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Pi and Buffon's Matches - Numberphile
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- čas přidán 11. 03. 2012
- Our Pi Playlist (more videos): bit.ly/PiPlaylist
Dr Tony Padilla's unique (and low budget) twist on the Buffon's Needle experiment to learn the true value of Pi.
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"We're gonna do Buffon's needle trick! So this is named after Georges-Louis Leclerc..."
Brilliant.
Same as John Churchill, duke of Malrborough, or Arthur Wellesley, duke of Wellington. If you ask someone who defeated Napoleon at Waterloo, almost everyone will say Wellington, not Arthur Wellesley :)
I saw the first part of this video, paused it, and spent two hours of my work day figuring out why it works. Another Numberphilefull day.
For simplicity, lets say the length of the needle (or matchstick) is 1, and the distance between the lines is 2. So, the whole sheet of paper can be partitioned into bands of width 2 centered on each line. And there's a smaller band centered on each line which marks the range of positions where a needle of a given orientation will overlap the line. On average, this smaller band has width 2/pi, because that's the average width of a needle in the direction perpendicular to the lines (from the average of the absolute value of the cosine or sine function.) Thus the ratio of total needles to overlapping needles is just 2 over (2/pi), which is pi.
Personally, I find this way of explaining it a bit more intuitive than giving the double integral, although it amounts to the same thing.
Yeah, I know, the video is from seven years ago, but maybe someone is reading new comments in 2019?
One "coolriosity": if he count 51 or 53 matches crossing the lines, that would give 3,196 and 3,075 as results. So the 52 matches found was EXACTLY how much it should be there. One can accurately guess the number before even count it.. Lets try: 163/PI = 51,88 = 52!!!
First calculation I did as well :)
Your aproach is actually really cool. When I show this trick to someone, I'll try to guess how many matches cross to show it off, instead of simply showing how the result is close to pi
So with 355 matches, 113 should cross the line and give about 6 digits of pi accurately.
Does it also work with concentric circles of radii that are multiples of 2L?
"We're gonna use matches instead because we're cheap"
$400 calculator. lol
Well it's a tablet with a (probably) free calculator app.
The length of the video is 6:28. Numberphile & Tau Conspiracy Theory? An bunch of other videos have tau lengths too.
Ah! But it's 4.49 in the French revolutionary time.
Terry Moore I don't know what you mean. Can you explain please? Thanks!
Max Buskirk I don't know for sure, but judging how the during the french revolution they tried to standardize everything, it is probably just base 100 rather than 60. The length of a second is probably different though. Just speculation.
Max Buskirk
After the revolution the French proposed a reformed calendar and clock along with the metric system. There were 10 hours of 100 minutes in a day, and an hour was 100 seconds. It didn't catch on, though, but there are still some anitique clocks using this system.
Thanks for explaining, Wonder why it didn't work though.
That was probably the most interesting thing i didn't understand
Where can I find an infinite number of match sticks...
Hmm... WALMART
Find em in match boxes
There are simulators on the web, but 100,000 matches made it pretty slow and the result is still not-so-accurate. So I preffer the 22/7 approximation or the digits I have got in memory: 3.14159265359 and these numbers are more than enough to calculate spaceship trajectory.
I think each person is given one at the infinite hotel (if you didn't watch the paradoxes video you won't understand the reference).
in Hilbert's hotel. Ask them to put 1 match into every room and then go over and ask every customer to give you one
okay i expected Buffon saving penalties by doing maths with pi, but still a good pi video though :D
I feel like they fudged this for demonstration purposes, because 52 matches was exactly the best number out of 163 to get closest to pi. 163/51 = 3.196. 163/52 = 3.135. 163/53 = 3.075.
They did work out mathematically that the odds were that they would get 52. So I guess they could have but it's more likely that they would have just gotten 52 as a result.
My Account With No Shame Chance of getting 52 is much smaller than not getting 52.
My Account With No Shame Despite the fact that chance of getting any paticular result in less than getting 52. Thus your assumption is false.
