A Beautiful Way to Calculate π: Buffon’s Needle Problem

Ali
Ali
Nov 6, 2020 · 4 min read
Image for post
Image for post

Sometimes I find myself thinking, what if we did not have the unique number “pi”? Probably, everything would change. Our beautiful mathematics turn strange; maybe even the earth would go awry and wouldn’t orbit the sun. If the world is still not so bad, it is because of that notorious constant number pi.

By the way, I assumed we all know what pi is since middle school. Our teacher showed us that “pi” (π) is an irrational number, which is the ratio of a circle’s circumference (2πr) to its diameter (2r). In other words, when you measure a circular object like a coin or a cup, it will always turn out that your circle is a little more than three times its width around. That is literally true for all circles of any size.

Person of Interest, Season 2 Episode 11 “2 Pi R

Pi s a fascinating never-ending number and more famous and influential than other never-ending numbers “e” and “square root of two.” It is because pi can show up in the strangest of places. We can see pi in the meandering path of rivers. Hans-Henrik Stølum proved that the average sinuosity of rivers around the world is pi.

In other words, when you divide a rivers’ actual length by its’ straight route from its source to mouth, you get a number that is very close to pi. Pi also tells us which colors should appear in a rainbow. But these are not the only exciting ways to find pi in nature.

Image for post
Image for post
1st Photo by Joshua Fuller on Unsplash, 2nd Photo by Mickael Tournier on Unsplash

When children had no internet, they liked to play tossing a coin on the floor and seeing the coin cross a line or not. After a while, a guy, a French philosopher Georges-Louis Leclerc, was good at math to determine whether the coin would cross a line. He just wanted to generalize that game. Remarkable idea!

Image for post
Image for post
Georges Louis Leclerc, Comte de Buffon | Wikimedia

First, he started dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines on the paper. Then he tried his experiment with many needles many times. And he did it. He got a remarkable result. The probability was directly related to the never-ending number pi value because two times the number of needles he dropped divided by the number of needles crossing a line was almost equal to pi all the time. So he made a formula:

P: the probability | n: the number of needles | c: the number of needles crossing a line. Then; P = 2n/c

For instance, get some toothpicks and then draw some lines on cardboard. But the gaps between lines should be equal, and the width between the gaps should be equal to the length of a toothpick. When you are ready, you can throw your toothpicks on the cardboard randomly. Sometimes when you drop the toothpick, it will cut a line, and sometimes it falls between lines.

Assume that you throw 100 toothpicks. When you count the number of toothpicks that cross the lines, you will probably see that 60–65 needles will cross the lines. If you apply the formula and do the calculation, you get (100 x 2)/63 = 3.17 or about 64%. I know it is not perfect because of the number of toothpicks. With more toothpicks, you can get better accuracy.

An Italian mathematician Mario Lazzarini threw needles almost 4000 times to perform this experiment 117 years ago. And he got pi with perfect accuracy. He got the first six decimal places of pi.

Even though there are no circles, it is still a beautiful approximation of pi. I know it seems awkward, and this geometrical probability blows your mind, but it is entirely true. It’s stunningly accurate.

If you don’t believe me, you can throw thousands of toothpicks and test the experiment yourself, or you can check the A Monte Carlo simulation below to see. The gif shows the pi estimation with a different number of toothpicks.

Image for post
Image for post
Throwing 1000 needles to estimate pi. Source: reddit

Today, we call this experiment “Buffon’s Needle problem” to honor that French philosopher Georges-Louis Leclerc. And math people use integral geometry to explain this relation. You can find the proof below.

Image for post
Image for post

However, Mathematics

Life is good for only two things, discovering mathematics…

Ali

Written by

Ali

Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better. alikayaspor@gmail.com

However, Mathematics

Life is good for only two things, discovering mathematics and teaching mathematics.

Ali

Written by

Ali

Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better. alikayaspor@gmail.com

However, Mathematics

Life is good for only two things, discovering mathematics and teaching mathematics.

Medium is an open platform where 170 million readers come to find insightful and dynamic thinking. Here, expert and undiscovered voices alike dive into the heart of any topic and bring new ideas to the surface. Learn more

Follow the writers, publications, and topics that matter to you, and you’ll see them on your homepage and in your inbox. Explore

If you have a story to tell, knowledge to share, or a perspective to offer — welcome home. It’s easy and free to post your thinking on any topic. Write on Medium

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store