Calculating Pi with Darts
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- เผยแพร่เมื่อ 13 มี.ค. 2015
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Pi can be calculated using a random sample of darts thrown at a square and circle target. The problem with this method lies in attempting to throw "randomly." We explored different ways to overcome our errors.
A million thanks to Derek Muller of Veritasium for his help with this video. / veritasium . Also a huge thank you to Dan, Virginia, Lara and Cyrus for providing a yard.
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If you're wondering what happened at 3:34 Physics Girl threw a dart over the fence into the neighbours yard right after I indicated this was my main concern with the over-the-shoulder technique.
Throw darts at board. Blindly chose a square less than half the area of the board, draw circle within square, count & calculate. This because aiming at the board will always cause more to be near the center.
You should have edited the video so that happened at 3:14
Ha! Just the other day I remembered this trick from math class
Hey guys, you are missing the hole point. Check out Physics Girl, she is hot. Nicely done Veritasium......
Marianne Corvellec, how do you measure the error in manual error counting? Count it multiple times, and get a distribution based on that?
Are there standard techniques for estimating the error rate in a given image recognition algorithm?
This is awesome! I love how you solved the lack of randomness problem by facing the pattern away from you. Brilliant!
Hey look the boss is back
Hey sup
I really like to see a collaboration video in which vsause, varitasium, physics girl, the science asylum, minute physics, and many other
Big fan of the science asylum
I believe that there would still be an unconscious bias to aim for the center of the paper, thereby skewing (and skewering) the results.
@@nikhilhumane5540 Similar thing happened in Mathvengers on Flammable Maths
Out of curiosity, I wrote a program to "calculate" pi in this way. I chose two random numbers between -1 and 1 and treated them as (x,y) coordinates. If that coordinate was inside a unit circle (magnitude
Vincent Killion I decided to calculate another value using many more darts and runs. I used 1000000 darts over 1000 runs. The value I got was 3.14164. Still only 3 digits of accuracy.
+Vincent Killion Would be interesting to make it throw 10 - 100 darts every millisecond on a very, very big square and watch the number get closer and closer to pi. Don't know how easy that is to program though.
+Vincent Killion This is called a Monte Carlo simulation. They are very helpful when solving problems that are difficult to do analytically, such as proportions between or areas under oddly shaped curves or surfaees.
+Vincent Killion I had to try this for myself. You can reduce the floating point precision error by a factor of 4 by using only one corner of the circle (x,y in the range of 0 to 1). Then just check if(x*x + y*y < 1). This was often enough to give me an extra digit, though it's still only usually 4 digits at most with similar sample sizes. Cool idea though, thanks!
+Uejji I actually read an article a while back where this exact method (Monte Carlo of random numbers inside/outside of a circle to estimate Pi) was used to test how uniformly distributed various pseudorandom number generators are.
Was this math? or chemistry :)
I see what you did there. Haha.
Physics Girl I saw pure chemistry...do this again!
Mmm didn't know the guy, will check out his videos,thks
It's statistics en.wikipedia.org/wiki/Monte_Carlo_method
Arthur Metasov If only my Professor would have let me throw darts at some PK models instead of doing Monte Carlo on R Studio for my undergrad research project...
"trying" to be random is the best way to be sure that the results will not be random.
+Eric Rowland Exactly. True randomness may or may not have clustering. That's like, the definition. A perfectly uniform distribution is, well, statistically predictable.
But I don't think randomness is what you need. I believe an even spread is better, since you are, in the end, using area.
+Knightofmines Correct. They don't need a random distribution of darts. They need a statistically perfectly uniform distribution of darts.
You are so deep I broke my shovel.
MadEngineer That pun tho...
Awesome video! I've always thought about doing this experiment in pubs where random throws are bound to happen over time.
Hi i like your video's to i am subscribe to you
I love you so much
please notice me
Hey look the boss is back!
This comment should be highlighted!
Presh, we want a face reveal.
