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`\pi` Day 2021 Art Posters - A forest of `\pi` (a Lindenmayer system)


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2021 `\pi` reminds us that good things grow for those who wait.' edition.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2019 `\pi` has hundreds of digits, hundreds of languages and a special kids' edition.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2018 `\pi` day stitches street maps into new destinations.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2017 `\pi` day imagines the sky in a new way.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` approximation day wonders what would happen if about right was right.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` day sees digits really fall for each other.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2015 `\pi` day maps transcendentally.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` approx day spirals into roughness.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` day hypnotizes you into looking.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` day

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2013 `\pi` day is where it started

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Circular `\pi` art and other distractions

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.

Most of the art is available for purchase as framed prints and, yes, even pillows. Sleep's never been more important — I take custom requests.

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)


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca buy artwork
768 digits of `\pi` as a forest of trees. Underwater technicolor edition. (BUY ARTWORK)

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca buy artwork
768 digits of `\pi` as a forest of trees. Scorched technicolor edition. (BUY ARTWORK)

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca buy artwork
768 digits of `\pi` as a forest of leafless trees. Desolate edition. (BUY ARTWORK)

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.

The 2015 π Day art takes a Mondrain perspective on π. The art was used in a collaboration with Max Cooper for his track Transcendental Tree Map from the album Yearning for the Infinite. Animation by Nick Cobby and myself. Watch the full show at the Barbican Centre.

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.

The 2020 π Day art celebrates the digits of π with piku (パイク) — poetry inspired by haiku. They serve as the form for The Outbreak Poems. On our 2022 Pi Day album "three one four: a number of notes", a piku accompanies each track.

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.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Each tree grows from a set of consecutive digits sampled from the digits of `\pi` up to the next 9. The first tree, shown here, grows from 314159. Each of the digits determine how many branches grow at each fork in the tree — the branches here are colored by their corresponding digit to illustrate this. Leaves encode the digits in a left-to-right order. The digit 9 spawns a flower on one of the branches of the previous digit.

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.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The colors of a flower are determined by the first digit of the next tree and the penultimate digit of the current tree. If the current tree only has one digit, then that digit is used.

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`.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca buy artwork
768 digits of `\pi` as a forest of trees. (BUY ARTWORK)
The 2019 π Day art celebrates digits of π with hundreds of languages and alphabets. If you're a kid at heart—rejoice—there's a special edition for you!

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.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Somewhere in the middle of the `\pi` forest. Feeling lost? Orient yourself with friendly labels.

shhh, the trees are sleeping

When the lights go out, it's harder to tell what's going on.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca buy artwork
768 digits of `\pi` as a forest of leafless trees. Bat cave edition. (BUY ARTWORK)

And if you really want a deep dive, check out the underwater edition.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca buy artwork
768 digits of `\pi` as a forest of trees. Underwater edition. (BUY ARTWORK)

Sometimes it's cloudy and sad in the forest.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca buy artwork
768 digits of `\pi` as a forest of leafless trees. Desolate edition. (BUY ARTWORK)

But it's best to see all the posters to make sure you don't miss anything.

The 2018 π Day art celebrates the 30th anniversary of π day and connects friends stitching road maps from around the world. Pack a sandwich and let's go!

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).


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The first 46 digits of `\pi` grow as 8 trees. The double 9 at digit 45 creates a flower.

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.

If you like space, you will love this. The 2017 π Day art imagines the digits of π as a star catalogue with constellations of extinct animals and plants. The work is featured in the article Pi in the Sky at the Scientific American SA Visual blog.

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.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Trees 79 to 90 bring us to the end of the Feynman Point — its 6 9's in a row show up as 5 flowers. The petals of the first Feynman Flower and center of the last Feynman Flower take after surrounding digits.

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.

The 2016 π Day art imagines the digits of Pi as physical masses collapsing under gravity and is featured in the articles The Gravity of Pi and The Boundless Beauty of Pi at the Scientific American [SA Visual blog](http://blogs.scientificamerican.com/sa-visual.

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.


