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# numbers: curious

PNAS Cover: Earth BioGenome Project

# visualization + design

The 2022 Pi Day art is a music album “three one four: a number of notes” . It tells stories from the very beginning (314…) to the very (known) end of π (...264) as well as math (Wallis Product) and math jokes (Feynman Point), repetition (nn) and zeroes (null).

# $\pi$ Approximation Day 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

The never-repeating digits of $\pi$ can be approximated by $22/7 = 3.142857$ to within 0.04%. These pages artistically and mathematically explore rational approximations to $\pi$. This 22/7 ratio is celebrated each year on July 22nd. If you like hand waving or back-of-envelope mathematics, this day is for you: $\pi$ approximation day!

Want more math + art? Discover the Accidental Similarity Number. Find humor in my poster of the first 2,000 4s of $\pi$.

There are two kinds of $\pi$ Approximation Day posters.

The first uses the Archimedean spiral for its design, which I've used before for other numerical art. The second packs warped circles, whose ratio of circumference to average diameter is $22/7$ into what I call $\pi$-approximate circular packing.

As you probably know, the ratio of the circumference of a circle to its diameter is $\pi$. $$C / d = \pi$$

For $\pi$ approximation day, let's ask what would happen if $$C / d = 22/7$$

where now $C$ is the circumference of some shape other than a circle. What could this shape be?

A good place to start is to think about an ellipse. I've done this before in the 22/7 Universe article, in which I considered an ellipse with a major axis of $r+\delta$ and a minor axis of $r$ and solved for $\delta$ such that the circumference of the ellipse divided by $2 r$ would be $22/7$. Doing so means numerically solving the equation $$\frac{C(r,r+\delta)}{2r} = 22/7$$

where $r + \delta$ is the major axis, $r$ is the minor axis and $C(r,r+\delta)$ is the circumference of the ellipse. Substituting the expression for the circumference, $$4(r+\delta) \int_0^{\pi/2} \sqrt { 1 - \left(1-\frac{r}{(r+\delta)^2}\right)\sin^2 \theta } d \theta = 2 r \frac{22}{7}$$

If we set $r=1$ and solve it turns out that only a very minor deformation is required and $\delta = 0.0008$. You can verify this at Wolfram Alpha.

I wanted to make some art based on the shape of the this ellipse, but a deformation of 0.08% is not perceptible. So I came up with a slightly different approach to how I define the original circumference-to-diameter ratio.

Instead of treating the diameter as $r$ and using $r + \delta$ as the major axis, I now define the diameter as twice the average radius, or $2r + \delta$. This means that the equation to solve is $$\frac{C(r,r+\delta)}{2r+\delta} = 22/7$$

As before, setting $r=1$ and substituting the expression for the circumference of an ellipse, we get $$4(1+\delta) \int_0^{\pi/2} \sqrt { 1 - \left(1-\frac{1}{(1+\delta)^2}\right)\sin^2 \theta } d \theta = (2+\delta) \frac{22}{7}$$

and solving this for $\delta$ find $$\delta = 0.083599769...$$

You can verify this at Wolfram Alpha.

This is a more useable approach since an 8% warping of a circle can be easily perceived.

The ratio of the circumference of a circle, $C(r)$, to its dimameter, $2r$, is $\pi$. If we warp the circle by 8%, the corresponding ratio, if we use twice the average radius as the diameter, is 22/7. This deformation can be easily identified.

Below is matrix of perfect circles along side the 8% deformed circles.

A matrix of perfect circles and ones which have been stretched by 8% along one axis and then randomly rotated. The deformed circles embody the $\pi$ approximation of 22/7.

The art posters are based on a packing of these deformed circles.

Warped circles, packed.
Even more warped circles, packed.

By superimposing perfect circles on the warped circles, fun patterns appear.

Superposition of perfect and warped circles, packed.

## perfect vs approximate packing

If you pack perfect circles perfectly, the area occupied by the circles is $\pi/4 = 78.5%$.

What is the area occupied by perfect packing of warped and randomly rotated (like in the posters) circles?

## color scheme

To motivate choice of colors, I chose images with a 1970's feel.

Images used for color schemes. The colors of each image were grouped into clusters—8 for the first two images and 6 for the third—to obtain proportions of representative colors.

Using my color summarizer, I analyzed each image for its representative colors. Using these colors and their proportions, I colored the perfect and warped circles.

Packed warped circles colored in proportion to color schemes derived from the images above.