Many things are 52, such as number of playing cards in a full deck of cards, it has 5 divisors discluding it self and the 2nd divisor down from 52 is 13 which is a prime. That would also make it 5 and 2 (52) with divisor properties
Chances of getting 52 from 163 match stick tosses, C(163,52)x(1/Pi)^52×(1-1/Pi)^111 =0.0669 = 6.69%.
That was fantastic. This is one of my favorite numberphiles videos. The fact that he actually did it all out with integrals was very nice.
every video length on this channel is a famous number.
A friend of mine brought a rice dish to the Pi Day party, and when I asked him why, he told about this method of estimating pi, but with grains of rice instead of matches. I thought that was pretty clever (we were supposed to bring food that had to do with pi, which to most of us meant "shaped like a circle").
Awesome! Looks like the duality expresses itself through Pi.
@NesZaStudios glad you're enjoying the films
we had to work on a project about this experiment in school and everything on the internet was just crap so we just copied the easiest formula from wikipedia to at least have something to say (yeah i know, not that clever) but this video actually made it a lot more understandable. At least now I understand the approach of Buffon. I still don't quite understand the whole formula but it's a good start, thank you very much (:
I keep clicking on these videos and going "This is the last one!" 7 minutes later: *clicks video* This is the last one!
I know like these guys alot more. Mainly because they have really interesting videos etc. But I also see an LFC Banner and LFC posters at 3:37 :3
***** You never walk alone!
I'm just starting calculus and was feeling discouraged about the practicality of math. Your videos have given me a richer understanding of math and I have started to apply myself more. Please don't stop what you're doing and thank you for your help.
are you still alive
Very neat..... Thanks for explaining where the Pi comes into play... amusing!
With 163 matches, if 51,52 and 53 crossed we would get 3.1961, 3.1346 and 3.0755 respectively. You got the closest possible (52) on the first go! Did you backward engineer it or was it totally random?
well statistically, 52 is the most likely outcome
Isn't this another argument fot replacing PI with TAU? You would then draw the lines ONE MATCH APART, which would make ths even more beautiful.
I can't see the other replies for some reason, so sorry if this was pointed out already, but the lines one length apart would give you pi/2, not tau.
Nate Reniger I was just thinking that. I was thinking "WOW!" and then I saw your post. Very accurate.
Lasse Trevland He's just really passionate about pi
+Manjyot Sandhar Yeah but tau would make things easier to calculate. Pi has interesting properties, but Tau is more practical.
Pawel X the video length is tau
First watched this when it was uploaded, then again 4 years later, now again 5 years after that and it's still beautiful. See you again in a few years future self, I hope you got your work visa, finished your bcito course and is now on your way to your residence.
what?
@HeyJD123 glad you liked it
how do you know that Px=1/l and Pϑ= 2/π?
look for: "The probability density function of the continuous uniform distribution" and you will get your answer
@@serek987654321 thanks for the hints
@@ankitahalder7245 😁 it's been a while
Am I the only one who expected Gianluigi Buffon to pop up in the video and save some penalties with save/not save ratio of 3,14 :D
no
I love how you always make your pi vids a multiple of pi :)
That is amazing. 52/163 is as close as you can get to pi using only 163 matches. Well done sir.
Lol should have used 22 matches instead
Nice!
+Hamed Khatiz
22/7 is equal to 3.142857 with the last five digits repeating. There is an interesting video about 1/7 called "Cyclic Numbers" on this channel, by the way.
Point is, that number is even closer to π.
22/7, not 7/22
+GroovingPict
Right, my bad. Thank you for pointing that out.
Any multiple of 355 matches would have been even better. (355/113)
suppose there are an infinite amount of match sticks and you counted how many match sticks land on a line which would be infinity and you divide that infinity by the amount of sticks( infinite amount of sticks ). there for infinity divided by infinity equals pi.
illuminati confirmed
not exactly. it is senseless to count exactly how much crossed the line in the first place. you could use a limit though
Buffon himself got more accurate estimates than he should have when taking account of random variation. He didn't use a fixed sample size, but stopped when he had 6 correct digits.
There is a much more beautiful proof that doesn't use calculus. It's based on the number of crossings (which is 0 or 1 if the needle is shorter than the line spacing).
