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visualization + math

# $\pi$ Day 2015 Art Posters

2021 $\pi$ reminds us that good things grow for those who wait.' edition.
2019 $\pi$ has hundreds of digits, hundreds of languages and a special kids' edition.
2018 $\pi$ day stitches street maps into new destinations.
2017 $\pi$ day imagines the sky in a new way.

2016 $\pi$ approximation day wonders what would happen if about right was right.
2016 $\pi$ day sees digits really fall for each other.
2015 $\pi$ day maps transcendentally.
2014 $\pi$ approx day spirals into roughness.

2014 $\pi$ day hypnotizes you into looking.
2014 $\pi$ day
2013 $\pi$ day is where it started
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.

Not a circle in sight in the 2015 $\pi$ day art. Try to figure out how up to 612,330 digits are encoded before reading about the method. $\pi$'s transcendental friends $\phi$ and $e$ are there too—golden and natural. Get it?

This year's $\pi$ day is particularly special. The digits of time specify a precise time if the date is encoded in North American day-month-year convention: 3-14-15 9:26:53.

The art has been featured in Ana Swanson's Wonkblog article at the Washington Post—10 Stunning Images Show The Beauty Hidden in $\pi$.

We begin with a square and progressively divide it. At each stage, the digit in $pi$ is used to determine how many lines are used in the division. The thickness of the lines used for the divisions can be attenuated for higher levels to give the treemap some texture.

Representing a number using a tree map. Each digit of the number is used to successively divide a shape, such as a square. (zoom)

This method of encoding data is known as treemapping. Typically, it is used to encode hierarchical information, such as hard disk spac usage, where the divisions correspond to the total size of files within directories.

At each level of the tree map, more digits are encoded. Shown here are tree maps for $pi$ for the first 6 levels of division. (zoom)

This kind of treemap can be made from any number. Below I show 6 level maps for $pi$, $phi$ (Golden ratio) and $e$ (base of natural logarithm).

At each level of the tree map, more digits are encoded. Shown here are tree maps for $pi$ for the first 6 levels of division. (zoom)

The number of digits per level, $n_i$ and total digits, $N_i$ in the tree map for $pi$, $phi$ and $e$ is shown below for levels $i = 1 .. 6$.

$PI PHI e i n_i N_i n_i N_i n_i N_i 1 1 1 1 1 1 1 2 4 5 2 3 3 4 3 15 20 9 12 19 23 4 98 118 59 71 96 119 5 548 666 330 401 574 693 6 2962 3628 1857 2258 3162 3855 7 16616 20244 10041 12299 17541 21396 8 91225 111469 9 500861 612330$

## Dividing the map

In all the treemaps above, the divisions were made uniformly for each rectangle. With uniform division, the lines that divide a shape are evenly spaced. With randomized division, the placement of lines is randomized, while still ensuring that lines do not coincide.

