Distractions and amusements, with a sandwich and coffee.
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.
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.
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).
By mapping the familiar rainbow onto the curve you can see how higher order curves "crinkle" (to borrow Bryan's term) around the square.
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.
My cover design on the 11 April 2022 Cancer Cell issue depicts depicts cellular heterogeneity as a kaleidoscope generated from immunofluorescence staining of the glial and neuronal markers MBP and NeuN (respectively) in a GBM patient-derived explant.
LeBlanc VG et al. Single-cell landscapes of primary glioblastomas and matched explants and cell lines show variable retention of inter- and intratumor heterogeneity (2022) Cancer Cell 40:379–392.E9.
Browse my gallery of cover designs.
My cover design on the 4 April 2022 Nature Biotechnology issue is an impression of a phylogenetic tree of over 200 million sequences.
Konno N et al. Deep distributed computing to reconstruct extremely large lineage trees (2022) Nature Biotechnology 40:566–575.
Browse my gallery of cover designs.
My cover design on the 17 March 2022 Nature issue depicts the evolutionary properties of sequences at the extremes of the evolvability spectrum.
Vaishnav ED et al. The evolution, evolvability and engineering of gene regulatory DNA (2022) Nature 603:455–463.
Browse my gallery of cover designs.
Celebrate `\pi` Day (March 14th) and finally hear what you've been missing.
“three one four: a number of notes” is a musical exploration of how we think about mathematics and how we feel about mathematics. It tells stories from the very beginning (314…) to the very (known) end of π (...264) as well as math (Wallis Product) and math jokes (Feynman Point), repetition (nn) and zeroes (null).
The album is scored for solo piano in the style of 20th century classical music – each piece has a distinct personality, drawn from styles of Boulez, Feldman, Glass, Ligeti, Monk, and Satie.
Each piece is accompanied by a piku (or πku), a poem whose syllable count is determined by a specific sequence of digits from π.
Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, 2016 `\pi` Day, 2017 `\pi` Day, 2018 `\pi` Day, 2019 `\pi` Day, 2020 `\pi` Day and 2021 `\pi` Day.
My design appears on the 25 January 2022 PNAS issue.
The cover shows a view of Earth that captures the vision of the Earth BioGenome Project — understanding and conserving genetic diversity on a global scale. Continents from the Authagraph projection, which preserves areas and shapes, are represented as a double helix of 32,111 bases. Short sequences of 806 unique species, sequenced as part of EBP-affiliated projects, are mapped onto the double helix of the continent (or ocean) where the species is commonly found. The length of the sequence is the same for each species on a continent (or ocean) and the sequences are separated by short gaps. Individual bases of the sequence are colored by dots. Species appear along the path in alphabetical order (by Latin name) and the first base of the first species is identified by a small black triangle.
Lewin HA et al. The Earth BioGenome Project 2020: Starting the clock. (2022) PNAS 119(4) e2115635118.
As part of the COVID Charts series, I fix a muddled and storyless graphic tweeted by Adrian Dix, Canada's Health Minister.
I show you how to fix color schemes to make them colorblind-accessible and effective in revealing patters, how to reduce redundancy in labels (a key but overlooked part of many visualizations) and how to extract a story out of a table to frame the narrative.