Martin Krzywinski / Genome Sciences Center / Martin Krzywinski / Genome Sciences Center / - contact me Martin Krzywinski / Genome Sciences Center / on Twitter Martin Krzywinski / Genome Sciences Center / - Lumondo Photography Martin Krzywinski / Genome Sciences Center / - Pi Art Martin Krzywinski / Genome Sciences Center / - Hilbertonians - Creatures on the Hilbert Curve
Lips that taste of tears, they say, are the best for kissing.Dorothy Parkerget crankymore quotes

filling space: beautiful

EMBO Practical Course: Bioinformatics and Genome Analysis, 5–17 June 2017.

visualization + design

Like paths? Got your lines twisted in a bunch?
Take a look at my 2014 Pi Day art that folds Pi.

Hilbert Curve Art, Hilbertonians and Monkeys

I collaborated with Scientific American to create a data graphic for the September 2014 issue. The graphic compared the genomes of the Denisovan, bonobo, chimp and gorilla, showing how our own genomes are almost identical to the Denisovan and closer to that of the bonobo and chimp than the gorilla.

Martin Krzywinski @MKrzywinski

Here you'll find Hilbert curve art, a introduction to Hilbertonians, the creatures that live on the curve, an explanation of the Scientific American graphic and downloadable SVG/EPS Hilbert curve files.

Hilbert curve

There are wheels within wheels in this village and fires within fires!
— Arthur Miller (The Crucible)

The Hilbert curve is one of many space-filling curves. It is a mapping between one dimension (e.g. a line) and multiple dimensions (e.g. a square, a cube, etc). It's useful because it preserves locality—points that are nearby on the line are usually mapped onto nearby points on the curve.

The Hilbert curve is a line that gives itself a hug.

It's a pretty strange mapping, to be sure. Although a point on a line maps uniquely onto the curve this is not the case in reverse. At infinite order the curve intersects itself infinitely many times! This shouldn't be a surprise if you consider that the unit square has the same number of points as the unit line. Now that's the real surprise! So surprising in fact that it apparently destabilized Cantor's mind, who made the initial discovery.

Bryan Hayes has a great introduction (Crinkly Curves) to the Hilbert curve at American Scientist.

If manipulated so that its ends are adjacent, the Hilbert curve becomes the Moore curve.

constructing the hilbert curve

The order 1 curve is generated by dividing a square into quadrants and connecting the centers of the quadrants with three lines. Which three connections are made is arbitrary—different choices result in rotations of the curve.

Hilbert curve. / Martin Krzywinski @MKrzywinski
First 8 orders of the space-filling Hilbert curve. Each square is 144 x 144 pixels. (zoom)

The order 6 curve is the highest order whose structure can be discerned at this figure resolution. Though just barely. The length of this curve is about 64 times the width of the square, so about 9,216 pixels! That's tight packing.

By order 7 the structure in the 620 pixel wide image (each square is 144 px wide) cannot be discerned. By order 8 the curve has 65,536 points, which exceeds the number of pixels its square in the figure. A square of 256 x 256 would be required to show all the points without downsampling.

Two order 10 curves have 1,048,576 points each and would approximately map onto all the pixels on an average monitor (1920 x 1200 pixels).

A curve of order 33 has `7.38 * 10^19` points and if drawn as a square of average body height would have points that are an atom's distance from one another (`10^{-10}` m).

mapping the line onto the square

By mapping the familiar rainbow onto the curve you can see how higher order curves "crinkle" (to borrow Bryan's term) around the square.

Hilbert curve. / Martin Krzywinski @MKrzywinski
First 8 orders of the space-filling Hilbert curve. Each square is 144 x 144 pixels. (zoom)

properties of the first 24 orders of the Hilbert curve

1 4 3 1.5
2 16 15 3.75
3 64 63 7.875
4 256 255 15.9375
5 1,024 1,023 31.96875
6 4,096 4,095 63.984375
7 16,384 16,383 127.9921875
8 65,536 65,535 255.99609375
9 262,144 262,143 511.998046875
10 1,048,576 1,048,575 1023.9990234375
11 4,194,304 4,194,303 2047.99951171875
12 16,777,216 16,777,215 4095.99975585938
13 67,108,864 67,108,863 8191.99987792969
14 268,435,456 268,435,455 16383.9999389648
15 1,073,741,824 1,073,741,823 32767.9999694824
16 4,294,967,296 4,294,967,295 65535.9999847412
17 17,179,869,184 17,179,869,183 131071.999992371
18 68,719,476,736 68,719,476,735 262143.999996185
19 274,877,906,944 274,877,906,943 524287.999998093
20 1,099,511,627,776 1,099,511,627,775 1048575.99999905
21 4,398,046,511,104 4,398,046,511,103 2097151.99999952
22 17,592,186,044,416 17,592,186,044,415 4194303.99999976
23 70,368,744,177,664 70,368,744,177,663 8388607.99999988
24 281,474,976,710,656 281,474,976,710,655 16777215.9999999

You can download the basic curve shapes for orders 1 to 10 and experiment yourself. Both square and circular forms are available.


news + thoughts

Happy 2017 `\pi` Day—Star Charts, Creatures Once Living and a Poem

Tue 14-03-2017

on a brim of echo,

capsized chamber
drawn into our constellation, and cooling.
—Paolo Marcazzan

Celebrate `\pi` Day (March 14th) with star chart of the digits. The charts draw 40,000 stars generated from the first 12 million digits.

