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data visualization + art

Aleph 2 — official music video

MAX COOPER'S ALEPH 2 | From Max Cooper's album Yearning for the Infinite. Music by Max Cooper. Animation by Martin Krzywinski.

Aleph 2 — live at the Barbican in London

MAX COOPER'S ALEPH 2 | From Max Cooper's album Yearning for the Infinite. Music by Max Cooper. Animation by Martin Krzywinski.
My animation of 5 dimensions sets the stage for Max Cooper's track Ascent from his new album Unspoken Words.
Another mathy collaboration with Max!

Infinity with Max Cooper — In Six Minutes

Great fleas have little fleas upon their backs to bite ’em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on;
While these again have greater still, and greater still, and so on.
—Augustus de Morgan

1 · The story of infinity in 6 minutes

In collaboration with Max Cooper, we tell a story about infinity in 6 minutes (and 34 seconds).

What's the story about? Natural numbers, integers, rationals, reals, power sets and the cardinality of the continuum — a real visual parade of numbers.

But math is a serious matter — so we include two of Cantor's diagnoal proofs in the video. Watch as we show that the size of the set of rational numbers is the same as the set of natural numbers (making them countable) but that the size of the set of real numbers is larger (making them uncountable).

2 · Aleph 2 — Yearning for the Infinite

In science one tries to tell people,
in such a way as to be understood by everyone,
something that no one ever knew before.
But in poetry, it’s the exact opposite.
—Paul Dirac, Mathematical Circles Adieu by H. Eves [quoted]

The style of Aleph 2 is that of a low-fi terminal. We used the Classic Console font by Csaba Széll, which I extended to include more set theory symbols (download extended font).

The animation was closely synchronized to the music, all the while trying to avoid looking too much like something from the Matrix movie.

3 · Video walkthrough

To help you interpret what you are seeing, I walk you through the video and introduce you some advanced concepts of set theory. This way, next time you see an equation like `|\mathbb{R}| = | \mathbb{P}(\mathbb{N}) | = 2^{\aleph_0}`, you'll sprout a smile of joy.

Yearning for the Infinite - Max Cooper and Martin Krzywinski - Visualization of Infinity and Pi / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
We begin counting the natural numbers, whose cardinality is `|\mathbb{N}| = \aleph_0`. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Yearning for the Infinite - Max Cooper and Martin Krzywinski - Visualization of Infinity and Pi / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A bijection between the naturals and integers demonstrating that both have the same cardinality. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Yearning for the Infinite - Max Cooper and Martin Krzywinski - Visualization of Infinity and Pi / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A bijection between the naturals and rationals. We traversing the table of rationals in a zig-zag manner. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Yearning for the Infinite - Max Cooper and Martin Krzywinski - Visualization of Infinity and Pi / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
We alter the digits on the diagonal to create a real that is nowhere in the list. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Yearning for the Infinite - Max Cooper and Martin Krzywinski - Visualization of Infinity and Pi / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Power set elements of naturals (left) and reals (right). The size of these sets is separated by one step in the hierarchy of Aleph numbers, if the Continuum Hypothesis is true. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)
Yearning for the Infinite - Max Cooper and Martin Krzywinski - Visualization of Infinity and Pi / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Aleph remains. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)

These pages describe the system I build to generate the animation and the mathematics behind infinity, including sets, cardinality, countability, $\aleph$ and the Continuum Hypothesis.

I also present the animation system I built for the video, which was coded from scratch.

4 · My other collaborations with Max

If you like infinity, you might like dimensions too.

Check out my video for Max Cooper's Ascent, which animates 5-dimensional cubes doing their confusing things.

My animation of 5 dimensions sets the stage for Max Cooper's track Ascent from his new album Unspoken Words.
news + thoughts

Nasa to send our human genome discs to the Moon

Sat 23-03-2024

We'd like to say a ‘cosmic hello’: mathematics, culture, palaeontology, art and science, and ... human genomes.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
SANCTUARY PROJECT | A cosmic hello of art, science, and genomes. (details)
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
SANCTUARY PROJECT | Benoit Faiveley, founder of the Sanctuary project gives the Sanctuary disc a visual check at CEA LeQ Grenoble (image: Vincent Thomas). (details)
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
SANCTUARY PROJECT | Sanctuary team examines the Life disc at INRIA Paris Saclay (image: Benedict Redgrove) (details)

Comparing classifier performance with baselines

Sat 23-03-2024

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.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Comparing classifier performance with baselines. (read)

Unfortunately, baselines are often overlooked and, in the presence of a class imbalance5, 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 20.

Happy 2024 π Day—
sunflowers ho!

Sat 09-03-2024

Celebrate π Day (March 14th) and dig into the digit garden. Let's grow something.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2024 π DAY | A garden of 1,000 digits of π. (details)

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.

Martin Krzywinski | contact | Canada's Michael Smith Genome Sciences CentreBC Cancer Research CenterBC CancerPHSA
Google whack “vicissitudinal corporealization”
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