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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.

news + thoughts

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).

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
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

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
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.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
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.

Annals of Oncology cover

Wed 14-09-2022

My cover design on the 1 September 2022 Annals of Oncology issue shows 570 individual cases of difficult-to-treat cancers. Each case shows the number and type of actionable genomic alterations that were detected and the length of therapies that resulted from the analysis.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
An organic arrangement of 570 individual cases of difficult-to-treat cancers showing genomic changes and therapies. Apperas on Annals of Oncology cover (volume 33, issue 9, 1 September 2022).

Pleasance E et al. Whole-genome and transcriptome analysis enhances precision cancer treatment options (2022) Annals of Oncology 33:939–949.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
My Annals of Oncology 570 cancer cohort cover (volume 33, issue 9, 1 September 2022). (more)

Browse my gallery of cover designs.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A catalogue of my journal and magazine cover designs. (more)

Survival analysis—time-to-event data and censoring

Fri 05-08-2022

Love's the only engine of survival. —L. Cohen

We begin a series on survival analysis in the context of its two key complications: skew (which calls for the use of probability distributions, such as the Weibull, that can accomodate skew) and censoring (required because we almost always fail to observe the event in question for all subjects).

We discuss right, left and interval censoring and how mishandling censoring can lead to bias and loss of sensitivity in tests that probe for differences in survival times.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Survival analysis—time-to-event data and censoring. (read)

Dey, T., Lipsitz, S.R., Cooper, Z., Trinh, Q., Krzywinski, M & Altman, N. (2022) Points of significance: Survival analysis—time-to-event data and censoring. Nature Methods 19:906–908.

3,117,275,501 Bases, 0 Gaps

Sun 21-08-2022

See How Scientists Put Together the Complete Human Genome.

My graphic in Scientific American's Graphic Science section in the August 2022 issue shows the full history of the human genome assembly — from its humble shotgun beginnings to the gapless telomere-to-telomere assembly.

Read about the process and methods behind the creation of the graphic.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
3,117,275,501 Bases, 0 Gaps. Text by Clara Moskowitz (Senior Editor), art direction by Jen Christiansen (Senior Graphics Editor), source: UCSC Genome Browser.

See all my Scientific American Graphic Science visualizations.


© 1999–2022 Martin Krzywinski | contact | Canada's Michael Smith Genome Sciences CentreBC Cancer Research CenterBC CancerPHSA