((Read it in the voice of Honest Trailers' narrator))
STARRING
-Dianna from Physics Girl
-Derek from Veritasium
-Pi from Math,
-Darts and Dart Trajectories from Physics,
- "How our brains are biased towards hitting the central target" from ?Psychology?
AND calculator apps
CALCULATING PI WITH DARTS.
It's a good combination of things!
Aditya Khanna I heard Pi was a real diva on set. Just because the movie was all about her, she thought she could talk down to everybody.
Aditya Khanna i
That was actually a really elegant way of bypassing the distribution bias of the thrown darts... I wish I'd thought of that in the first 3 minutes.
I thought of the same thing at the start. To be more accurate you could use importance sampling while using a bivariate normal to approximate the distribution of the scatter in the dart impacts.
I think we naturally aim for the center of objects which would add a certain amount of bias to the process.
That's why the trick is to use a much smaller sample area than the target. It's why their last trial worked so much better than the others.
Also using multiple sample areas would allow to estimate the degree of bias, to check whether the generating process is random or not.
My approach would be to get drunk first and throw darts next.
Cut the square to exact size. Weigh it. Cut out the circle from the square. Weight it. Calculate pi from the weight ratio.
just like Matt Parker on his standupmaths chanel
But wood would be hard to cut without loosing the weight in wood chips and sawdust. The saw has some width and you cut a lot of material along the line, of which you don't know which side it belongs to.
I believe we can make it happen here in the comments, the technology is revealing :D
Yes, that would have been effective, but c'mon, this was probably much more fun.
I know a few people are thinking it, and there are probably some comments like it.
But I think these two would be perfect together. Heck, It looks like they are having fun together already.
you guys would have really smart children
Use shotgun instead of darts :D 100% random
+Ficraft no. Shotgun will generate pellets centered around the center of the bore.
+Patrick Ireland sawn off shotgun from 40 meters.
An Uzi would be more efficient and random :D
An uzi would be much less "random"
yeah!
It's great to see you guys collaborating, I love your enthusiasm. I subscribed to your channel recently (after I found your video about mirrors) and had already been following veritasium for some time. Great work with that experiment!
Have you considered the error bars due to counting statistics? For a Poisson distribution, sigma = SQRT(n). This can be used as a simple approximation by applying this only to the numerator. However, correct Baysenian errors are the way to go. Anyhow...ALL your results are within one sigma. They are all fine. -- Professor Hemmick and his Intro Physics Class
so proud of you for this observation, keep it up daddy
Any activity can be justified if you shout "FOR SCIENCE" when you do it.
Determination LVL OVER 9000! To be able to actually do all of that in two days must require that much willpower :) I hope you had fun!
Yes, much willpower. But it was a lot of fun!
Physics Girl Much willpower. So Pi. Very fun. Wow.
Physics Girl
So are we likely to see any more Physics Girl / Veritsium collaboration?
I'm so happy to see you two teaming up like this
You have come a long way in such a short amount of time doing these youtube videos Please keep doing what you're doing
It should be great to spent a hole day with you, guys. Since I've discovered Physics Girl and Veritasium, I can be sure that no matter how bad and sad could my day be, I'll ways have a happy moment with you. You can really transmit your excitement about science. Thank you for what you do.
Derek's got so much game
4:57 best laugh in science history
I am seriously wondering how you do not have any more subscribers... You have so many views and all, your subjects are original and your explanations are great. And great video!
That a very round about way to spiral in on the value of pi, but I like (1) the method and (2) showing that perseverance pays off. Great job, Physics Girl and Vetitasium!
Physics girl + veratasium = best couple ever! You two should totally get together :)
I ship it
My new OTP
Dianrek
100%
Deriana
Physics Girl, you are my inspiration! Please keep up the awesome work you do and I'm telling all my friends to subscribe to you 'cause you need to get to million subs soon!
This was such a fun and clever pi day video! I have seen the mathematical proof of this method before but never watched people actually try it. Plus watching two of my favorite science TH-camrs throwing darts was great. :) Happy Pi Day to you too!
