Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - contact me Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca on Twitter Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Lumondo Photography Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Pi Art Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Hilbertonians - Creatures on the Hilbert Curve
Here we are now at the middle of the fourth large part of this talk.Pepe Deluxeget nowheremore quotes

In Silico Flurries: Computing a world of snow. Scientific American. 23 December 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 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

Optimal experimental design

Tue 31-07-2018
Customize the experiment for the setting instead of adjusting the setting to fit a classical design.

The presence of constraints in experiments, such as sample size restrictions, awkward blocking or disallowed treatment combinations may make using classical designs very difficult or impossible.

Optimal design is a powerful, general purpose alternative for high quality, statistically grounded designs under nonstandard conditions.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Optimal experimental design. (read)

We discuss two types of optimal designs (D-optimal and I-optimal) and show how it can be applied to a scenario with sample size and blocking constraints.

Smucker, B., Krzywinski, M. & Altman, N. (2018) Points of significance: Optimal experimental design Nature Methods 15:599–600.

Background reading

Krzywinski, M., Altman, N. (2014) Points of significance: Two factor designs. Nature Methods 11:1187–1188.

Krzywinski, M. & Altman, N. (2014) Points of significance: Analysis of variance (ANOVA) and blocking. Nature Methods 11:699–700.

Krzywinski, M. & Altman, N. (2014) Points of significance: Designing comparative experiments. Nature Methods 11:597–598.

The Whole Earth Cataloguer

Mon 30-07-2018
All the living things.

An illustration of the Tree of Life, showing some of the key branches.

The tree is drawn as a DNA double helix, with bases colored to encode ribosomal RNA genes from various organisms on the tree.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The circle of life. (read, zoom)

All living things on earth descended from a single organism called LUCA (last universal common ancestor) and inherited LUCA’s genetic code for basic biological functions, such as translating DNA and creating proteins. Constant genetic mutations shuffled and altered this inheritance and added new genetic material—a process that created the diversity of life we see today. The “tree of life” organizes all organisms based on the extent of shuffling and alteration between them. The full tree has millions of branches and every living organism has its own place at one of the leaves in the tree. The simplified tree shown here depicts all three kingdoms of life: bacteria, archaebacteria and eukaryota. For some organisms a grey bar shows when they first appeared in the tree in millions of years (Ma). The double helix winding around the tree encodes highly conserved ribosomal RNA genes from various organisms.

Johnson, H.L. (2018) The Whole Earth Cataloguer, Sactown, Jun/Jul, p. 89

Why we can't give up this odd way of typing

Mon 30-07-2018
All fingers report to home row.

An article about keyboard layouts and the history and persistence of QWERTY.

My Carpalx keyboard optimization software is mentioned along with my World's Most Difficult Layout: TNWMLC. True typing hell.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
TNWMLC requires seriously flexible digits. It’s 87% more difficult than using a standard Qwerty keyboard, according to Martin Krzywinski, who created it (Credit: Ben Nelms). (read)

McDonald, T. (2018) Why we can't give up this odd way of typing, BBC, 25 May 2018.

Molecular Case Studies Cover

Fri 06-07-2018

The theme of the April issue of Molecular Case Studies is precision oncogenomics. We have three papers in the issue based on work done in our Personalized Oncogenomics Program (POG).

The covers of Molecular Case Studies typically show microscopy images, with some shown in a more abstract fashion. There's also the occasional Circos plot.

I've previously taken a more fine-art approach to cover design, such for those of Nature, Genome Research and Trends in Genetics. I've used microscopy images to create a cover for PNAS—the one that made biology look like astrophysics—and thought that this is kind of material I'd start with for the MCS cover.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Cover design for Apr 2018 issue of Molecular Case Studies. (details)

Happy 2018 `\tau` Day—Art for everyone

Wed 27-06-2018
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
You know what day it is. (details)

Universe Superclusters and Voids

Mon 25-06-2018

A map of the nearby superclusters and voids in the Unvierse.

By "nearby" I mean within 6,000 million light-years.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The Universe — Superclustesr and Voids. The two supergalactic hemispheres showing Abell clusters, superclusters and voids within a distance of 6,000 million light-years from the Milky Way. (details)