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Lips that taste of tears, they say, are the best for kissing.Dorothy Parkerget crankymore quotes

pi: exciting


DNA on 10th — street art, wayfinding and font


visualization + design

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The 2019 Pi Day art celebrates digits of `\pi` with hundreds of languages and alphabets. If you're a kid at heart—rejoice—there's a special edition for you!

The art of Pi (`\pi`), Phi (`\phi`) and `e`


Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2019 `\pi` has hundreds of digits, hundreds of languages and a special kids' edition.

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2018 `\pi` day

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2017 `\pi` day

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` approximation day

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` day

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2015 `\pi` day

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` approx day

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` day

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2013 `\pi` day

Pi Art Posters
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Circular `\pi` art

Numbers are a lot of fun. They can start conversations—the interesting number paradox is a party favourite: every number must be interesting because the first number that wasn't would be very interesting! Of course, in the wrong company they can just as easily end conversations.

is π normal?

It is not yet known whether the digits of π are normal—determining this is an important problem in mathematics. In other words, is the distribution of digit frequencies in π uniform? Do each of the digits 0–9 appear exactly 1/10th of the time, does every two-digit string appear exactly 1/100th of the time and so on for every finite-length string1?

1 One interesting finite-length string is the 6-digit Fenyman Point (...999999...) which appears at digit 762 in π. The Feynman Point was the subject of 2014 `\pi` Day art.

This question can be posed for different representations of π—in different bases. The distribution frequencies of 1/10, 1/100, and so on above refer to the representation of π in base 10. This is the way we're used to seeing numbers. However, if π is encoded as binary (base 2), would all the digits in 11.00100100001111... be normal? The table below shows the first several digits of π in each base from 2 to 16, as well as the natural logarithm base, `e`.

base, `b``\pi_b`base, `b``\pi_b`
211.00100100001111 103.14159265358979
310.01021101222201 113.16150702865A48
43.02100333122220 123.184809493B9186
53.03232214303343 133.1AC1049052A2C7
63.05033005141512 143.1DA75CDA813752
73.06636514320361 153.21CD1DC46C2B7A
83.11037552421026 163.243F6A8885A300
`e`10.10100202000211
source: virtuescience.com

Because the digits in the numbers are essentially random (this is a conjecture), the essence of the art is based on randomness.

A vexing consequence of π being normal is that, because it is non-terminating, π would contain all patterns. Any word you might think of, encoded into numbers in any way, would appear infinitely many times. The entire works of Shakespeare, too. As well, all his plays in which each sentence is reversed, or has one spelling mistake, or two! In fact, you would eventually find π within π, but only if you have infinite patience.

This is why any attempts to use the digits of `\pi` to infer meaning about anything is ridiculous. The exact opposite of what you find is also in `\pi`.

Stoneham's constant

A number can be normal in one base, but another. For example, Stoneham's constant,

`\alpha_{2,3} = 1/2 + 1/(2^{3^1} 3^1) + 1/(2^{3^2} 3^2) + 1/(2^{3^3} 3^3) + ... + 1/(2^{3^k} 3^k) + ... `

is 0.54188368083150298507... in base 10 and 0.100010101011100011100011100... in base 2.

Stoneham's constant is provably normal in base 2. In some other bases, such 6, Stoneham's constant is provably not normal.

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news + thoughts

Yearning for the Infinite — Aleph 2

Mon 18-11-2019

Discover Cantor's transfinite numbers through my music video for the Aleph 2 track of Max Cooper's Yearning for the Infinite (album page, event page).

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Yearning for the Infinite, Max Cooper at the Barbican Hall, London. Track Aleph 2. Video by Martin Krzywinski. Photo by Michal Augustini. (more)

I discuss the math behind the video and the system I built to create the video.

Hidden Markov Models

Mon 18-11-2019

Everything we see hides another thing, we always want to see what is hidden by what we see.
—Rene Magritte

A Hidden Markov Model extends a Markov chain to have hidden states. Hidden states are used to model aspects of the system that cannot be directly observed and themselves form a Markov chain and each state may emit one or more observed values.

Hidden states in HMMs do not have to have meaning—they can be used to account for measurement errors, compress multi-modal observational data, or to detect unobservable events.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Hidden Markov Models. (read)

In this column, we extend the cell growth model from our Markov Chain column to include two hidden states: normal and sedentary.

We show how to calculate forward probabilities that can predict the most likely path through the HMM given an observed sequence.

Grewal, J., Krzywinski, M. & Altman, N. (2019) Points of significance: Hidden Markov Models. Nature Methods 16:795–796.

Background reading

Altman, N. & Krzywinski, M. (2019) Points of significance: Markov Chains. Nature Methods 16:663–664.

Hola Mundo Cover

Sat 21-09-2019

My cover design for Hola Mundo by Hannah Fry. Published by Blackie Books.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Hola Mundo by Hannah Fry. Cover design is based on my 2013 `\pi` day art. (read)

Curious how the design was created? Read the full details.

Markov Chains

Tue 30-07-2019

You can look back there to explain things,
but the explanation disappears.
You'll never find it there.
Things are not explained by the past.
They're explained by what happens now.
—Alan Watts

A Markov chain is a probabilistic model that is used to model how a system changes over time as a series of transitions between states. Each transition is assigned a probability that defines the chance of the system changing from one state to another.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Markov Chains. (read)

Together with the states, these transitions probabilities define a stochastic model with the Markov property: transition probabilities only depend on the current state—the future is independent of the past if the present is known.

Once the transition probabilities are defined in matrix form, it is easy to predict the distribution of future states of the system. We cover concepts of aperiodicity, irreducibility, limiting and stationary distributions and absorption.

This column is the first part of a series and pairs particularly well with Alan Watts and Blond:ish.

Grewal, J., Krzywinski, M. & Altman, N. (2019) Points of significance: Markov Chains. Nature Methods 16:663–664.

1-bit zoomable gigapixel maps of Moon, Solar System and Sky

Mon 22-07-2019

Places to go and nobody to see.

Exquisitely detailed maps of places on the Moon, comets and asteroids in the Solar System and stars, deep-sky objects and exoplanets in the northern and southern sky. All maps are zoomable.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
3.6 gigapixel map of the near side of the Moon, annotated with 6,733. (details)
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
100 megapixel and 10 gigapixel map of the Solar System on 20 July 2019, annotated with 758k asteroids, 1.3k comets and all planets and satellites. (details)
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
100 megapixle and 10 gigapixel map of the Northern Celestial Hemisphere, annotated with 44 million stars, 74,000 deep-sky objects and 3,000 exoplanets. (details)
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
100 megapixle and 10 gigapixel map of the Southern Celestial Hemisphere, annotated with 69 million stars, 88,000 deep-sky objects and 1000 exoplanets. (details)

Quantile regression

Sat 01-06-2019
Quantile regression robustly estimates the typical and extreme values of a response.

Quantile regression explores the effect of one or more predictors on quantiles of the response. It can answer questions such as "What is the weight of 90% of individuals of a given height?"

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Quantile regression. (read)

Unlike in traditional mean regression methods, no assumptions about the distribution of the response are required, which makes it practical, robust and amenable to skewed distributions.

Quantile regression is also very useful when extremes are interesting or when the response variance varies with the predictors.

Das, K., Krzywinski, M. & Altman, N. (2019) Points of significance: Quantile regression. Nature Methods 16:451–452.

Background reading

Altman, N. & Krzywinski, M. (2015) Points of significance: Simple linear regression. Nature Methods 12:999–1000.