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visualization + design
`\pi` Approximation Day Art Posters
The never-repeating digits of `\pi` can be approximated by 22/7 = 3.142857 to within 0.04%. These pages artistically and mathematically explore rational approximations to `\pi`. This 22/7 ratio is celebrated each year on July 22nd. If you like hand waving or back-of-envelope mathematics, this day is for you: `\pi` approximation day!
Equip your imagination.
What would circles look like if `\pi`=22/7?
Tiny loop, Folded dimensions and solidland
Imagine that the circle had a tiny loop at one of its points. The circumference of this loop would be added to the circumference of the circle, but the loop would be so small that we would never notice it.
This is reminiscent of how string theories describe higher dimensions—as tiny loops at each point in space, except in my example the loop is only at one point.
This idea originated with Klein, who explained the fourth dimension as a curled up circle of a very small radius. Another way in which this curling-up is used is to say that the fifth dimension is a curled up Planck length, as explained in this Imagining 10 Dimensions video.
flatlanders and solidlanders
If this idea is difficult to wrap your head around, you're not alone. We cannot think of additional dimensions in the regular spatial sence since we have no means of experiencing such phenomena. We can however imagine how flatlanders might explain the 3rd dimension, since we can perceive it. They would draw the curled up circles in their plane because they would not have the experience of drawing with perspective mimicking our 3rd dimension.
We would draw their explanation as shown on the right in the figure above, borrowing from our concept of the 3rd spatial dimension. Now imagine showing our explanation to a flatlander. They would not see the same thing as you—the circles would not intuitively imply the higher dimension to them.
This is analogous to why we cannot draw folded up dimensions. We are merely solidlanders—flatlanders in 3d space. Creatures that can perceive more spatial dimensions would use us as examples of diminished perceptual ability.
relativistic speeds, frames of reference and length contraction
Another way to imagine how a circle might look is a little more realistic. The theory of special relativity tells us that when we travel at speed relative to another object the dimensions of that object appear contracted to us in the direction of motion.
This contraction is always present, but essentially imperceptible unless we're travelling fast enough. For example, in order for a 1 meter object to appear contracted by the length of a hydrogen molecule (0.3 nm) we would have to be travelling at 7.3 km/s (Wolfram Alpha calculation)!
How fast would we have to be going to compress the circle sufficiently so that its circumference and radius ratio embody the `22/7` approximation of `\pi`? Pretty fast, it turns out. If we travel at just over 12,000 km/sec (0.04 times the speed of light, Wolfram Alpha calculation), the circle will compress as shown in the figure above, and the ratio of its circumference to the radius along direction of motion will make `\pi` appear to be `22/7`.
This compression in length would be barely perceptible to us. Below are both circles, shown overlapping, with `delta` being the extra length in radius required.
The value of `\delta`, which is 0.0008049179155 (if `r = 1`), can be calculated by considering the perimeter of an ellipse. The fact that `\delta` is small shouldn't be surprising since `22/7` is an excellent approximation of `\pi`, good to 0.04%.
Calculating the parameter of an ellipse is more complicated than calculating it for a circle because it uses something called an elliptic integral. This integral has no analytical solution and requires numerical approximation. Luckily, we have computers.
We can use the expression shown above for the perimeter of the ellipse to determine how much the circle needs to be deformed. Let's write `a = r + \delta` (original radius with slight deformation `\delta`) and `b=r`. Since `22/7 > \pi` we know that `\delta > 0`.
It remains to solve the equation below for a value of `\delta` that will yield a ratio of circumference to `r` of `2 \times 22/7`.
To make things simpler, let set `r=1`. Solving the equation numerically, I find $$\delta = 0.0008049179155$$
You can verify this solution at Wolfram Alpha.
the meaning of full-circle
After all this, we come full-circle to the meaning of full-circle.
You might ask why I didn't change the definition of `\pi` to `22/7` in the upper limit of the integral. After all, why not make the approximation exercise more faithful to the approximation?
It turns out that if I did that I would get `\delta=0`, which brings us back to the original circle. How is this possible?
Technically, this is because the integral returns the upper limit as its answer if the eccentricity is zero (i.e., `E(x,0)=x`).
Intuitively, this is because changing the upper limit of the integral actually redefines the angle of a full revolution. Now, full-circle isn't `2 \pi` radians, but `2 \times 22/7`. Given that the ratio of the circumference of a circle to its radius is exactly the size, in radians, of a full revolution, we don't need to change the shape of the circle if we're willing to change what a full revolution means.
news + thoughts
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.
Analyzing outliers: Robust methods to the rescue
Robust regression generates more reliable estimates by detecting and downweighting outliers.Outliers can degrade the fit of linear regression models when the estimation is performed using the ordinary least squares. The impact of outliers can be mitigated with methods that provide robust inference and greater reliability in the presence of anomalous values.
We discuss MM-estimation and show how it can be used to keep your fitting sane and reliable.
Greco, L., Luta, G., Krzywinski, M. & Altman, N. (2019) Points of significance: Analyzing outliers: Robust methods to the rescue. Nature Methods 16:275–276.
Background reading
Altman, N. & Krzywinski, M. (2016) Points of significance: Analyzing outliers: Influential or nuisance. Nature Methods 13:281–282.
Two-level factorial experiments
To find which experimental factors have an effect, simultaneously examine the difference between the high and low levels of each.Two-level factorial experiments, in which all combinations of multiple factor levels are used, efficiently estimate factor effects and detect interactions—desirable statistical qualities that can provide deep insight into a system.
They offer two benefits over the widely used one-factor-at-a-time (OFAT) experiments: efficiency and ability to detect interactions.
Since the number of factor combinations can quickly increase, one approach is to model only some of the factorial effects using empirically-validated assumptions of effect sparsity and effect hierarchy. Effect sparsity tells us that in factorial experiments most of the factorial terms are likely to be unimportant. Effect hierarchy tells us that low-order terms (e.g. main effects) tend to be larger than higher-order terms (e.g. two-factor or three-factor interactions).
Smucker, B., Krzywinski, M. & Altman, N. (2019) Points of significance: Two-level factorial experiments Nature Methods 16:211–212.
Background reading
Krzywinski, M. & Altman, N. (2014) Points of significance: Designing comparative experiments.. Nature Methods 11:597–598.