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Distractions and amusements, with a sandwich and coffee.

Trance opera—Spente le Stelle
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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.

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.

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.

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.

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

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

In the past, I've been tangentially involved in fashion design. I've also been more directly involved in fashion photography.

It was now time to design my first ... pair of socks.

In collaboration with Flux Socks, the design features the colors and relative thicknesses of Rogue olympic weightlifting plates. The first four plates in the stack are the 55, 45, 35, and 25 competition plates. The top 4 plates are the 10, 5, 2.5 and 1.25 lb change plates.

The perceived weight of each sock is 178.75 lb and 357.5 lb for the pair.

The actual weight is much less.

*Find patterns behind gene expression and disease.*

Expression, correlation and network module membership of 11,000+ genes and 5 psychiatric disorders in about 6" x 7" on a single page.

Design tip: Stay calm.

More of my American Scientific Graphic Science designs

Gandal M.J. et al. Shared Molecular Neuropathology Across Major Psychiatric Disorders Parallels Polygenic Overlap *Science* **359** 693–697 (2018)

We discuss the many ways in which analysis can be confounded when data has a large number of dimensions (variables). Collectively, these are called the "curses of dimensionality".

Some of these are unintuitive, such as the fact that the volume of the hypersphere increases and then shrinks beyond about 7 dimensions, while the volume of the hypercube always increases. This means that high-dimensional space is "mostly corners" and the distance between points increases greatly with dimension. This has consequences on correlation and classification.

Altman, N. & Krzywinski, M. (2018) Points of significance: Curse(s) of dimensionality *Nature Methods* **15**:399–400.