Estimate π with Pasta?

How can we estimate Pi by throwing straight noodles onto a square grid? Why would pi be here? Solution to Buffon's needle by generalizing to Buffon's noodle using only probability theory and no calculus. More links below

Links to my follow up videos:
* Computer simulations in Python/Google collab:    • Computer simulations of Buffon's Need...  
* Detailed proof of proportionality:    • Proofs for Buffon's Noodle Problem: A...  

0:00 How to estimate pi with pasta
0:44 Formula for pi
1:10 Computer simulations
1:25 The Buffon Needle Problem
1:48 One weird trick...cook the spaghetti
2:19 The Buffon Noodle Problem
3:11 Experimental vs Theoretical averages
4:50 Straight vs Curvy noodles
6:05 Average Number of Crossings is proportional to length
6:26 Using a circular noodle to calculate C
7:20 Full proof of why the formula estimates pi
8:57 Bonus proof: Barbier's Theorem


Other YouTube videos about Buffon's needle
*Classic Numberphile video on Buffon's needle (using calculus!):    • Pi and Buffon's Matches - Numberphile  
[Note that the Numberphile uses horizontal planks instead of a square grid (which gives a factor of 2) and their matches are half as long as the noodles I use, which gives another factor of 2. So their overall formula is Pi = Num Matches/Num Crossings (differs by a factor of 4 compared to this video)]

*MIT OpenCourseware with concise calculus solution:    • S09.1 Buffon's Needle & Monte Carlo S...  
[Like Numberphile they use horizontal planks so it differs by a factor of 2 from my video, Pi=2 Num Needles/Num Crossings for length 1 needles]

*MindYourDecisions video on Buffon's needle:    • Surprising π from probability! Buffon...  
[Includes the calculus proof and "Barbier's proof" which is similar to Buffon's noodle proof. Same factor of 2 difference because of using horizontal planks as last two videos, Pi=2 Num Needles/Num Crossings]

*Mathologer video on shapes of constant width and Barbier's theorem:    • New Reuleaux Triangle Magic  
[They talk about the other geometric proof of Barbier's theorem at 8:49]


Wikipedia Links:
*Buffon's Needle Problem: https://en.wikipedia.org/wiki/Buffon%...
[Note that the traditional version of Buffon's needle use infinite horizontal planks instead of square grid which gives a factor of 2 difference in the estimation formula, Pi = 2* num noodles/num crossings for that version]

*Buffon's Noodle: https://en.wikipedia.org/wiki/Buffon%...

*Monte Carlo methods: https://en.wikipedia.org/wiki/Monte_C...
[Buffon's needle is a first example of a Monte Carlo method: estimating something by using randomness to your advantage!]

*Geometric probability: https://en.wikipedia.org/wiki/Geometr...
[Other problems similar in flavor to Buffon's needle]

*Expected value (also known as "average value") https://en.wikipedia.org/wiki/Expecte...
[The key to our solution was manipulating expected values, and in particular using the fact they are linear!]

*Central Limit theorem https://en.wikipedia.org/wiki/Central...
[Explains how big the typical difference between theoretical and experimental averages are]

*Law of large numbers https://en.wikipedia.org/wiki/Law_of_...
[Another sense in which experimental averages converge to theoretical averages]

*Barbier's Theorem: https://en.wikipedia.org/wiki/Barbier...

*Special examples of constant width shapes: https://en.wikipedia.org/wiki/Reuleau...

Music by Vincent Rubinetti
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjw...
Thanks Vincent for generously giving permission to use your wonderful music for my video!

This video was made for the 3Blue1Brown Summer of Math Exposition (SoME1) (see https://www.3blue1brown.com/blog/some1). Thanks for organizing this excellent competition! See
   • 2021 Summer of Math Exposition results   for the winners announcement!

#3blue1brown #SoME1 #pi #probability #math #maths #MonteCarlo #whypi #animation