238462643383279, 50288419, 7169, 39, 9, 3751058209, and 749.
The digits control how the tree grows — but there is also a good amount of botanical variation. Below I outline the growth process — see the methods section for details.
the rules of the forest
branches
The first digit of a tree controls how many branches grow from the trunk of the tree. For example, the first tree's first digit is 3, so you see 3 branches growing from the trunk.
The next digit's branches grow from the end of a branch of the previous digit in left-to-right order. This process continues until all the tree's digits have been used up.
The branching exception is 0, which terminates the current branch — 0 branches grow!
leaves and flowers
The tree's digits themselves are drawn as circular leaves, color-coded by the digit.
The leaf exception is 9, which causes one of the branches of the previous digit to sprout a flower! The petals of the flower are colored by the digit before the 9 and the center is colored by the digit after the 9, which is on the next tree. This is how the forest propagates.
Leaves are placed at the tips of branches in a left-to-right order — you can "easily" read them off. Additionally, the leaves are distributed within the tree (without disturbing their left-to-right order) to spread them out as much as possible and avoid overlap. This order is deterministic.
The leaf placement exception are the branch set that sprouted the flower. These are not used to grow leaves — the flower needs space!
special cases — the forest's children
The digit subset "09" is very special. By the rules above, since 0 terminates the branch and 9 grows a flower, we get a flower on the ground — the tree doesn't get to grow but (luckily) flowers to propagates to the next tree.
Two or more 9's in a row generate a series of flowers. The digit forest poster ends in 5 flowers — these are the Feynman Flowers — created by the 999999 at digit 762, which is called the Feynman Point in `\pi`.
a digit nature walk
The rules of the forest are complicated. The labels below the trees help you orient yourself in the stream of digits. Flowers on the ground have no label.
shhh, the trees are sleeping
When the lights go out, it's harder to tell what's going on.
And if you really want a deep dive, check out the underwater edition.
Sometimes it's cloudy and sad in the forest.
But it's best to see all the posters to make sure you don't miss anything.
How it started
The first digit set is 314159 and the 3141 can be read off from the colored leaves. Left to right, these are: orange, red, yellow, red. The 5 is immediately before a 9, so it sprouts a flower. The petals are colored by the digit (5 is green) and the center by the first digit of the next tree (2 is dark orange).
Some trees are smaller than others. The tree for 79 only has a chance to grow 7 branches from the trunk before sprouting a flower.
How it's going
The artwork shows the forest up to the end of the Feynman Point, which is the first 999999 in `\pi`. It happens at digit 762 and ends at digit 768.
I'll leave you to work out how the Feynman Point results in 5 Feynman Flowers and why the center of the last flower is a different color.
Deterministic but always changing
There is "random" variation in aspects of a tree, such as branch length, angle, and direction of growth. However, the randomness is deterministic — the identical same forest is always generated.
To achieve this, I used the digits of each tree and its predecessor (all but the first have one) to create a random number generator — a linear congruential generator.
If you stare into the forest long enough, you can see the branches sway and sway away.
The more digits in the tree (and its predecessor) the more "randomness" there is in the output of the generator. Two flowers in a row use "99" as the input to the generator, which is no randomness at all. But the generator from the first tree's "314159" offers lots of variation.
Each aspect of the tree that has variation has its own generator. There's more detail about this in the methods section.
Propensity score matching
I don’t have good luck in the match points. —Rafael Nadal, Spanish tennis player
In many experimental designs, we need to keep in mind the possibility of confounding variables, which may give rise to bias in the estimate of the treatment effect.
If the control and experimental groups aren't matched (or, roughly, similar enough), this bias can arise.
Sometimes this can be dealt with by randomizing, which on average can balance this effect out. When randomization is not possible, propensity score matching is an excellent strategy to match control and experimental groups.
Kurz, C.F., Krzywinski, M. & Altman, N. (2024) Points of significance: Propensity score matching. Nat. Methods 21:1770–1772.