I only see chemistry in this video
Chen Yao **salutes hat** good one
QuantumErikGaming fedora*
TheUltomato damn
huehuehue
@@Ej7600 what's wrong with a hat?
Several of my students have volunteered to help you in making a video to correct the treatment of the error bars...
Cool idea! I'm thoroughly impressed by the two days of dedication that went into making this.
I steel can't believe that you got Veritasium to do this with you. I love to watch his videos .
Happy Pi Day
Great work !!
*Everytime someone posts a Pi-day video, I am like "Oh, Vi Hart's gonna be all over this... "Why not Tau/8"!"*
"they seem to be centrally distributed"
That's exactly the reason
every realization of a random throw is the sum of multiple random variables with uniform distrubutions, which result in a normal distribution centered around the center of the board.
if your circle area is smaller than a standard deviation of that normal random variable you should get a better result, which is exactly what they did at the end by trying to eliminate the psychological factor if you subconsciously trying to aim at the center of the board.
yes, one can actually derive the normal distribution's pdf solving a functional equation that comes from the assumption that we are throwing darts at a bullseye, in which equal areas are more likely to be hit the closer they are to the center.
I liked at the end how you showed the date for when you made it and it was pie! ;D I love those little things people point out that is totally awesome! :D
Great job! Interesting. Excellent job of randomness acquisition with dart throwing... That was indeed a tough task. Very well done!
Looks like love.
3.14.16 is a better estimate
*****
Given two dates 3/14/15 and 3/14/16 (last year and this year), the latter is the better estimate of pi if its digits are taken as 3.1416.
Two dates bracket pi this way every 1000 years.
+Ariyan Ahmed I yield the issue.
Yes, on 3/14/15 at 9:26:53.8979323846264338327 A.M. In that extended sense a better estimate.
+Ariyan Ahmed Firstly, 9:62 is not a time of day. Secondly, George Steele was right, it's 3.141592653...
+Ariyan Ahmed 9:62 would be 10:02
+Ariyan Ahmed 3.141563?
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787083039069792077346722182562599661501421503068038447734549202605414665925201497442850732518666002132434088190710486331734649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007230558763176359421873125147120532928191826186125867321579198414848829164470609575270695722091756711672291098169091528017350671274858322287183520935396572512108357915136988209144421006751033467110314126711136990865851639831501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064204675259070915481416549859461637180270981994309924488957571282890592323326097299712084433573265489382391193259746366730583604142813883032038249037589852437441702913276561809377344403070746921120191302033038019762110110044929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915441101044682325271620105265227211166039666557309254711055785376346682065310989652691862056476931257058635662018558100729360659876486117910453348850346113657686753249441668039626579787718556084552965412665408530614344431858676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797270826683063432858785698305235808933065757406795457163775254202114955761581400250126228594130216471550979259230990796547376125517656751357517829666454779174501129961489030463994713296210734043751895735961458901938971311179042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043063321751829798662237172159160771669254748738986654949450114654062843366393790039769265672146385306736096571209180763832716641627488880078692560290228472104031721186082041900042296617119637792133757511495950156604963186294726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939780541934144737744184263129860809988868741326047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is a much better estimate
Another awesome way to find pi! Your excitement makes it so much more enjoyable =D
VERY COOL ! Just had my physics lab exam toady ahahha i'm not bored by these calculations and attempts to reduce systematic errors at all :D
What about using rain falling on a big target with the same shape, as the random generator ?
You could also just throw sand and weigh it.
Why are so many comments about their relationship status?
Two attractive people doing something together? Are you kidding?
LeiosOS dude they’re like two young lovers who like physics and do stuff together
If they even have a relationship that transcends basic fun you mean, I have no idea but if they want to bump uglies it is no business of ours.
Veritasium is married....
It's because the camera
Nice job addressing the bias. Since there's something of an inherent tendency, I think, to try to hit the target, turning it around so the circle's not visible was my first thought, but the final solution was very well done, indeed. Of course, since Derek's the one who drew it and had an idea of what it looked like on the hidden side, I think it's safe to attribute the residual error to him. :-)
Glad to see you two collaborating, hope he brings you a bigger audience!
