Why do colliding blocks compute pi?

Even prettier solution: • How colliding blocks a...
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/clacks-thanks
Home page: www.3blue1brown.com
Many of you shared solutions, attempts, and simulations with me this last week. I loved it! You all are the best. Here are just two of my favorites.
By a channel STEM cell: • A solution to 3Blue1Br...
By Doga Kurkcuoglu: bilimneguzellan.net/bouncing-c...
And here's a lovely interactive built by GitHub user prajwalsouza after watching this video: prajwalsouza.github.io/Experi...
NY Times blog post about this problem:
wordplay.blogs.nytimes.com/20...
The original paper by Gregory Galperin:
www.maths.tcd.ie/~lebed/Galpe...
For anyone curious about if the tan(x) ≈ x approximation, being off by only a cubic error term, is actually close enough not to affect the final count, take a look at sections 9 and 10 of Galperin's paper. In short, it could break if there were some point where among the first 2N digits of pi, the last N of them were all 9's. This seems exceedingly unlikely, but it quite hard to disprove.
Although I found the approach shown in this video independently, after the fact I found that Gary Antonick, who wrote the Numberplay blog referenced above, was the first to solve it this way. In some ways, I think this is the most natural approach one might take given the problem statement, as corroborated by the fact that many solutions people sent my way in this last week had this flavor. The Galperin solution you will see in the next video, though, involves a wonderfully creative perspective.
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
Timestamps
0:00 - Recap on the puzzle
1:10 - Using conservation laws
6:55 - Counting hops in our diagram
11:55 - Small angle approximations
13:04 - Summary
Thanks to these viewers for their contributions to translations
German: Greenst0ne
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
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Website: www.3blue1brown.com
Twitter: / 3blue1brown
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Facebook: / 3blue1brown

[@3blue1brown]

Some added notes (copied from the pinned commend to the next video): 1) Some people have asked about if the tan(x) ≈ x approximation, being off by only a cubic error term, is actually close enough not to affect the final count. It's actually a very interesting answer! I really went back and forth on whether or not to include this in the video but decided to leave it out to better keep things to the point. This difference between arctan(x) and x could be problematic for our final count if, at some point when you're looking at the first 2n digits of pi, the last n of them are all 9's. It seems exceedingly unlikely that this should be true. For example, among the first 100 million digits of pi, the maximal sequence of consecutive 9's has length 8, whereas you'd need a sequence of 50 million for things to break our count! Nevertheless, this is quite difficult to prove, related to the question of whether or not pi is a "normal" number, roughly meaning that it's digits behave like a random sequence. It was left as a conjecture in Galperin's paper on the topic. See sections 9 and 10 of that paper (linked in the description) for more details. 2) A word on terminology: I tend to use the word “phase space” to describe any space like the ones described in this video and the last, encoding some state of some system. You should know, though, that often in the context of mechanics, this term is reserved for the special case of a space which encodes both the positions and the momenta of all the objects involved. For example, in that setting, the “phase space” here would be four-dimensional, where the four coordinates represent the position and momentum of each pair of blocks. The term “configuration space”, in contrast, just refers to one where the coordinates describe the positions of all the objects involved, which is what we do next video.

[@diegovargas3377]

It easy t is 50 % of the 5 of the

[@Kernel15]

"For example, among the first 100 million digits of pi, the maximal sequence of consecutive 9's has length 8, whereas you'd need a sequence of 50 million for things to break our count!" As pi is irrational, doesn't that mean that every possible number can be found in there as there is no point at which the digits of pi would start to repeat? Sure, it would probably be too far in to matter by then, but yes, the last n digits of pi being 9 would be true at some point.

[@MAYOJAMESD]

@@Kernel15 I'm not sure how true this is. Just because a sequence of numbers is infinite does not mean that it contains every imaginable combination of numbers. For example, there is an infinity of numbers between 1 and 0. There's .0001, there's 1/3, you got pi over ten, etc. Yet, even given the infinite count of numbers you get between 0 and 1, you can never get any number larger than one. There is an infinite number of numbers, yet not all numbers are included. So even though Pi is irrational, I don't think it necessarily means that you get every possible combination of digits. In fact, since by the very definition of it being infinite we can never create all of its digits, there will always be some combination of digits that we cannot give a proof for being included in pi.

[@mustafasoykut2510]

Get 4000000 subscribers plz 3Blue1Brown

[@atlas0dan]

I'm wondering whether there is a relationship between the next digit of pi and the momentum of the two blocks after reaching the maximum number of collisions

[@SimonClark]

This is one of the most beautiful educational videos I've ever seen.

[@TMHUpNorth]

Oh look, it's Dr Simon Clark.

[@TheSharkyBoyCostyn]

Dr physicist, atmospheric physicist

[@niccy266]

Yooo Simon Clark

[@denisfan1055]

This is sooooooooooooooo confusing, Yet Amazing science/physics-wise.

[@benjiboi69_vlogz87]

Hi

[@mgsquared5204]

It’s interesting to note that in the 64 kg example at 0:28 it actually computes the first 5 digits of pi in binary. 2^(6-1) gives the first 6 digits of pi in base two! This method works for all bases!

