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visualizations
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lunch break
Distractions and amusements, with a sandwich and coffee.
visualization + design
The art of Pi (`\pi`), Phi (`\phi`) and `e`
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.
The art here is my attempt at transforming famous numbers in mathematics into pretty visual forms, start some of these conversations and awaken emotions for mathematics—other than dislike and confusion
Numerology is bogus, but art based on numbers can be beautiful. Proclus got it right when he said (as quoted by M. Kline in Mathematical Thought from Ancient to Modern Times)
Wherever there is number, there is beauty.
—Proclus Diadochus
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the numbers π, φ and e
The consequence of the interesting number paradox is that all numbers are interesting. But some are more interesting than others—how Orwellian!
All animals are equal, but some animals are more equal than others.
—George Orwell (Animal Farm)
Numbers such as `\pi` (or `\tau` if you're a revolutionary), `\phi`, `e`, `i = \sqrt{-1}`, and `0` have captivated imagination. Chances are at least one of them appears in the next physics equation you come across.
π φ e
= 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 ... = 1.61803 39887 49894 84820 45868 34365 63811 77203 09179 ... = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 ...
Of these three transcendental numbers, `\pi` (3.14159265...) is the most well known. It is the ratio of a circle's circumference to its diameter (`d = \pi r`) and appears in the formula for the area of the circle (`a = \pi r^2`).
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The Golden Ratio (`\phi`, 1.61803398...) is the attractive proportion of values `a > b` that satisfy `{a+b}/2 = a/b`, which solves to `a/b = {1 + \sqrt{5}}/2`.
The last of the three numbers, `e` (2.71828182...) is Euler's number and also known as the base of the natural logarithm. It, too, can be defined geometrically—it is the unique real number, `e`, for which the function `f(x) = e^x` has a tangent of slope 1 at `x=0`. Like `\pi`, `e` appears throughout mathematics. For example, `e` is central in the expression for the normal distribution as well as the definition of entropy. And if you've ever heard of someone talking about log plots ... well, there's `e` again!
Two of these numbers can be seen together in mathematics' most beautiful equation, the Euler identity: `e^{i\pi} = -1`. The tau-oists would argue that this is even prettier: `e^{i\tau} = 1`.
accidentally similar
Did you notice how the 13th digit of all three numbers is the same (9)? This accidental similarity generates its own number—the Accidental Similarity Number (ASN).
news + thoughts
Yearning for the Infinite — Aleph 2
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).
I discuss the math behind the video and the system I built to create the video.
Hidden Markov Models
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.
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
My cover design for Hola Mundo by Hannah Fry. Published by Blackie Books.
Curious how the design was created? Read the full details.
Markov Chains
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.
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
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.
Quantile regression
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?"
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.