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visualization + math

Pi Day 2022 — three one four: a number of notes — A musical journey into the digits of Pi


Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2021 `\pi` reminds us that good things grow for those who wait.' edition.

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2019 `\pi` has hundreds of digits, hundreds of languages and a special kids' edition.

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2018 `\pi` day stitches street maps into new destinations.

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2017 `\pi` day imagines the sky in a new way.


Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` approximation day wonders what would happen if about right was right.

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` day sees digits really fall for each other.

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2015 `\pi` day maps transcendentally.

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` approx day spirals into roughness.


Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` day hypnotizes you into looking.

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` day

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2013 `\pi` day is where it started

Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / 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.

3 There you go
1 Straight
4 Number me not
1 Scales
5 There is more of me
9 To forget than you can remember
—Emma Beauxis-Aussalet (314... piku)

Welcome to 2022 Pi Day: a celebration of `\pi` and mathematics (and music).

Buy the Album

The "three one four: a number of notes" album is available on Bandcamp, Spotify and Tidal.

Numberphile Podcast

Greg and I discuss the album on the Numberphile podcast.

Download the score

The album is fully scored for solo piano.

Artwork

The artwork for the album is the Wallis Sieve. It is a fractal whose area is `\pi/4`.

The sieve is generated by starting with a black square and dividing it into a grid of 3 × 3 squares and removing the middle square. On the second iteration, each of the squares is further divided into a grid of 5 × 5 squares, and the middle in each grid is removed. As this process continues (at iteration `n` we divide each square into `(2n+1)^2` smaller squares and remove the middle one), the black area (what remains after the middle squares are removed) approaches `\pi/4`.

The Wallis Sieve is said to "round the square" because by removing progressively smaller squares, we've rounded the big square into a quarter circle.

You can download a very high resolution level 5 Wallis Sieve. For normal viewing, the first three levels are visible and the fourth appears as dots. The fifth level is essentially invisible and, unless you're looking to zoom in interactively, there's little point in dividing futher.


Pi Day 2022 - three one four: a number of notes - A musical journey into the digits of Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The Wallis Sieve is a fractal whose area is `\pi/4`. On each iteration `n` a square is divided into `(2n+1)^2` squares and the middle one is removed.

Musical Influences

The album is inspired by 20th century classical music. Each track is a tribute to an influential composer from the era.

The methods section has detailed notes from Greg, the composer.

track 1 — 314... : the mid-century multiple serialists

Pierre Boulez (Piano Sonatas No. 1 and No. 2)

Karlheinz Stockhausen (Klavierstücke I-XVII)

track 2 — Feynman Point

Gyorgy Ligeti (Musica Ricercata 1–11)

track 3 — Wallis Product

Steve Reich (Piano phase score and visualization | Clapping Music Electric Counterpoint)

Phillip Glass (Mad Rush, String Quartet No. 3)

track 4 — nn

Erik Satie (Gnossienne No. 3, 3 Gymnopedies and 6 Gnossiennes)

track 5 — null

Morton Feldman (Piano Pieces | Triadic Memories for the Piano | Intermissions for Piano)

track 6 — ...264 : Bebop jazz

Wynton Kelly (On Green Dolphin St. | Autumn Leaves | If I Should Lose You)

Bud Powell (Wail)

Thelonious Monk (Underground)

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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