Thomas Ogden

# Pickup Sticks & Pi

We all learn at school that $\pi$ is the ratio of a circle’s circumference to its diameter, but $\pi$ turns up all over the place in mathematics. For example, did you know you can measure $\pi$ just by dropping sticks on a table and counting them? This is because $\pi$ has a role in statistics too. Try it for yourself, and below we’ll visit the story of how this remarkable trick was discovered.

Sticks Dropped Crossing a Line Estimate of $\pi$ Error
0 0 0 0%

## Buffon’s Needle

In 1777, a French philosopher called Georges-Louis Leclerc, Comte de Buffon, wrote out a very elegant theorem which turned out to be the earliest problem in geometric probability.

Buffon discovered that if you draw a set of equally-spaced parallel lines (say, $d$ centimetres apart) and drop sticks on them which are shorter than the spacing (say $l$ centimetres long, where $l$ is less than $d$), then the probability of a stick crossing a line is $\frac{2 l}{\pi d}.$

This means that if you drop lots of sticks randomly and count how many cross the parallel lines, you can calculate what $\pi$ is by rearranging the formula: $\pi = \frac{2ls}{c d}$ where $s$ is the number of sticks you drop and $c$ is the number that crossed a line.

Isn’t that remarkable? Buffon had worked out that you can calculate $\pi$ just by dropping a bunch of sticks on a table — no circles required! The problem is known as Buffon’s Needle.

As usual in statistics, we want to take a big sample to get an estimate close to the true probability. If I ask 5 people ‘are you left handed?’ and you ask 500 people the same, we know that your bigger sample is likely to give us the better idea of what proportion of people are left handed in the whole country. The same applies here — to get a good sample we need to drop a lot of sticks.

Perhaps that’s why it took a long time after Buffon’s discovery for somebody to actually try it themselves, but in 1901 the Italian mathematician Mario Lazzarini gave it a go. He span around and dropped over 3,400 sticks onto the floor, counted up the number that crossed over lines and estimated $\pi$ to be 3.1415929. We know $\pi$ to be 3.1415927…, so he was correct in the first 6 digits, an error of 0.000006%. That’s good! Suspiciously good in fact, and so improbable that it’s more likely he cheated. Hoist by his own probabilistic petard?

## Monte Carlo

What Lazzarini did we would today call a simulation. Simulations have become vastly easier and more important now that we can use computers to do the legwork for us. I’m sure a dizzy Lazzarini would’ve appreciated one. Simulations like this, where we use randomness (or, usually, pseudo-randomness) to give us a sample of results, are called Monte Carlo methods, after the European city famous for its casino.

Monte Carlo methods are very useful in physics, as we can model a large system with a huge number of possible states with a much smaller, but representative subset. It has important uses in thermal physics, molecular modelling, astrophysics and weather forecasting. And, as Buffon showed 200 years ago, can even be used to calculate $\pi$.

Finally, Buffon may have been the earliest human to discover this result but the natural world might have been first. Matthew Cobb pointed me to a fascinating paper that suggests ants use Buffon’s Needle to determine, before initiating an emigration, if a nest has sufficient area to house their colony.