`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Different combinations of variation can create some funky effects. Here I show the digit forest as imagined underwater, in a desert, in a drought and just hanging out in a bat cave.

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.

news + thoughts

How Analyzing Cosmic Nothing Might Explain Everything

Thu 18-01-2024

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.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
How Analyzing Cosmic Nothing Might Explain Everything. Text by Michael Lemonick (editor), art direction by Jen Christiansen (Senior Graphics Editor), source: SDSS

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

constellation figures

Alan MacRobert (Sky & Telescope), Paulina Rowicka/Martin Krzywinski (revisions & Microscopium)

stars

Hoffleit & Warren Jr. (1991) The Bright Star Catalog, 5th Revised Edition (Preliminary Version).

cosmology

H0 = 67.4 km/(Mpc·s), Ωm = 0.315, Ωv = 0.685. Planck collaboration Planck 2018 results. VI. Cosmological parameters (2018).

Error in predictor variables

Tue 02-01-2024

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.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Error in predictor variables. (read)

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 20.

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.

Convolutional neural networks

Tue 02-01-2024

Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry. – Richard Feynman

Following up on our Neural network primer column, this month we explore a different kind of network architecture: a convolutional network.

The convolutional network replaces the hidden layer of a fully connected network (FCN) with one or more filters (a kind of neuron that looks at the input within a narrow window).

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Convolutional neural networks. (read)

Even through convolutional networks have far fewer neurons that an FCN, they can perform substantially better for certain kinds of problems, such as sequence motif detection.

Derry, A., Krzywinski, M & Altman, N. (2023) Points of significance: Convolutional neural networks. Nature Methods 20:1269–1270.

Background reading

Derry, A., Krzywinski, M. & Altman, N. (2023) Points of significance: Neural network primer. Nature Methods 20:165–167.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of significance: Logistic regression. Nature Methods 13:541–542.

Neural network primer

Tue 10-01-2023

Nature is often hidden, sometimes overcome, seldom extinguished. —Francis Bacon

In the first of a series of columns about neural networks, we introduce them with an intuitive approach that draws from our discussion about logistic regression.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Neural network primer. (read)

Simple neural networks are just a chain of linear regressions. And, although neural network models can get very complicated, their essence can be understood in terms of relatively basic principles.

We show how neural network components (neurons) can be arranged in the network and discuss the ideas of hidden layers. Using a simple data set we show how even a 3-neuron neural network can already model relatively complicated data patterns.

Derry, A., Krzywinski, M & Altman, N. (2023) Points of significance: Neural network primer. Nature Methods 20:165–167.

Background reading

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of significance: Logistic regression. Nature Methods 13:541–542.

Cell Genomics cover

Mon 16-01-2023

Our cover on the 11 January 2023 Cell Genomics issue depicts the process of determining the parent-of-origin using differential methylation of alleles at imprinted regions (iDMRs) is imagined as a circuit.

Designed in collaboration with with Carlos Urzua.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Our Cell Genomics cover depicts parent-of-origin assignment as a circuit (volume 3, issue 1, 11 January 2023). (more)

Akbari, V. et al. Parent-of-origin detection and chromosome-scale haplotyping using long-read DNA methylation sequencing and Strand-seq (2023) Cell Genomics 3(1).

Browse my gallery of cover designs.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A catalogue of my journal and magazine cover designs. (more)

Science Advances cover

Thu 05-01-2023

My cover design on the 6 January 2023 Science Advances issue depicts DNA sequencing read translation in high-dimensional space. The image showss 672 bases of sequencing barcodes generated by three different single-cell RNA sequencing platforms were encoded as oriented triangles on the faces of three 7-dimensional cubes.

More details about the design.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
My Science Advances cover that encodes sequence onto hypercubes (volume 9, issue 1, 6 January 2023). (more)

Kijima, Y. et al. A universal sequencing read interpreter (2023) Science Advances 9.

Browse my gallery of cover designs.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A catalogue of my journal and magazine cover designs. (more)
Martin Krzywinski | contact | Canada's Michael Smith Genome Sciences CentreBC Cancer Research CenterBC CancerPHSA
Google whack “vicissitudinal corporealization”
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