Buffon’s pine needles
A little Christmas experiment for you to try at home.
A couple of days ago I was asked by a journalist friend of mine, who works for the Guardian, about a favourite science experiment I might do over the holidays. So I thought quite hard about it and probably went over the top for what she wanted, but this is what I came up with: Buffon’s pine needles.
Buffon’s needle is a simple and now classic way of approximating the value of π. It involves dropping a needle lots of times onto a sheet of lined paper (with the lines farther apart than the length of the needle) and counting the number of times the needle crosses one of the lines when it lands.
In its original form, first posed in the 18th century, Georges-Louis Leclerc, Comte de Buffon asked a question in probability theory about how often you would expect the needle to cross one of the lines. The theoretical answer, it turns out, has a factor of π in it. So, if you estimate the probability by doing an experiment, you can reverse engineer the formula to use the proportion of times the needle crosses a line to estimate π.
To make it festive though, instead of throwing a single sewing needle a few hundred times, we can try throwing a handful of pine needles a few tens of times. Grab yourself a bunch of pine needles off the tree – perhaps once you’ve taken the tree outside after Christmas you can sweep up the remaining needles and use those.
You need to choose as many needles as you can that are roughly the same length, L. Let’s say you manage to find a total of T similar sized pine needles. You’re also going need a piece of paper which has lines ruled on it which are a distance W apart - further apart than the length of your needles. If your needles are about L=3cm long then your lines could be W=4 cm or W=5 cm apart. You need to scatter the pine needles randomly (without aiming) on top of your piece of ruled paper and then count the number of needles that cross one of the lines, C. We actually have floorboards which are regularly spaced so we just dropped pine needles onto the floor. Once you have counted you can plug your numbers into this formula:
To find your approximation of π.
Remember L is the length of your pine needles and W is the distance apart of your lines. T is the total number of pine needles you throw and C is the number of crosses you count in total.
Originally the experiment was done with just a single needle dropped hundreds of times, but who has the energy for that at Christmas. If you’re feeling energetic though and want to get a closer approximation then you can repeat the drop multiple times. A few hundred drops should get you a respectable ballpark approximation for π.
What I love about this is that it demonstrates how π crops up in really unexpected places. It feels almost like magic, but it’s just probability in action on your living room floor.
We ended up throwing a total of 500 pine needles onto our floor and counted a total of 86 crosses. Our pine needles were all about 3.1 cm long and the width of our floorboards was 13.8cm. Plugging this into the formula gave us an estimate of π as 2.53. Not perfect by any stretch (π is suppose to be about 3.14…), but definitely in the right ball park.
If you do end up having a go, let me know the value of the approximation you found for π in the comments.
Merry Christmas.
Should W be E or vice versa?