Wow the two of you should really do more vids together man
How'd you go about calculating the final result? The average of all pi-results from the final board? The video was great :))
and I ship it (provided that you're both single)
***** creepy, the both of you
Sorry, Veritasium is married with two kids...
Nice video guys. I was thinking that using darts was a little strange as you'd always intend to hit the centre of the target but with the pattern on day 2 you seemed to find a solution for this bias. Pretty accurate too!
This is how to learn and having fun!!!
You guy's are genius!
they look like a good couple
Pi day should be next year because if you round up you get 3.1416
Pi day is every year because you usually round it up to 3.14
Lol :) apple pi :P
Oh gosh! I missed pi day! Happy to see both of you in one vid. More please! :)
Great video! I am already subscribed to your channel. Looking forward to more videos :)
I'd have thought the very fact that you are trying not to miss the board means the closer to the center the more likely you are to hit it and the further away the less likely.
Why randomize? Why not superimpose a fine grid of dots over the square to ensure even distribution and the get that ratio?
+Eric Pommerer i thought the same thing
Impressive! I've got to say it's quite clever to combine the concept of probability into getting an approximate pi.
This is so brilliant! I really enjoy everything :) so awesome
And on the third day, physics girl said "Let there be Pi," and there was Pi, and she saw that the Pi was good.
Especially apple pie
you two are a good couple
You inspired me to do that experiment as well - although I wrote a program to do all the darting and calculations for me (so I didn't get to throw actual darts). I let the program throw 10^10 darts, and the difference I got between the real value and the experimental value is 1.5211E-05 (the standard deviation is 1.6422E-05). Awesome.
Did a simulated version of this experiment with my high school math students. As I recalling had 16 students (I teach at an alternative school for at-risk kids, so classes are typically 12 to 24 students). Each student created a spreadsheet and generated a thousand (x, y) pairs of uniformly random numbers from 0 to 1 inclusively that represented the impact point in the first quadrant of the unit-radius dartboard. We then calculated the length of a segment from (0, 0) to the point of impact. If the length of the segment was 1 or less, we scored a hit, otherwise t was a miss. We tallied the hits and like in baseball found a hitting average. I remember how the kids were so surprised by the precision from a random experiment. It was a great teaching moment about how individual events can be random but in their aggregation be very predictable.
You two would make a good couple.
I ship this. LOL
HAPPY PI DAY EVERYONE!
A clever idea and a nice challenge, I loved it.
I search and come back to this video just to see the best physical couple on youtube.
Omg half of these comments are just shippers 😂 but you would make a cute couple :P
Is there romance in the offing?
I've noticed a couple collabs now between these 2, & I've noticed the very flirtatious banter & the telling smiles & looks being exchanged.
💖
1. Be skeptical of a claim. (*Can* you calculate pi this way?)
2. Formulate and perform an experiment.
3. Check your results against accepted values.
4. Do they agree? If not, re-evaluate *your* experiment to remove bias.
5. Repeat steps 3-5 until you are convinced your experiment is correct.
6. Here you confirmed the result, but maybe you find a situation in which the original hypothesis doesn't hold.
7. Science!
This is why I thoroughly enjoy your videos. You took a weekend(?) of of hanging out with friends, added a simple activity, and turned it into real science. Science doesn't have to be boring (and it nearly never is!).
That works as long as you aren't Hawkeye, or exceptionally good at darts. I very much like the blind repeating pattern to achieve randomness. Nicely done. In computers, getting a random distribution of points inside a circle just means taking a random x,y and discarding the ones that lie outside the circle. What is inside the circle is uniformly random. I use this method to generate a random spherical starfield, avoiding the annoying extra collection of points at the corners.
i ship you with Derek!
I ship this soooooo hard
Awesome! I would have never guessed it would have worked so well!
That was fun to watch. Keep it up.
They totally bonin'
For SURE
the cutest nerds ever
Peter I am sure they made Pi that night in different positions
i ship them both so hard.
