Friday, September 30, 2016

Mathematical Divination: Finding Pi With Nothing But Matchsticks & Graph Paper

As a beautiful fall day rustles by outside, a physics student stands in the classroom with an arm held out over his lab table, clutching a fistful of matches. He holds them tight, palm upward, over a sheet of graph paper, on which he's painstakingly drawn a series of parallel lines, separated by a distance just larger than the length of the matchsticks. With an uncertain frown, he looks around at his peers, some of whom are already hunched over the tables, busy counting. With a shrug, the student tosses the fistful of matches up into the air, trying desperately to strike a balance between control and chaos—he's got to land as many of them on the page as he can, while still ensuring that they end up oriented at a suitably random scatter of angles.

He sits down on the cold metal seat of the lab stool, leans over the table and counts, scratching tally marks into the paper of his notebook. Satisfied, he sits up and looks to the blackboard, copying the equations on it. He substitutes in his numbers, the product of his careful tossing and counting, and starts punching figures into his calculator, transcribing and reducing the fraction on his page to simpler and simpler numbers, until he arrives at a ratio. He enters it in to the calculator with anticipation and, upon seeing the result—3.1—grins and leans back. There's still work to do, error bars to calculate, but for now he's proud to have pulled a decent approximation of a fundamental constant out of nothing but sticks and lines on a page.

An image floats to the top of his mind—a bearded, wizened mystic sits before a fire, rattling stones around in a turtle shell before tossing them out into a circle drawn in the dirt. The student's grin widens with the recognition that he's doing modern divination, applying esoteric insights to chaos, and discerning something deep about the universe in doing so. It feels like arcane power.

So how does he do it? How do you pull pi, the ratio of a circle's radius to its circumference, out of a random scatter of matchsticks? The exercise is called Buffon's Needle, named in honor of the 18th-century French count who is credited with being the first to explore a similar problem mathematically. The original problem, which presumably struck Buffon after knocking a sewing kit onto a hardwood floor, asks about the probability that a randomly dropped needle will cross one of the long edges of a floorboard.

The answer, it turns out, can be determined mathematically using the length of the needle, the width of the floorboards, and the value of pi—there's some integral calculus involved as well, but the upshot of the more complex math is a simple formula for the probability that a needle will cross a line:

...where d is the distance between the lines and l is the length of the needles. Pi shows up in this formula because the needles land at random angles between 0 and 360° (or 2*pi, in radians) with respect to the direction of the parallel lines. The exercise described above, however, uses a slight rearrangement of this formula:
...allowing us to find a decent approximation of pi by measuring the probability for ourselves and back-calculating! If we throw m matches, and c of them cross lines, the value of P is just c/m, as below:
It's never quite perfect, but theoretically the more matches you use, the closer you'll get to the actual value of pi. Grab some graph paper and toothpicks, and try it for yourself!

Stephen Skolnick

1 comment:

  1. I did this in high school (using 10,000 toothpicks). I didn't even win a runner-up award because nobody understood my exhibit -- even though I explained it very thoroughly.