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visualization
**+** math

On March 14th celebrate `\pi` Day. Hug `\pi`—find a way to do it.

For those who favour `\tau=2\pi` will have to postpone celebrations until July 26th. That's what you get for thinking that `\pi` is wrong. I sympathize with this position and have `\tau` day art too!

If you're not into details, you may opt to party on July 22nd, which is `\pi` approximation day (`\pi` ≈ 22/7). It's 20% more accurate that the official `\pi` day!

Finally, if you believe that `\pi = 3`, you should read why `\pi` is not equal to 3.

For the 2014 `\pi` day, two styles of posters are available: folded paths and frequency circles.

The folded paths show `\pi` on a path that maximizes adjacent prime digits and were created using a protein-folding algorithm.

The frequency circles colourfully depict the ratio of digits in groupings of 3 or 6. Oh, look, there's the Feynman Point!

This year's Pi Day art expands on the work from last year, which showed Pi as colored circles on a grid. For those of you who really liked this minimalist depiction of π , I've created something slightly more complicated, but still stylish: Pi digit frequency circles. These are pretty and easy to understand. If you like random distribution of colors (and circles), these are your thing.

But to take drawing Pi a step further, I've experimented with folding its digits into a path. The method used is the same kind used to simulate protein folding. Research into protein folding is very active — the 3-dimensional structure of proteins is necessary for their function. Understanding how structure is affected by changes to underlying sequence is necessary for identifying how things go wrong in a cell.

Enough with proteins, you're here for the art.I will be using the replica exchange Monte Carlo algorithm to create folded paths (download code).

The choice of mapping between digit (0-9) and state (polar, hydrophobic) is arbitrary. I have chosen to assign the prime digits (2, 3, 5, 7) as hydrophobic. Another way can be to use perfect squares (1, 2, 4, 9). I construct the path by assigning each digit to a path node. One can partition π into two (or more) digit groupings (31, 41, 59, 26, ...) as well.

Want more math + art? Look at 2013 Pi Day art, discover the Accidental Similarity Number and other number art. Find humor in my poster of the first 2,000 4s of Pi.

The quality of the path will depend on how hard you look. Each time the folding simulation is run you run the chance of finding a better solution. For the 64 digits of
π
shown above, I ran the simulation 500 times and found over 200 paths with the same low energy. It's interesting to note that the path with `E=-22`

was found in <1 second and it took most of the computing time to find the next move.

Below I show 100 paths of 64-digits with `E=-23`

, sorted by their aspect ratio.

Running the simulation for 64 digits is very practical — it takes only a few minutes. In a sectino below, I show you how to run your own simulation.

Let's fold more digits! How about 768 digits — all the way to "...999999". This is the famous The Feynman Point in π where we see the first set of six 9s in row. This happens surprisingly early — at digit 762. In this sequence there are 298 prime digits with the other 470 being composite.

I have chosen not to emphasize the start and end of the path — finding them is part of the fun (You are haven't fun, aren't you?). The end is easier to spot — the 6 9s stand out. Finding the start, on the other hand, is harder.

The Feynman Point is a specific instance of repeating digits, which I call
(*d*,*n*)
points.

You can read more about these locations, where I have enumerated all such locations in the first 268 million digits of π .

Below is a list of the 20 best paths that I've been able to find. They range from `E=-223`

to `E=-219`

. I annotate each path with a few geometrical properties, such as width, height, area and so on. In some of the art these properties annotate the path (energy x×y r cm,cmabs).