For each poster of these color schemes, two poster versions are available. In one, the perfect cirlces are shown with warped circles as a clip mask. In the other, warped circles are shown, clipped by perfect circles.

# Cancer Cell cover

Sat 23-04-2022

My cover design on the 11 April 2022 Cancer Cell issue depicts depicts cellular heterogeneity as a kaleidoscope generated from immunofluorescence staining of the glial and neuronal markers MBP and NeuN (respectively) in a GBM patient-derived explant.

LeBlanc VG et al. Single-cell landscapes of primary glioblastomas and matched explants and cell lines show variable retention of inter- and intratumor heterogeneity (2022) Cancer Cell 40:379–392.E9.

My Cancer Cell kaleidoscope cover (volume 40, issue 4, 11 April 2022). (more)

Browse my gallery of cover designs.

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

# Nature Biotechnology cover

Sat 23-04-2022

My cover design on the 4 April 2022 Nature Biotechnology issue is an impression of a phylogenetic tree of over 200 million sequences.

Konno N et al. Deep distributed computing to reconstruct extremely large lineage trees (2022) Nature Biotechnology 40:566–575.

My Nature Biotechnology phylogenetic tree cover (volume 40, issue 4, 4 April 2022). (more)

Browse my gallery of cover designs.

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

# Nature cover — Gene Genie

Sat 23-04-2022

My cover design on the 17 March 2022 Nature issue depicts the evolutionary properties of sequences at the extremes of the evolvability spectrum.

Vaishnav ED et al. The evolution, evolvability and engineering of gene regulatory DNA (2022) Nature 603:455–463.

My Nature squiggles cover (volume 603, issue 7901, 17 March 2022). (more)

Browse my gallery of cover designs.

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

# Happy 2022 $\pi$ Day—three one four: a number of notes

Mon 14-03-2022

Celebrate $\pi$ Day (March 14th) and finally hear what you've been missing.

“three one four: a number of notes” is a musical exploration of how we think about mathematics and how we feel about mathematics. It tells stories from the very beginning (314…) to the very (known) end of π (...264) as well as math (Wallis Product) and math jokes (Feynman Point), repetition (nn) and zeroes (null).

Listen to $\pi$ in the style of 20th century classical music. (details)

The album is scored for solo piano in the style of 20th century classical music – each piece has a distinct personality, drawn from styles of Boulez, Feldman, Glass, Ligeti, Monk, and Satie.

Each piece is accompanied by a piku (or πku), a poem whose syllable count is determined by a specific sequence of digits from π.

Check out art from previous years: 2013 $\pi$ Day and 2014 $\pi$ Day, 2015 $\pi$ Day, 2016 $\pi$ Day, 2017 $\pi$ Day, 2018 $\pi$ Day, 2019 $\pi$ Day, 2020 $\pi$ Day and 2021 $\pi$ Day.

# PNAS Cover — Earth BioGenome Project

Fri 28-01-2022

My design appears on the 25 January 2022 PNAS issue.

My PNAS cover design captures the vision of the Earth BioGenome Project — to sequence everything. (more)

The cover shows a view of Earth that captures the vision of the Earth BioGenome Project — understanding and conserving genetic diversity on a global scale. Continents from the Authagraph projection, which preserves areas and shapes, are represented as a double helix of 32,111 bases. Short sequences of 806 unique species, sequenced as part of EBP-affiliated projects, are mapped onto the double helix of the continent (or ocean) where the species is commonly found. The length of the sequence is the same for each species on a continent (or ocean) and the sequences are separated by short gaps. Individual bases of the sequence are colored by dots. Species appear along the path in alphabetical order (by Latin name) and the first base of the first species is identified by a small black triangle.

Lewin HA et al. The Earth BioGenome Project 2020: Starting the clock. (2022) PNAS 119(4) e2115635118.

# The COVID charts — hospitalization rates

Tue 25-01-2022

As part of the COVID Charts series, I fix a muddled and storyless graphic tweeted by Adrian Dix, Canada's Health Minister.

I show you how to fix color schemes to make them colorblind-accessible and effective in revealing patters, how to reduce redundancy in labels (a key but overlooked part of many visualizations) and how to extract a story out of a table to frame the narrative.

Clear titles introduce the graphic, which starts with informative and non-obvious observations of the relationship between age, number of comorbidities, vaccination status and hospitalization rates. Supporting the story is a tidy table that gives you detailed statistics for each demographic. (more)