3.139 is totally amazing. Nice work!
wow!!! you are an awesome physics teacher! great job! Go! Go! Go!
It would be cool if they had pi day in the year 1592.
March 14th didn't cease to exist in 1592 so certainly they did
Why don't you guys create a computer program that does this randomly? You can increase the samples by thousands to get more accurate results.
Psh.
Boring.
This was clearly inspired by monte carlo simulations
That would be silly. If you're going for accuracy, you don't need to be random at all, you just need to have a lot of dots, they can be in rows and columns. If you want to be more practical only draw the grids densely near the perimeter of the circle, and weigh those dots less accordingly.
You need a "uniform distribution." Either a random spread or a normally distributed pattern will work in the limit of infinite dot density.
Wow ! This is an awesone idea !
I am going to try this at home !
cool! and nice to see 2 of my fav youtubers working together
how many layers of irony are you on
You guys should date. Then get married. Then make beautiful physics babies.
one word come to my mind here - "sapiosexual"
Doesn't exist
+TheDominickel I mean he used the word right there, so that probably exists...
+DodecagonYT Regarding the definition, I doubt it applies here. Her look is quite fine too.
+TheDominickel what if a person is blind? Or maybe just not shallow... Though I must admit pretty much everyone has this bias
You will tend to have more darts inside the circle when it is the center of the target you're aiming at! In fact it'll be a 2D Gaussian distribution. What you want is a constant distribution, which is closer to what you did at the end of the video. Another solution would be to just make a 2D grid of dots spaced 1cm apart and that would have worked very well! If you had infinitely many dense dots, then you'd be computing the exact area and would get the exact answer.
I’m really going to have to start watching these videos before I start drinking. I’ll definitely bookmark this one for sober viewing. Exceptional video, PhysicsGirl!!!
yeah
3/14/15
month 14 yes yes
In English they say March, 14...
+Paulo Santiago maybe he not learn english.
+danish N.S Maybe LOL
You both are wrong, I have been studying english for almost 7 years, I just didn't know how dates are written or said.
And danish N.S, maybe the one that "not learn english" is you :).
+Cochu U Such a basic stuff... But okay, everything is fine now. No need to be angry, right? I'm not angry and you are not too, right? ^-^
This can be done much more accurately with a computer simulation. In fact, I wrote a small program after watching this video, which selects 131,072 pixels randomly from a 2,048x2,048 pixel square to. After taking the average value of 5,000 iterations of the simulation, the value I got was 3.141590, or within 0.00006%. Interesting idea!
I loove it seeing these big science and math channels collaborate on a single video.
I always had a feeling and knew you would eventually do a video with veritasium
they would have beautiful, really smart babies
What if you got a square container and a cylindrical container, then put the cylindrical one inside the square one and wait for it to rain. Then take the ratio of the volumes of water multiplied by 4.
Great idea
Nice Demo! Monte Carlo simulation 😊
The more darts we throw, the closer our estimate of π will be to its true value.
oh cool! I never thought i would see you 2 together! cool project
I think you mean happy tau/2 day
Didn't know what tau was till now. Thanks!!
The funny thing I find about tau is that it is exactly 2pi. But its symbol is exactly pi/2 😂
Diana Derek fanfiction?
Amazing featuring!
I've watched everything from Veritasium and seen many collaborations with Derek, how come I didn't learn of the existence of your channel before MinutePhysics shared your video on mouth clouds...
You guys are so cute together, I'm NERDING OUT JEALOUS of your chemistry. Nice!
In the year 3141 it will be Pi Year!
or it already passed in year 314
> Then *a thousand* years after that it's gonna be 31415
Are you sure about that?
> Calm down.
The only way I have to be more calm would be to die.
> I was exaggerating dude.
But that's actually much more then one tousand…
This proves counting is quite difficut to you…
My life does not depend on your counting abilities.
Happy pi day everyone. And btw the video was great. This was actually my first physics girl video. I had been watching Veritasium's videos 😊👍👌👊
The way you try to eliminate systematical error is brilliant.