Nasa to send our human genome discs to the Moon
We'd like to say a ‘cosmic hello’: mathematics, culture, palaeontology, art and science, and ... human genomes.
Comparing classifier performance with baselines
All animals are equal, but some animals are more equal than others. —George Orwell
This month, we will illustrate the importance of establishing a baseline performance level.
Baselines are typically generated independently for each dataset using very simple models. Their role is to set the minimum level of acceptable performance and help with comparing relative improvements in performance of other models.
Unfortunately, baselines are often overlooked and, in the presence of a class imbalance, must be established with care.
Megahed, F.M, Chen, Y-J., Jones-Farmer, A., Rigdon, S.E., Krzywinski, M. & Altman, N. (2024) Points of significance: Comparing classifier performance with baselines. Nat. Methods 21:546–548.
Happy 2024 π Day—
sunflowers ho!
Celebrate π Day (March 14th) and dig into the digit garden. Let's grow something.
How Analyzing Cosmic Nothing Might Explain Everything
Huge empty areas of the universe called voids could help solve the greatest mysteries in the cosmos.
My graphic accompanying How Analyzing Cosmic Nothing Might Explain Everything in the January 2024 issue of Scientific American depicts the entire Universe in a two-page spread — full of nothing.
The graphic uses the latest data from SDSS 12 and is an update to my Superclusters and Voids poster.
Michael Lemonick (editor) explains on the graphic:
“Regions of relatively empty space called cosmic voids are everywhere in the universe, and scientists believe studying their size, shape and spread across the cosmos could help them understand dark matter, dark energy and other big mysteries.
To use voids in this way, astronomers must map these regions in detail—a project that is just beginning.
Shown here are voids discovered by the Sloan Digital Sky Survey (SDSS), along with a selection of 16 previously named voids. Scientists expect voids to be evenly distributed throughout space—the lack of voids in some regions on the globe simply reflects SDSS’s sky coverage.”
voids
Sofia Contarini, Alice Pisani, Nico Hamaus, Federico Marulli Lauro Moscardini & Marco Baldi (2023) Cosmological Constraints from the BOSS DR12 Void Size Function Astrophysical Journal 953:46.
Nico Hamaus, Alice Pisani, Jin-Ah Choi, Guilhem Lavaux, Benjamin D. Wandelt & Jochen Weller (2020) Journal of Cosmology and Astroparticle Physics 2020:023.
Sloan Digital Sky Survey Data Release 12
Alan MacRobert (Sky & Telescope), Paulina Rowicka/Martin Krzywinski (revisions & Microscopium)
Hoffleit & Warren Jr. (1991) The Bright Star Catalog, 5th Revised Edition (Preliminary Version).
H0 = 67.4 km/(Mpc·s), Ωm = 0.315, Ωv = 0.685. Planck collaboration Planck 2018 results. VI. Cosmological parameters (2018).
constellation figures
stars
cosmology
Error in predictor variables
It is the mark of an educated mind to rest satisfied with the degree of precision that the nature of the subject admits and not to seek exactness where only an approximation is possible. —Aristotle
In regression, the predictors are (typically) assumed to have known values that are measured without error.
Practically, however, predictors are often measured with error. This has a profound (but predictable) effect on the estimates of relationships among variables – the so-called “error in variables” problem.
Error in measuring the predictors is often ignored. In this column, we discuss when ignoring this error is harmless and when it can lead to large bias that can leads us to miss important effects.
Altman, N. & Krzywinski, M. (2024) Points of significance: Error in predictor variables. Nat. Methods 21:4–6.
Background reading
Altman, N. & Krzywinski, M. (2015) Points of significance: Simple linear regression. Nat. Methods 12:999–1000.
Lever, J., Krzywinski, M. & Altman, N. (2016) Points of significance: Logistic regression. Nat. Methods 13:541–542 (2016).
Das, K., Krzywinski, M. & Altman, N. (2019) Points of significance: Quantile regression. Nat. Methods 16:451–452.