A multiplier, such as $phi$ (Golden Ratio), can be used to control the division. In this case, the first division is made at 1/$phi$ (0.62/0.38 split) and the remaining rectangle (0.38) is further divided at $/$phi$(0.24/0.14 split). The divisions of each shape can be influenced by another number and the level at which the division is performed. (zoom) Using a non-uniform multipler is one way to embed another number in the art. When a multiplier like$phi$is used, divisions at the top levels create very large rectangles. To attenuate this, the effect of the multiplier can be weighted by the level. Regardless of what multiplier is used, the first level is always uniformly divided. Division at subsequent levels incorporates more of the multiplier effect. The orientation of the division can be uniform (same for a layer and alternating across layers), alternating (alternating across and within a layer) or random. This modification has an effect only if the divisions are not uniform. The divisions of each shape can be influenced by another number and the level at which the division is performed. (zoom) ## Adjusting line thickness To emphasize the layers, a different line thickness can be used. When lines are drawn progressively thinner with each layer, detail is controlled and the map has more texture. When all lines have the same thickness, it is harder to distinguish levels. The divisions of each shape can be influenced by another number and the level at which the division is performed. (zoom) You could see this as a challenge! Below I show the treemaps for$pi$,$phi$and$e$with and without stroke modulation. The divisions of each shape can be influenced by another number and the level at which the division is performed. (zoom) When displayed at a low resolution (the image below is 620 pixels across), shapes at higher levels appear darker because the distance between the lines within is close to (or smaller) than a pixel. By matching the line thickness to the image resolution, you can control how dark the smallest divisions appear. The divisions of each shape can be influenced by another number and the level at which the division is performed. (zoom) ## Adding color Adding color can make things better, or worse. Dropping color randomly, without respect for the level structure of the treemap, creates a mess. We can rescue things by increasing the probability that a given rectangle will be made transparent—this will allow the color of the rectangle below to show through. Additionally, by drawing the layers in increasing order, smaller rectangles are drawn on top of bigger ones, giving a sense of recursive subdivision. The divisions of each shape can be influenced by another number and the level at which the division is performed. (zoom) Because the color is assigned randomly, various instances of the treemap can be made. The maps below have the same proportion of colors and transparency (same as the first image in second row in the figure above) and vary only by the random seed to pick colors. Different instances of 5 level$pi$treemaps. The proportion of transparent, white, yellow, red and blue shapes is 20:1:1:1:1. (zoom) ## Coloring using adjacency graph The color assignments above were random. For each shape the probability of choosing a given color (transparent, white, yellow, red, blue) was the same. Color choice for a shape can also be influenced by the color of neighbouring shapes. To do this, we need to create a graph that captures the adjacency relationship between all the shapes at each level. Below I show the first 4 levels of the$pi$treemap and their adjacency graphs. In each graph, the node corresponds to a shape and an edge between nodes indicates that the shapes share a part of their edge. Shapes that touch only at a corner are not considered adjacent. Different instances of 5 level$pi$treemaps. The proportion of transparent, white, yellow, red and blue shapes is 20:1:1:1:1. (zoom) One way in which the graphs can be used is to attempt to color each layer using at most 4 colors. The 4 color theorem tells us that only 4 unique colors are required to color maps such as these in a way that no two neighbouring shapes have the same color. It turns out that the full algorithm of coloring a map with only 4 colors is complicated, but reasonably simple options exist.. For the maps here, I used the DSATUR (maximum degree of saturation) approach. Different instances of 5 level$pi` treemaps. The proportion of transparent, white, yellow, red and blue shapes is 20:1:1:1:1. (zoom)

The DSATUR algorithm works well, but does not guarantee a 4-color solution. It performs no backtracking. If you look carefully, one of the rectangles in the 4th layer (top right quadrant in the graph) required a 5th color (shown black).

news + thoughts

# Neural network primer

Mon 06-02-2023

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.

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.

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.

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.

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.

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.

A catalogue of my journal and magazine cover designs. (more)

# Regression modeling of time-to-event data with censoring

Mon 21-11-2022

If you sit on the sofa for your entire life, you’re running a higher risk of getting heart disease and cancer. —Alex Honnold, American rock climber

In a follow-up to our Survival analysis — time-to-event data and censoring article, we look at how regression can be used to account for additional risk factors in survival analysis.

We explore accelerated failure time regression (AFTR) and the Cox Proportional Hazards model (Cox PH).

Nature Methods Points of Significance column: Regression modeling of time-to-event data with censoring. (read)

Dey, T., Lipsitz, S.R., Cooper, Z., Trinh, Q., Krzywinski, M & Altman, N. (2022) Points of significance: Regression modeling of time-to-event data with censoring. Nature Methods 19.

# Music video for Max Cooper's Ascent

Tue 25-10-2022

My 5-dimensional animation sets the visual stage for Max Cooper's Ascent from the album Unspoken Words. I have previously collaborated with Max on telling a story about infinity for his Yearning for the Infinite album.

I provide a walkthrough the video, describe the animation system I created to generate the frames, and show you all the keyframes

Frame 4897 from the music video of Max Cooper's Asent.

The video recently premiered on YouTube.

Renders of the full scene are available as NFTs.

# Gene Cultures exhibit — art at the MIT Museum

Tue 25-10-2022

I am more than my genome and my genome is more than me.

The MIT Museum reopened at its new location on 2nd October 2022. The new Gene Cultures exhibit featured my visualization of the human genome, which walks through the size and organization of the genome and some of the important structures.

My art at the MIT Museum Gene Cultures exhibit tells shows the scale and structure of the human genome. Pay no attention to the pink chicken.