Using the fact that the mean of a sum is the sum of the means, the average number of crossings doesn't depend on the shape of the needle--it could even be a shoelace. And the average number is proportional to the length of the needle. To see this imagine the needle chopped into a large number of tiny pieces. The whole is the sum of the parts. Next suppose the needle is bent into a circle which just fits between the lines. If d is the line spacing the length of the needle is d x pi. There will (almost) always be two crossings, so the average number is 2. Therefore the average number per unit length is 2/(d pi) giving (2 l)/(d pi) for a needle of length l.
love how the lines dividing the paper into 2-match-length-segments conveniently turned into lines dividing the sheet into steps for the math portion of the video.
Apparently, 71 pretzels is not a large enough sample size. I got 3.476...
Lol
+Darren D. the thickness may play a role in that. That's probably why the problem was initially defined with needles
I'm curious how you would calculate tau using this method. Do you just use l instead of 2l? Another reason to use tau imo lol.
You would just multiply pi by 2...
Do'be Eeeval Why?
Using the length of one match instead of two would actually give you the golden ratio.
+The Doctor. simply The Doctor. ... Why?
+MrAlvarogame l have actuallu no idea, l read it on some book about the golden ratio.
Video length is 2pi. I love this channel.
What a beautiful experiment.
Very interesting. An error analysis would have been useful.
Why Px =1/L ??? why P@=2/pi???
Seriously, why explain in detail the integral calculation and not even the variables?
chocolatechocochoco Since the probabilty of both the center of the match and the angle it makes is uniform (and constant). For the center of the match you have to find which constant this is such that the integral from 0 to L (the sum of all probabilities) of this constant is equal to 1 (In probabilty theory the sum of all probabilites has to be equal to 1, in other words if you have a set of events, one of them must happen each time): Integral[from 0 to L] of C = 1 ==> C = 1/L. Similarly for the angle: Integral[from 0 to pi/2] of C = 1 ==> C = 2/pi.
Thor Tunge
Ok thanks a lot, but it was not obvious at all in this video
This is the coolest method for calculating Pi that I have ever seen.
Note that 52 is the closest to pi you can get with 163. 51 and 53 are both farther from pi. So to get any closer, you need a lot more matches. 320/102 is only slightly closer at 3.1372549.
3200/1019 is 3.14033366045.
If you were to use degrees instead of radians you would get a different answer. So does this mean that the universe uses radians?
Radians don’t have units is the short and sweet of it.
no, you wouldn't get a different answer.
1 pi = 3,1415...
1 degree = pi/180 = 0,0174...
regardless which unit you use, after conversion you will get the same value
I fucking love this channel ! ! !
Keep up the good work dude! ! ! ! ! !
@urcritic cheers... Tony will be pleased!
The most deserving 'big channel' ever.
did anyone saw the liverpool fc scarf
I'm sorry but I'm calling bs on him picking up EXACTLY 52 matches. had he picked up 51; 163/51=~3.196 and had he picked up 53; 163/53=~3.075. What are the odds of him picking up the exact number that would give him the closest possible approximation to pi? Im not arguing with the fact that this method gives you pi, but im saying that the distribution of the number of matchsticks he would pick up has a mean of 52 but it also has a standard deviation that would mean the vast majority of the time you wont pick up 52
He was just lucky, I guess. But you are absolutely right.
***** Look, we know that 52 is special. The chances of getting 52 are very small. If you then pick up the nails and count them getting 52 is ridiculously unlikely but getting anything else other than 52 is expected.
Its like someone with a deck of cards who shuffles randomly, then says he's gonna pull out the 6 of hearts. It is expected that he would pull something that isn't the 6 hearts. ie it wont be surprising pulling out a 10 or queen of some suit. It would be surprising if he pulled out the 6 hearts because we have already given it special status. Same as picking up 52 nails
rgqwerty63
Whether he cheated or not the question you should be asking yourself is not how he picked up fifty two matches but why he did not multiply the numerator by 2 or divide by the length of the spacing * the numbers of needles that cross the line but we still have the value pi.
In most text book I have read the formula says that the total needles must be multiplied by 2 then divided by the number of needles that touch the line * the spacing.
