You can approximate the value of Pi by dropping needles on the ground

Happy B day! March 14 is the date on which rational people celebrate this irrational number, because 3/14 contains the first three digits of pi. And hey, it’s worth a day. By definition, it’s the ratio of a circle’s circumference and diameter, but it shows up in all sorts 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 have calculated it to 314 trillion decimal places and haven’t gotten to the end. At this point, I’m inclined to accept that. I mean, NASA only uses the first 15 decimal places to navigate spacecraft, and that’s more than enough for terrestrial applications.

The coolest thing for me is that there are many ways to approximate this value, which I’ve written about in the past. For example, you can do this by swinging a block on a spring. But perhaps the craziest method was demonstrated in 1777 by Georges Louis Leclerc, Comte de Buffon.

Decades ago, Buffon posed this question as a probability question in geometry: Imagine that you have a floor with parallel lines separated by a distance. D. On this floor, I dropped a set of needles along its length to. What is the probability that the needle crosses one of the parallel lines?

The photo 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 the line and the other does not. Okay, but what are the chances? This is not the most trivial problem, but let’s consider just one needle that fell. We only care about two values ​​- the distance (S) from the farthest end of the needle to the line, and the angle of the needle (θ) relative to the vertical (see chart below). if S Less than half the distance between the lines, we get the needle crossing. As you can see, you will get a higher probability with a smaller probability S Or smaller θ.