Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / - contact me Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / on Twitter Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / - Lumondo Photography Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / - Pi Art Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / - Hilbertonians - Creatures on the Hilbert CurveMartin Krzywinski / Canada's Michael Smith Genome Sciences Centre / - Pi Day 2020 - Piku
This love's a nameless dream.Cocteau Twinstry to figure it outmore quotes

pi: fun

Scientific graphical abstracts — design guidelines

visualization + design

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski buy artwork
81 digits of `\pi` as a forest of trees: standard, bat cave and underwater editions. ( BUY ARTWORK )
Martin Krzywinski @MKrzywinski
The 2021 Pi Day art celebrates the digits of `\pi` with a forest! Visit the bat cave and underwater ecosystems for the full experience.

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

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.

Martin Krzywinski @MKrzywinski
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 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 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 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.

Martin Krzywinski @MKrzywinski
A `\pi` day music video!: Transcendental Tree Map premieres on 2020 Pi Day from Max Cooper's Yearning for the Infinite. Animation by Nick Cobby and myself. Watch live from Barbican Centre.

The forest was generated using an Lindenmayer system — this part of the process is relatively easy.

Integrating variation into how the trees look while keeping a clean look-and-feel took a lot of experimentation — there are many free parameters in this section.

Lindenmayer system

A Lindenmayer system is a kind of parallel rewriting system. The system starts with some initial state, typically represented by a string. For example FX.

On each iteration, rules are applied to each characters in the state, which replaces the character with some other character. For example, suppose we replace X with [-FX][+FX]. Starting with the axiom FX The first three rounds of rewriting gives each give us


It doesn't look like much, I know, but the magic happens when you interpret these characters in the context of turtle graphics. Specifically, let

F move forward distance d
- turn left &theta degrees;
+ turn right &theta degrees;
[ spawn a new turtle at this position facing at current angle
] restore previous turtle

If we set `\theta = 15\deg` then the shapes for each iteration are

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski
The first three rounds of rewriting of a simple L-system with axiom FX and rewrite rule X = [-FX][+FX].

We can interpret F as a branch and X a point were new branches grow. You can already see that by level 3 two of the branches — there's nothing in the system that prevents this, since each turtle operates independently.

Rules can be as simple or as complicated as you want and new characters to the string can be added to correspond to different turtle actions.

The `\pi` forest Lindenmayer system

The rules for the `\pi` forest are quite simple. The system starts with the axiom fX (grow a trunk and then branch off) and each digit corresponds to a different branching rule — the digit gives the number of branches with zero terminating a branch.

X 0 []
X 1 [---FX][++h]
X 2 [----FX][+++FX]
X 3 [-----FX][-FX][++++FX]
X 4 [------FX][---FX][++FX][+++++FX]
X 5 [-------FX][----FX][-FX][+++FX][+++++++FX]
X 6 [--------FX][------FX][---FX][+FX][+++++FX][++++++++FX]
X 7 [--------FX][-----FX][--FX][FX][+++FX][++++++FX][+++++++++FX]
X 8 [---------FX][------FX][---FX][-FX][++FX][++++FX][+++++++FX][+++++++++FX]

where each – or + correspond to a left or right turn of 8 degrees. Any turns

The digit 9 has a special rule that sprouts a flower at the previous branching point and starts a new tree.

Also, at least step branches grow

f ff
g gg

with the f step being a fraction `(1-1/\pi)^2 \approx 46%` of the length of F and g being `(1-1/\pi)^4 \approx 22%` of the length of F. These shorter branch allow me to have a shorter trunk (axiom is fX and not FX) and a very short branch used in the rule for the digit 1, which branches off to one side while growing a little stubby branch to the other side.

There is one additional rule in which the turtle is mirrored at a branching point when a digit and the next digit have different parity (one odd and one even).

X = MX

The mirror operator makes the current turtle interpret any left turns as right turns, and vice versa.

The reason for the multiple turns in the rules is for convenience &mdsah; it allows me to define the angles of the branches as a multiple of a fixed value.

The system is grown for 6 iterations.

The first tree

Here is the first tree of the forest – for digits 314159. It ends in a 9 because the rule for 9 is to grow a flower, terminate the current tree and start a new tree with the next set of digits (up to the next 9, and so on).

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski
The first tree in the `\pi` forest. Its digits are 314159 -- they end at the first 9 in `\pi`.

You can trace the digits for the tree left-to-right along the leaves. The flower of the 9 will appear on the branch of the digit immediately before it — in this case, a 5. The color of the flower petal is green (5) and the center is dark orange (first digit of the next tree).

The L-system string for this tree is below. See if you can discover what symbol is used for the flower.


adding variation

To add variation to the tree, I take the tree's digits and create linear congruential generators for each of the following: branch turn angle, branch growth rate, thickness tapering and turtle angle when growing a branch.

Each generator is constructed as follows. The digits of the tree `d_1d_2d_3...d_n` are split into `a = d_1d_2d_3...d_{n-1}`, `c = d_n` and `m = d_1d_2d_3...d_n`. The generator is seeded with `X_0 = d_1` and then the next digit is `X_1 = (a X_0 + c) \; \textrm{mod} \; m`. Each output `X_i` is transformed to `y \times (2X_i/m - 1)`. Because some of the trees have a large number of digits, the operations are performed with an arbitrary precision library.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski
The output of the linear congruential generator (LCG) for the first tree. The first 1,000 samples are shown below a lag plot of two successive samples. As with all LCGs, you get a banding pattern.