My maths calculations
===========
Buffon's needle problem can be performed empirically by first drawing parallel lines one unit of length apart on a plane surface and then randomly dropping needles of unit length on the surface. Multiply the length 'l' of matchstick by 2 and use this number as the numerator of a a ratio. Count the probability of the number of needles that cross the line times the spacing of the line d. Use this count for the dominator.
l = assume that the average matchstick length is 40 mm
d = spacing of the grid was 2 matchstick therefore spacing 2*40 = 80mm
p = 52/163 = 0.3190184049079
pi = 2*l /p*d
pi = 2*40/0.3190184049079*80 = 3.13461538
What is really interesting is he did not use the value 2 nor did he multiply it by the line spacing but yet the answer is relatively the same.
MathsForever I think it is better to use the line spacing length of 2x because a slightly interested, non-math person would find less math involved and not be a skeptic. This just uses 1 fraction to calculate.
***** I'm a bit late to this discussion, but you must note that the number of matches that cross the line do not follow a uniform distribution, where each number has the same probability of being selected, such as rolling a dice. Thus, it is much more likely to pick 52 than say 30 or 62.
And this is not the only video with special length, it applys to many of their videos :)
about to watch 30 mins of video about pie! followed by other numberphile videos no doubt!
@asimath Px describes the probability that it will fall a distance x from a line. Importantly, this distance can be at most L (if you go bigger, you're just closer to a different line.) So, you assume you have an equal probability for 0
Tau minutes long. Love it!
This is amazing.... calculating pi without ever increasingly circular polygons, or actual measurements. just matchsticks and a board.
as he mentioned they used this method to estimate pi when the did not know the value.
also they were demonstrating an unusual method that is surprisingly accurate
@thewii552 well you're on the right channel!
@rauffux glad to hear it
Happy Pi Day. I love these videos.
probability is found by working the area under the curve of the probability function. so essentially you can find the probability of something getting a value between 2 points but integerating, as definite integrals give area under a curve.
More of Dr. Tony!!!!
Wow you guys sure got a lot of skill pressing ctrl+c and ctrl+v!
When I grow up I want to be just like you fellas.
P(theta) is the probability density function (pdf) of the (acute) angle a match makes with the parallel lines. A pdf is a function of a continuous variable, upon integrating which over a range, gives the probability that the value of the variable is within that range. Obviously integrating the pdf over all possible outcomes must give 1. Since the angle the match makes with the lines is completely random, the pdf is a constant and has value 2/pi because the whole range is pi/2.
X's in maths (in England anyway) are written like that, so as to be easily distinguishable from multiplications. It's the same with complex numbers, where 'Z' is written with a bar through it, so it isn't confused with 2's or 7's if written poorly.
Sine (and cosine and etc.) is used in practically everything; from measuring waves to navigation to approximating celestial objects and movement of particles. In it's very basic, it's the ratio of a right triangle hypothenuse (longest side) divided by the side opposite of the angle you're dealing with. the sine, cosine, and tangent of any right triangle is always the same. Sin(60 degrees) will always be sqrt(3)/2. Sin(45 degrees) = sqrt(2)/2
Numberphile and Vihart are getting me excited for when I get to learn radiants!
seems like a fun way to find rational approximations of pi
I like how much of a risk taker this professor is. He was rubbing matches on paper!!
One match length apart would actually give 163/(2*52) = pi/2 = tau/4. Which makes sense since it is the first quarter turn that we are interested in.
That is quite geniuous.
It is possible for the match to be at an angle of 0-pi, but the probability of it hitting a line as we move from 0->pi/2 increases the same as if it were moving from pi->pi/2. 1/2 of the matches will land at an angle of 0-pi/2 and 1/2 of the matches will land at an angle of pi/2-pi. Therefore, we sum them to get the entirety of the matches and the angle (whether it is pi/4 or 3pi/4) does not matter.
I'm 1 minute into the video. I've just realized that I read about this trick in a great book I have that's called _The Greatest Puzzles Ever Solved: 200 enigmas that have challenged mankind, from the dawn of time to the present day_. I don't remember how it goes though, and I'm interested to see it explained elsewhere, so I'm gonna finish the video now. ^^
Once again, I was fascinated by the wonders of Maths .. thought I do not understand the 'difficult maths part'...
