After last year’s once-in-a-lifetime π Day (3/14/15 according to the odd U.S. convention for writing dates), this year could seem a bit of a letdown to American math nerds.

But for those of us who prefer rounding to truncation, 3/14/16 is still something to celebrate. And I, for one, am willing to settle for this year’s 5-significant-figure π date. The logic is simple:

 

 

Last year, for the occasion, I showed how to calculate π with Dynamo using Machin’s Formula. This year, let’s try something different.

Buffon’s Needle

Did you know you can calculate π by dropping a box full of needles on the floor? Basically, the theory goes that π is related to the probability that a randomly placed long object (say, a needle) intersects one of a series of parallel lines on the floor. Here’s the setup:

The probability that the needle touches a line is related to the ratio of the length of the needle (ℓ) to the distance between the parallel lines (d).

 

 

Maybe you can see it with the diagram, but the probability, according to Georges-Louis Leclerc, Comte de Buffon, is related to the ratio of distance between the lines (d) and the component of the needle that is perpendicular to the parallel lines (ℓ cosθ). Specifically, you can write the probability based on the ratio of the length of the needle to the distance between the lines, x = ℓ / d.

 

 

And if you set the length of the needle to be the same as the distance between the lines (x = 1), the math gets super easy:

 

 

And, remember, this is an exercise in probability, so we need a large sample size to get a good answer.

… so we should drop a lot of needles!

 

 

Download the dyn file here: Buffon’s Needle.dyn. More on Buffon’s Needle at Wolfram Math World.