Mathematical Divination: Finding Pi With Nothing But Matchsticks & Graph Paper
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:
Dale Boggs said...
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.
Friday, September 30, 2016 at 7:36 PM