The output will be distributed in the range `[-y,y)`. The distribution will depend on the values of `a`, `c` and `m`. It might be reasonably uniform for some combinations and absolutely non-uniform for others. But here, non-uniform doesn't mean horrible — just another rule in the digit forest.

Each type of variation starts with an identical generator.

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski
The effect of successively adding different types of variation. Shown are the first 5 trees in the forest with no leaves or flowers. A row contains all the variation in the previous row plus the type noted on the row.

The parameters that influence the amount of variation are irrelevant — I tweaked them until I had something that looked "pretty good".

winds of change are blowing

You might have noticed that because a branch growth step is doubled at each iteration, a branch is composed of 64 turtle steps. This was done so that there's lots of opportunities to sample variation along a branch. And also for boring technical reasons: SVG does not allow a path with varying thickness.

This means that variation can build up along a branch, especially if we allow the turtle to veer quite a bit as the branch grows. Remember, a branch begins to grows in a particular direction but can change course along its growth.

Below I show what happens when I crank up every variation type by 200% and by 300% (e.g. 300% moar branch growth rate jitter!). Things add up pretty quickly — I'm looking at you 79! What are you doing? Who knew the digit forest could be so complicated?

`\pi` Day 2021 Art Posters - A forest of Pi (a Lindenmayer system)
 / Martin Krzywinski @MKrzywinski
The effect of increasing variation of all types to 200% and 300%.

once more with ecosystem flavour

Exploring the space of variation is fun — even at 1am as I'm trying to finish this.

Below are some scenarios to get the imagination going.

The underwater scenario uses an effect in which turtle angle during branch growth is increased and constrained to be mostly up and no branch thickness change.

The prickly forest has a larger branch angle and larger branch length decay as a function of depth.

The dry forest has extreme thickness tapering and branch growth angle variation increases with each branch depth.

Finally, the bat cave is similar to the underwater case except that it has a much narrower branching angle and branch length now increases with branch depth. And one more change that's easy to spot.

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

So as you can see, the canonical digit forest is just one of a wide range of possible numerical environments.


news + thoughts

Happy 2021 `\pi` Day—
A forest of digits

Sun 14-03-2021

Celebrate `\pi` Day (March 14th) and finally see the digits through the forest.

Martin Krzywinski @MKrzywinski
The 26th tree in the digit forest of `\pi`. Why is there a flower on the ground?. (details)

This year is full of botanical whimsy. A Lindenmayer system forest – deterministic but always changing. Feel free to stop and pick the flowers from the ground.

Martin Krzywinski @MKrzywinski
The first 46 digits of `\pi` in 8 trees. There are so many more. (details)

And things can get crazy in the forest.

Martin Krzywinski @MKrzywinski
A forest of the digits of '\pi`, by ecosystem. (details)

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 and 2019 `\pi` Day.

Testing for rare conditions

Tue 16-03-2021

All that glitters is not gold. —W. Shakespeare

The sensitivity and specificity of a test do not necessarily correspond to its error rate. This becomes critically important when testing for a rare condition — a test with 99% sensitivity and specificity has an even chance of being wrong when the condition prevalence is 1%.

We discuss the positive predictive value (PPV) and how practices such as screen can increase it.

Martin Krzywinski @MKrzywinski
Nature Methods Points of Significance column: Testing for rare conditions. (read)

Altman, N. & Krzywinski, M. (2021) Points of significance: Testing for rare conditions. Nature Methods 18

Standardization fallacy

Tue 09-02-2021

We demand rigidly defined areas of doubt and uncertainty! —D. Adams

A popular notion about experiments is that it's good to keep variability in subjects low to limit the influence of confounding factors. This is called standardization.

Unfortunately, although standardization increases power, it can induce unrealistically low variability and lead to results that do not generalize to the population of interest. And, in fact, may be irreproducible.

Martin Krzywinski @MKrzywinski
Nature Methods Points of Significance column: Standardization fallacy. (read)

Not paying attention to these details and thinking (or hoping) that standardization is always good is the "standardization fallacy". In this column, we look at how standardization can be balanced with heterogenization to avoid this thorny issue.

Voelkl, B., Würbel, H., Krzywinski, M. & Altman, N. (2021) Points of significance: Standardization fallacy. Nature Methods 18:5–6.

Graphical Abstract Design Guidelines

Fri 13-11-2020

Clear, concise, legible and compelling.

Making a scientific graphical abstract? Refer to my practical design guidelines and redesign examples to improve organization, design and clarity of your graphical abstracts.

Martin Krzywinski @MKrzywinski
Graphical Abstract Design Guidelines — Clear, concise, legible and compelling.

"This data might give you a migrane"

Tue 06-10-2020

An in-depth look at my process of reacting to a bad figure — how I design a poster and tell data stories.

Martin Krzywinski @MKrzywinski
A poster of high BMI and obesity prevalence for 185 countries.

He said, he said — a word analysis of the 2020 Presidential Debates

Thu 01-10-2020

Building on the method I used to analyze the 2008, 2012 and 2016 U.S. Presidential and Vice Presidential debates, I explore word usagein the 2020 Debates between Donald Trump and Joe Biden.

Martin Krzywinski @MKrzywinski
Analysis of word usage by parts of speech for Trump and Biden reveals insight into each candidate.