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Distractions and amusements, with a sandwich and coffee.

Trance opera—Spente le Stelle
• be dramatic
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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.

This year's `\pi` day art collection celebrates not only the digit but also one of the fundamental forces in nature: gravity.

In February of 2016, for the first time, gravitational waves were detected at the Laser Interferometer Gravitational-Wave Observatory (LIGO).

The signal in the detector was sonified—a process by which any data can be encoded into sound to provide hints at patterns and structure that we might otherwise miss—and we finally heard what two black holes sound like. A buzz and chirp.

The art is featured in the Gravity of Pi article on the Scientific American SA Visual blog.

All the art was processed while listening to Roses by Coeur de Pirate, a brilliant female French-Canadian songwriter, who sounds like a mix of Patricia Kaas and Lhasa. The lyrics Oublie-moi (Forget me) are fitting with this year's theme of gravity.

Mais laisse-moi tomber, laisse-nous tomber

Laisse la nuit trembler en moi

Laisse-moi tomber, laisse nous tomber

Cette fois

But let me fall, let us fall

Let the night tremble in me

Let me fall, let us fall

This time

The gravitational force between two masses `m_1` located at `(x_1,y_1)` and `m_2` located at `(x_2,y_2)` is given by

$$F = \frac{G m_1 m_2}{r^2} \tag{1} $$

where `r` is the distance between the masses given by

$$r = \sqrt{ \Delta x ^2 + \Delta y ^2 } = \sqrt{ (x_2-x_1)^2 + (y_2-y_1)^2 } \tag{2} $$

The force is directed along the vector formed by `r` and can be decomposed into `x` and `y` components using \begin{align} F_x &= F \frac{ \Delta x}{r} = F \frac{x_2-x_1}{r} \tag{3} \\ F_y &= F \frac{ \Delta y}{r} =F \frac{y_2-y_1}{r} \tag{4} \end{align}

The acceleration of each mass can be obtained using `F = ma` and similarly decomposed into `x` and `y` components \begin{align} a_{1x} &= \frac { F_{1x} }{ m_1} = \frac{G m_2 \Delta x}{r^3} \tag{5} \\ a_{1y} &= \frac { F_{1y} }{ m_1} = \frac{G m_2 \Delta y}{r^3} \tag{6} \\ a_{2x} &= \frac { F_{2x} }{ m_2} = -\frac{G m_1 \Delta x}{r^3} \tag{7} \\ a_{2y} &= \frac { F_{2y} }{ m_2} = -\frac{G m_1 \Delta y}{r^3} \tag{8} \end{align}

When there are `n` masses in the system, the acceleration of mass `i` is the sum of the accelerations due to all other masses \begin{align} a_{ix} &= \sum_{i \ne j} \frac{G m_j \Delta x_{ij}}{r_{ij}^3} \tag{9} \\ a_{iy} &= \sum_{i \ne j} \frac{G m_j \Delta y_{ij}}{r_{ij}^3} \tag{10} \end{align}

The equations of motion for the masses over a period of time `\Delta t` are

\begin{align} \Delta v_x &= \Delta t a_x \tag{11} \\ \Delta v_y &= \Delta t a_y \tag{12} \\ \Delta x &= \Delta t \left( v_x + a_x \frac{\Delta t}{2} \right) \tag{13} \\ \Delta y &= \Delta t \left( v_y + a_y \frac{\Delta t}{2} \right) \tag{14} \end{align}

There are various ways in which the numerical simulation can be performed. The Euler, Verlet, Runge-Kutta methods are perhaps the most common. I use the Verlet approach.

Using the equations of motions above, the Verlet simulation goes as follows

- calculate acceleration, `a_1` (eq 9,10)
- update position (eq 13,14)
- calculate new acceleration, `a_2` (eq 9,10)
- update velocity using `(a_1+a_2)/2` (eq 7,8)

The masses are initially uniformly distributed on a circle and given a zero initial velocity or a normally distributed random velocity.

I ran about 10,000 individual simulations with different values of `n` and `k` and collected ones that stood out as pretty.

The size of a mass is taken to be `s = m^{1/3}`. When two masses, `m_1` and `m_2` come within a distance of `\left( s_1 + s_2 \right)(1-z)` of each other, they collide. Here `z` is a collision margin parameter that I set to either `z=0` or `z=0.25`.

During the collision, a new body is created with mass `M = m_1 + m_2` given a speed that conserves momentum in the collision. \begin{align} v_x &= \frac{m_1 v_{1x} + m_2 v_{2x} }{M} \\ v_y &= \frac{m_1 v_{1y} + m_2 v_{2y} }{M} \end{align}

For my simulation, the following values are used

- `G = 100`
- mass for each digit, `d` is `(1+d)^k`
- masses placed on circle with radius `216`
- when randomized, `(v_x,v_y) \sim N(0,1)`
- `\Delta t = 0.01`
- simulation runs for up to 100,000 steps
- canvas size is `1440 \times 1440`

*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 stateâ€”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.

*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 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.

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

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.

Altman, N. & Krzywinski, M. (2016) Points of significance: Analyzing outliers: Influential or nuisance. Nature Methods 13:281–282.

Two-level factorial experiments, in which all combinations of multiple factor levels are used, efficiently estimate factor effects and detect interactionsâ€”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.

Krzywinski, M. & Altman, N. (2014) Points of significance: Designing comparative experiments.. Nature Methods 11:597–598.