Approaching Perfect Approximations

Graphic by Jake Slaughter

This year has thrown a lot of things into question; common yardsticks like graduation year or personal identifiers like “college student” have changed so suddenly I struggle to know where I stand. I know this isn’t true for everyone, but I have always counted on the reliability of math—the understanding that each problem has correct answers that can be proven beyond a doubt, even if multiple paths can be taken to get there.

Take 𝝅, for instance: an irrational number we have all grown up learning about, which is most often associated with circles. The ratio between a circle’s circumference and its radius is 𝝅, as is the relationship between the radius and the area of a circle by the equation =Ar2. 𝝅 can be experimentally determined by measuring circles, or the circles themselves can be approximated by using regular polygons. 

But what about without circles? In 1777, Buffon proposed a thought experiment: 

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

If the length of the needle (L) is shorter than the distance between the lines (d), you can calculate the probability of intersection with geometric probability. Whether or not a given needle falls across a line can be determined by two things: the distance the center of the needle is from the closest line and the angle the needle makes with the lines. When calculating the joint probability distribution function, both of these independent variables need to be accounted for. By integrating the distribution function over the full range of possible values for both these parameters, we find the equation:

P(intersection) = 2L / 𝝅d

Probability can take on two different functions. It can be the exact, calculated, and theoretical probability that would be found using the above equations (i.e. plug in values for L, d and 𝝅 to find the probability of intersection). Alternatively, it can be an experimental probability derived from running the test over and over again.

So, looking at the needle problem, if I define the experimental probability as p (the ratio of needles that cross to total dropped needles), I can rewrite the equation:

𝝅 ≈ 2L / dp

Through experimentation or simulation, this equation can be used to approximate 𝝅. It can only ever be an approximation because p is an experimental probability, but as more nails are dropped, to infinity in theory, p gets closer and closer to P(intersection), giving a better and better approximation of 𝝅. 

Imagine that you’re dropping three-inch nails onto your floor of wooden planks, each six inches wide. According to our equation, this would mean that each needle has a probability of  2(3)(6), or 1/𝝅, to cross a line between two planks. Now imagine that you dropped 10,000 nails. The number of nails that you would expect to cross lines is 10,000 ✕ 1/𝝅, or 3,183. In this case, is approximately equal to the ratio of total nails dropped to nails that crossed lines, giving us an approximation of 3.14169. The more nails you drop, the closer the approximation becomes.

As we all navigate the new paths before us in these uncertain times, I am comforted by the fact that we can still arrive at meaningful answers through different rationales. As I continue my gap year, I’ve begun mentally simulating what this upcoming semester might have in store for me: imagining all the different jobs I could take on, destinations I could try to travel to, and people I could try to meet up with. Like randomly dropping needles onto planks in order to approximate 𝝅, I feel myself haphazardly throwing plans at the plane of my future, hoping to arrive closer to an answer amidst the multitude, hoping something will become clearer through the repeated attempts. Even if this year hasn’t presented me with a clear circle, and I’m not sure where my future will take me, I find solace in this thought experiment. Though definitive answers might now be out of reach, at least we can approximate a solution.

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