# e - energy, as positive number # x,y - path width and height # r - aspect ratio = x/y # area - area (x*y) # cm - center of mass |(sum(x),sum(y))|/n and |(sum(|x|),sum(|y|))|/n # dend - distance between start and end of path 0 e 223 size 37 51 r 0.725 area 1887 cm 1.9 13.4 dend 24.4 1 e 222 size 36 44 r 0.818 area 1584 cm 17.3 18.8 dend 10.4 2 e 221 size 37 50 r 0.740 area 1850 cm 7.6 14.0 dend 16.3 3 e 221 size 70 36 r 1.944 area 2520 cm 1.0 17.3 dend 30.1 4 e 221 size 41 55 r 0.745 area 2255 cm 17.9 20.6 dend 29.5 5 e 221 size 50 49 r 1.020 area 2450 cm 20.8 22.1 dend 34.1 6 e 221 size 61 35 r 1.743 area 2135 cm 11.4 18.2 dend 15.0 7 e 221 size 53 45 r 1.178 area 2385 cm 14.7 18.1 dend 18.8 8 e 221 size 32 52 r 0.615 area 1664 cm 14.0 18.1 dend 33.8 9 e 220 size 46 70 r 0.657 area 3220 cm 26.6 27.8 dend 27.3 10 e 220 size 55 55 r 1.000 area 3025 cm 5.1 16.8 dend 15.0 11 e 220 size 58 34 r 1.706 area 1972 cm 9.3 14.6 dend 43.4 12 e 220 size 62 50 r 1.240 area 3100 cm 30.6 31.4 dend 33.4 13 e 220 size 41 45 r 0.911 area 1845 cm 15.4 17.6 dend 19.2 14 e 220 size 47 51 r 0.922 area 2397 cm 25.6 26.7 dend 16.0 15 e 220 size 38 52 r 0.731 area 1976 cm 13.1 15.9 dend 23.6 16 e 220 size 57 46 r 1.239 area 2622 cm 20.7 22.7 dend 51.7 17 e 220 size 43 57 r 0.754 area 2451 cm 21.3 23.3 dend 29.6 18 e 219 size 45 45 r 1.000 area 2025 cm 16.5 18.2 dend 33.1 19 e 219 size 51 46 r 1.109 area 2346 cm 16.0 19.2 dend 44.4

As you can see, the dimensions of the paths vary greatly. Low energy paths are not necessarily symmetrical. Paths with a small `cm`

are balanced around their center. Paths with `r`

≈1 are confined in a square boundary. Paths with small `dend`

have their start and end points close to one another.

The art would not be complete if we didn't somehow try to further force things into a circle! The path lattice is rectangular, but can be deformed into an ellipse or circle using the following transformation

` [(x'),(y')] = [(x sqrt(1-y^2/2)),(y sqrt(1-x^2/2)) ] `

news
**+** thoughts

*Nature is often hidden, sometimes overcome, seldom extinguished. —Francis Bacon*

In the first of a series of columns about neural networks, we introduce them with an intuitive approach that draws from our discussion about logistic regression.

Simple neural networks are just a chain of linear regressions. And, although neural network models can get very complicated, their essence can be understood in terms of relatively basic principles.

We show how neural network components (neurons) can be arranged in the network and discuss the ideas of hidden layers. Using a simple data set we show how even a 3-neuron neural network can already model relatively complicated data patterns.

Derry, A., Krzywinski, M & Altman, N. (2023) Points of significance: Neural network primer. *Nature Methods* **20**.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of significance: Logistic regression. Nature Methods **13**:541–542.

Our cover on the 11 January 2023 Cell Genomics issue depicts the process of determining the parent-of-origin using differential methylation of alleles at imprinted regions (iDMRs) is imagined as a circuit.

Designed in collaboration with with Carlos Urzua.

Akbari, V. *et al.* Parent-of-origin detection and chromosome-scale haplotyping using long-read DNA methylation sequencing and Strand-seq (2023) Cell Genomics 3(1).

Browse my gallery of cover designs.

My cover design on the 6 January 2023 Science Advances issue depicts DNA sequencing read translation in high-dimensional space. The image showss 672 bases of sequencing barcodes generated by three different single-cell RNA sequencing platforms were encoded as oriented triangles on the faces of three 7-dimensional cubes.

More details about the design.

Kijima, Y. *et al.* A universal sequencing read interpreter (2023) *Science Advances* **9**.

Browse my gallery of cover designs.

*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).

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**.

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

The video recently premiered on YouTube.

Renders of the full scene are available as NFTs.

*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.

© 1999–2023 Martin Krzywinski | contact | Canada's Michael Smith Genome Sciences Centre ⊂ BC Cancer Research Center ⊂ BC Cancer ⊂ PHSA