Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - contact me Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca on Twitter Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - Lumondo Photography Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - Pi Art Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - Hilbertonians - Creatures on the Hilbert CurveMartin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca - Pi Day 2020 - Piku
Here we are now at the middle of the fourth large part of this talk.Pepe Deluxeget nowheremore quotes

Martin Krzywinski / Canada's Michael Smith Genome Sciences Centre / mkweb.bcgsc.ca
The Outbreak Poems — artistic emissions in a pandemic


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 mkweb.bcgsc.ca

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 mkweb.bcgsc.ca
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 mkweb.bcgsc.ca
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

orderpointssegmentslength
`n``4^n``4^{n-1}``2^n-2^{-n}`
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.

VIEW ALL

news + thoughts

Virus Mutations Reveal How COVID-19 Really Spread

Mon 04-05-2020

Genetic sequences of the coronavirus tell story of when the virus arrived in each country and where it came from.

Our graphic in Scientific American's Graphic Science section in the June 2020 issue shows a phylogenetic tree based on a snapshot of the data model from Nextstrain as of 31 March 2020.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Virus Mutations Reveal How COVID-19 Really Spread. Text by Mark Fischetti (Senior Editor), art direction by Jen Christiansen (Senior Graphics Editor), source: Nextstrain (enabled by data from GISAID).

Cover of Nature Cancer April 2020

Mon 27-04-2020

Our design on the cover of Nature Cancer's April 2020 issue shows mutation spectra of patients from the POG570 cohort of 570 individuals with advanced metastatic cancer.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Each ellipse system represents the mutation spectrum of an individual patient. Individual ellipses in the system correspond to the number of base changes in a given class and are layered by mutation count. Ellipse angle is controlled by the proportion of mutations in a class within the sample and its size is determined by a sigmoid mapping of mutation count scaled within the layer. The opacity of each system represents the duration since the diagnosis of advanced disease. (read more)

The cover design accompanies our report in the issue Pleasance, E., Titmuss, E., Williamson, L. et al. (2020) Pan-cancer analysis of advanced patient tumors reveals interactions between therapy and genomic landscapes. Nat Cancer 1:452–468.

Modeling infectious epidemics

Wed 06-05-2020

Every day sadder and sadder news of its increase. In the City died this week 7496; and of them, 6102 of the plague. But it is feared that the true number of the dead this week is near 10,000 ....
—Samuel Pepys, 1665

This month, we begin a series of columns on epidemiological models. We start with the basic SIR model, which models the spread of an infection between three groups in a population: susceptible, infected and recovered.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Modeling infectious epidemics. (read)

We discuss conditions under which an outbreak occurs, estimates of spread characteristics and the effects that mitigation can play on disease trajectories. We show the trends that arise when "flattenting the curve" by decreasing `R_0`.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Modeling infectious epidemics. (read)

This column has an interactive supplemental component that allows you to explore how the model curves change with parameters such as infectious period, basic reproduction number and vaccination level.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Modeling infectious epidemics. (Interactive supplemental materials)

Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: Modeling infectious epidemics. Nature Methods 17:455–456.

The Outbreak Poems

Sat 04-04-2020

I'm writing poetry daily to put my feelings into words more often during the COVID-19 outbreak.

Small hours
of
the world and me.

Read the poems and learn what a piku is.

Deadly Genomes: Genome Structure and Size of Harmful Bacteria and Viruses

Tue 17-03-2020

A poster full of epidemiological worry and statistics. Now updated with the genome of SARS-CoV-2 and COVID-19 case statistics as of 3 March 2020.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Deadly Genomes: Genome Structure and Size of Harmful Bacteria and Viruses (zoom)

Bacterial and viral genomes of various diseases are drawn as paths with color encoding local GC content and curvature encoding local repeat content. Position of the genome encodes prevalence and mortality rate.

The deadly genomes collection has been updated with a posters of the genomes of SARS-CoV-2, the novel coronavirus that causes COVID-19.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Genomes of 56 SARS-CoV-2 coronaviruses that causes COVID-19.
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Ball of 56 SARS-CoV-2 coronaviruses that causes COVID-19.
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The first SARS-CoV-2 genome (MT019529) to be sequenced appears first on the poster.

Using Circos in Galaxy Australia Workshop

Wed 04-03-2020

A workshop in using the Circos Galaxy wrapper by Hiltemann and Rasche. Event organized by Australian Biocommons.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Using Circos in Galaxy Australia workshop. (zoom)

Download workshop slides.

Galaxy wrapper training materials, Saskia Hiltemann, Helena Rasche, 2020 Visualisation with Circos (Galaxy Training Materials).