Martin Krzywinski @MKrzywinski
12,000,000 digits of `\pi` interpreted as a star catalogue. (details)

The 80 constellations are extinct animals and plants. Here you'll find old friends and new stories. Read about how Desmodus is always trying to escape or how Megalodon terrorizes the poor Tecopa! Most constellations have a story.

Martin Krzywinski @MKrzywinski
Find friends and stories among the 80 constellations of extinct animals and plants. Oh look, a Dodo guardings his eggs! (details)

This year I collaborate with Paolo Marcazzan, a Canadian poet, who contributes a poem, Of Black Body, about space and things we might find and lose there.

Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day and and 2016 `\pi` Day.

Data in New Dimensions: convergence of art, genomics and bioinformatics

Tue 07-03-2017

Art is science in love.
— E.F. Weisslitz

A behind-the-scenes look at the making of our stereoscopic images which were at display at the AGBT 2017 Conference in February. The art is a creative collaboration with Becton Dickinson and The Linus Group.

Its creation began with the concept of differences and my writeup of the creative and design process focuses on storytelling and how concept of differences is incorporated into the art.

Oh, and this might be a good time to pick up some red-blue 3D glasses.

BD Genomics 3D art exhibit - AGBT 2017 / Martin Krzywinski @MKrzywinski
A stereoscopic image and its interpretive panel of single-cell transcriptomes of blood cells: diseased versus healthy control.

Interpreting P values

Thu 02-03-2017
A P value measures a sample’s compatibility with a hypothesis, not the truth of the hypothesis.

This month we continue our discussion about `P` values and focus on the fact that `P` value is a probability statement about the observed sample in the context of a hypothesis, not about the hypothesis being tested.

Martin Krzywinski @MKrzywinski
Nature Methods Points of Significance column: Interpreting P values. (read)

Given that we are always interested in making inferences about hypotheses, we discuss how `P` values can be used to do this by way of the Benjamin-Berger bound, `\bar{B}` on the Bayes factor, `B`.

Heuristics such as these are valuable in helping to interpret `P` values, though we stress that `P` values vary from sample to sample and hence many sources of evidence need to be examined before drawing scientific conclusions.

Altman, N. & Krzywinski, M. (2017) Points of Significance: Interpreting P values. Nature Methods 14:213–214.

Background reading

Krzywinski, M. & Altman, N. (2017) Points of significance: P values and the search for significance. Nature Methods 14:3–4.

Krzywinski, M. & Altman, N. (2013) Points of significance: Significance, P values and t–tests. Nature Methods 10:1041–1042.

...more about the Points of Significance column

Snellen Charts—Typography to Really Look at

Sat 18-02-2017

Another collection of typographical posters. These ones really ask you to look.

Martin Krzywinski @MKrzywinski
Snellen charts designed using physical constants, Braille and elemental abundances in the universe and human body.

The charts show a variety of interesting symbols and operators found in science and math. The design is in the style of a Snellen chart and typset with the Rockwell font.

Essentials of Data Visualization—8-part video series

Fri 17-02-2017
Martin Krzywinski @MKrzywinski

In collaboration with the Phil Poronnik and Kim Bell-Anderson at the University of Sydney, I'm delighted to share with you our 8-part video series project about thinking about drawing data and communicating science.

Martin Krzywinski @MKrzywinski
Essentials of Data Visualization: Thinking about drawing data and communicating science.

We've created 8 videos, each focusing on a different essential idea in data visualization: encoding, shapes, color, uncertainty, design, drawing missing or unobserved data, labels and process.

The videos were designed as teaching materials. Each video comes with a slide deck and exercises.

P values and the search for significance

Mon 16-01-2017
Little P value
What are you trying to say
Of significance?
—Steve Ziliak

We've written about P values before and warned readers about common misconceptions about them, which are so rife that the American Statistical Association itself has a long statement about them.

This month is our first of a two-part article about P values. Here we look at 'P value hacking' and 'data dredging', which are questionable practices that invalidate the correct interpretation of P values.

Martin Krzywinski @MKrzywinski
Nature Methods Points of Significance column: P values and the search for significance. (read)

We also illustrate how P values can lead us astray by asking "What is the smallest P value we can expect if the null hypothesis is true but we have done many tests, either explicitly or implicitly?"

Incidentally, this is our first column in which the standfirst is a haiku.

Altman, N. & Krzywinski, M. (2017) Points of Significance: P values and the search for significance. Nature Methods 14:3–4.

Background reading

Krzywinski, M. & Altman, N. (2013) Points of significance: Significance, P values and t–tests. Nature Methods 10:1041–1042.

...more about the Points of Significance column