resources
visualizations
presentations
lunch break
Distractions and amusements, with a sandwich and coffee.
visualization + design
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.
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.
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.
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.
properties of the first 24 orders of the Hilbert curve
| order | points | segments | length |
| `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
Optimal experimental design
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.
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
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.
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
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.
McDonald, T. (2018) Why we can't give up this odd way of typing, BBC, 25 May 2018.
Molecular Case Studies Cover
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.
Happy 2018 `\tau` Day—Art for everyone
Universe Superclusters and Voids
A map of the nearby superclusters and voids in the Unvierse.
By "nearby" I mean within 6,000 million light-years.
