You Can Approximate Pi by Dropping Needles on the Floor
Happy Pi Day! March 14 is the date that otherwise rational people celebrate this irrational number, because 3/14 contains the first three digits of pi. And hey, pi deserves a day. By definition, it’s the ratio of the circumference to the diameter of a circle, but it appears in all kinds of places that seem to have nothing to do with circles, from music to quantum mechanics.
Pi is an infinitely long decimal number that never repeats. How do we know? Well, humans calculated it with 314 trillion decimal places and didn’t get to the end. At this point, I’m inclined to accept it. I mean, NASA only uses the first 15 decimal places to navigate spacecraft, and that’s more than enough for terrestrial applications.
The cool thing, for me, is that there are many ways to get closer to this value, which I’ve written about before. For example, you can do this by oscillating a mass on a spring. But perhaps the craziest method of all was proven in 1777 by George Louis Leclerc, Count of Buffon.
Decades earlier, Buffon had asked this question as a question of probability in geometry: imagine that you have a floor with parallel lines separated by a distance. d. On this floor you drop a bunch of needles long L. What is the probability that a needle crosses one of the parallel lines?
A picture will help you understand what is happening. Let’s say I drop just two needles on the floor (feel free to replace the needles with something safer, like toothpicks). Also, to make things easier later, we can say that the needle length and line spacing are equal (d = L).
You can see that one of the needles crosses a line and the other does not. OK, but what are the chances? It’s not the most trivial problem, but let’s think about a single needle falling. We only care about two values: distance (x) from the farthest end of the needle to a line, and the angle of the needle (θ) relative to a perpendicular (see the diagram below). If x is less than half the spacing between the lines, we get a needle crossing. As you can see, you will get a higher probability with a smaller x or a smaller one θ.
