Cheerful Pi Day! March 14 is the day when 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 is the ratio of the circumference and diameter of a circle, but it appears in all kinds of places that seem to have nothing to do with wheels, from music to quantum mechanics.
Pi is an infinitely long decimal number that never repeats. How do we know? Well, people calculated it to 314 trillion decimal places and didn’t quite get there. At this point I’m willing to accept it. That is, NASA only uses the first 15 decimal places for spacecraft navigation, and that is more than enough for terrestrial applications.
The frosty thing for me is that there are many ways to approximate this value, as I’ve written about in the past. You can do this, for example, via a vibrating mass on a spring. But perhaps the craziest method of all was proven in 1777 by Count de Buffon, George Louis Leclerc.
Decades earlier, Buffon had posed this as a question about probability in geometry: imagine you had a floor with parallel lines separated by a distance D. You drop a bunch of needles of the appropriate length onto this floor L. What is the probability that the needle will cross one of the parallel lines?
The photo will aid you understand what is happening. Let’s assume I only drop two needles on the floor (you can replace the needles with something safer, like toothpicks). Also, to make things easier later, we can say that the needle length and the line spacing are equal (d = L).
You can see that one of the needles crosses the line and the other does not. OK, but what are the chances? This is not a minor problem, but let’s think about one dropped needle. we are only interested in two values - distance (X) from the distal end of the needle to the line and the angle of the needle (AND) relative to the perpendicular (see diagram below). If X is less than half the space between the lines, we get a needle cut. As you can see, with a lower probability you can get a higher probability X or smaller AND.