I gasped very loudly at 1:48.. my wife was like "OH MY GOD WHAT'S WRONG??" from the other room.
This is the most amazing fact about pi I know. That is amazing!
Mind. Blown.
I remember my old MSX-2 computer had a little basic program that replicated this experiment visually. You saw the lines and the matches on screen. That's how I know of this method.
This makes me think of the central limit theorem. If you have a large enough sample size, the sampling distribution of sample means will approach a normal distribution, and the formula of normal distribution has π in it...
I use this program to figure out a lot of the math they use on numberphile ;).
I absolutely love it. I got it to speak to me the name of the number range it takes to get any digits(the number 3 i.e.) to be in 99% of all the numbers. Geek much.
It was this experiment which is why one of my units of probability in my app is /π.
@SMFApples because there would be no chance to promote it and share it with the necessary people BEFORE pi day...
shops don't start selling Easter Eggs only on Easter Sunday!
correction for some people like me. the video length is tau. btw I love this channel to
I like how the text at 2:37 says 'pie' :)
I am really amused with myself that I thought pi was 22/7 for a long time. 😂😂😂 And then it dawned.
it's a common algebraic substitution... we draw it as two sort of curves to represent an unknown so it's not confused with a multiplication sign...
that sharpie looks very flowing on that paper. I could say that I wanted one but I wouldn't have anything useful to write or draw unless I decided to show someone something which is a great way to is a marker
I mean that it works.. it's mind blowing!
This is one of the coolest math vids I've ever seen. I'm a tauist, so would you get approximately 2π if you halved the distance between lines?
That was intended. When determining probability, you're going to use the inverse of the function.
He also wrote 1/x instead of x as he should of when determining probability.
I really love your explanations guys
it has a simple mathematical background but it is complicated to explain comment box. check out wikipedia for buffon's needle and probability density function.
That Liverpool scarf
Yes, take a 50p coin for example, it has a constant diameter and it has a centre, but it doesn't have a constant radius.
In this article, all sexagesimal digits are represented as decimal numbers, except where otherwise noted. [For example, 10 means ten and 60 means sixty.]
So not really. It's kind of based AS IF it were base 60, but when we say the numbers, and add them, we are using decimal.
For many it's not about understanding, it's about making mathematics as simple and elegant as possible. I think that the circle constant should be circumference/radius, since circles have a constant radius and there are other shapes can have a constant diameter. You even get 2π in the derivation of A=πr^2.
Imposing it in schools would definitely be a stupid idea though.
The urgent length of the video is the circumference if a circle with a radius of 1.
Why do they assume that x takes values only from 0 to L? Why disregard the possibility that the centre of the match is closer to the next line (i.e. x > l) when calculating the pdf of x? And the same for theta, why disregard the angles from pi/2 to pi?
I think the "perfect" or "complete" randomness nature of the distribution guarantees it's not a very special case and none of the matches randomly crossing any line certainly is a special case.
See how he pushes the matches to have a better distribution instead of just leaving several of them on top of each other. That would be a special case as well since certain positions and angles might be favored by the way you throw the matches from one box.
This is REALLY interesting. I love videos like this.
The accuracy is directly proportional to the number of matches (or needles). The number of lines doesn't really matter, as long as they have the same distance between them (2x[match length]).
@XaverSS There's no need to add a C, since the integrals are definite! A definite integral has a number as its output... On the other hand, an indefinite integral (antiderivative), outputs a function, which has undergone the reverse of the derivation process! So, C is needed to compensate for the lost constant in the process of derivation! [(c)'=0 ,c being a constant]
In contrast to what some people suggest, I don't think this works better with tau. Taking a distance of one match (instead of two) between the lines only decreases the ratio of line-crossing to not-line-crossing matches. If you want to get a ratio of tau, you need take a bigger distance between lines.
In case you disagree, please tell me where I have gone wrong
do a video about spigot algorithm for finding digits of pi without computing the previous digits
To get exactly pi, you would have to throw one match and remove it from the board before throwing another. Ideally, you don't want the matches to have any interaction with each other. You want each throw to be completely random and independent.