[@deldia]

Haha mental

[@magica3526]

yep! that's why it's powers of 100, to convert to base 10

[@OriEy7]

It actually does not work for base 7

[@fedoraguy6774]

@@OriEy7 it most certainly will

[@OriEy7]

@@fedoraguy6774 My comment wasn't very detailed. It only give the wrong digits when you only calculate a few (1 or 2 maybe?). But then it works like in does for base 2 and 10. I did simulate it a long time ago. This basically happens, because Pi in base 7 has a 'long' run of 6s: 3.06636...

[@______6879]

This video has lived rent free in my head for 4 years now. I literally think about this every 2-3 weeks. Congratulations

[@RegularFantaTheBest]

Lmao

[@vampy_noah1659]

Tbh I watched this vid like 2 years ago and did not quite understand the equation. Now that I am older and almost done with high school, I understand the conservation of energy and momentum and we literally just had it in our recent physics class. This is why I love physics, it always has a relation to something. Whether it is mathematical or irrational and just a fun fact. It blows my mind everyday

[@rando_guy]

same. it feels more amazing when start understanding what's going on

[@leastofmyworries971]

Once you take university calculus, this makes 100% sense.

[@rando_guy]

@@leastofmyworries971 yep and classical mechanics

[@MeMyself_andAI]

Mathematics is the language of our universe. The better you can speak it, the better you can communicate with it.

[@yourbigfan1777]

Same 😎

[@akshaytiwari9348]

You experience ultimate happiness when you see the links between two completely unrelated topics.

[@yourlordandsaviouryeesusbe2998]

That is correct. Atleast they *seem* completely unrelated before we investigate further.

[@PKIVV]

> Every conspiracy theorist

[@yuvrajsinghraj]

Everything is related we cannot see that easily

[@jaydonnell]

What do you know connection is the key to happiness

[@1996Pinocchio]

@@yourlordandsaviouryeesusbe2998 By invastigating, we divide. Before investigation, everything is one experience.

[@gogl0l386]

Grant I don't know if you're aware of this, but you're actually changing the world. The next generation of mathematicians will be a bunch of people inspired by you. You're absolutely a master of presenting and visualisinh beautiful proofs without the need of advanced mathematics.

[@aidanokeeffe7928]

Amen! If I never found this channel, I wouldn't be a math major today.

[@TheSkepticSkwerl]

I still don't know what theta is.

[@notnilc2107]

@@TheSkepticSkwerl Wait til you hear about different counting systems. I questioned my understanding of numbers for a whole year after I seeing the base 12 counting system video on numberphile.

[@pedronunes3063]

I was in the Brazilian Olympic Week (Semana Olímpica) where who won a medal in the OBM (Brazilian Mathematics Olympics) go to a place to have classes and meet other medalists (actually I didn't won a medal but I got so close that I was called) and there many people knew 3b1b's channel. So I guess... greatings from Brazil.

[@floridaman6982]

I littererly got interested in higher maths by his videos and just searching the web. It amazes me how this stuff was discovered before computers

[@SpiritmanProductions]

I'm not a maths expert but I do love these videos because I get an intuitive sense of understanding, even when I don't quite grasp all of the proof's finer points. And the animations are amazing. Great work, and thanks for posting.

[@Lolo_Antonio-FryEmUpFan]

I didn't understand anyone

[@Ar_sole_hair_fiddler]

You've got that "listen to this tape in your sleep" technique

[@blakebodycote1024]

if there's anything I've learnt from my maths, statistics, computer science and data science courses is that visualisations of the relationships between numbers is most definitely the future of teaching mathematics. It's videos like this that show how describing concepts with visual elements make teaching mathematics exponentially easier and I cannot wait to see how much more it becomes integrated with learning.

[@Cyba_IT]

Yup, totally agree. When I was at school decades ago there would be a diagram in the textbook and they'd explain the equation but it was hard to visualize exactly how it worked. Now kids have this.

[@these_handles_are_stupid]

Agreed, I am terrible at math and finished high school only just after visual elements started to become a thing, so it was not easy for me at all. A visual means of explanation makes math not only easier to understand but easier to appreciate and wonder at. It really is kinda cool sometimes. I wish I had this sort of thing in school. It would have made it feel so much less of a chore.

[@shadowmancer7040]

Sadly politics and lawyers and other worse things have instead derailed our education system into a glorified day care and we end up with high school graduates that can barely read, can't write, and can't do math. Woo common core!

[@austinhernandez2716]

​​@@shadowmancer7040Thank Republican Governor Sonny Perdue for that.

[@Leto_0]

​​@@shadowmancer7040 I say half the reason is the baby boomer generation's _wonderful_ hip new idea of "hands off parenting", where they sit back and let he television raise their children, and "give them the freedom to make their own mistakes". So nice of them to give us that freedom. Who needs a parental figure to teach them the value of education? Oh and while we're talking about role models... how do we expect teachers to inspire and educate students if they're paid so little that the only teachers left are bitter, apathetic, and generally not super intelligent? Just like with public servants, we NEED them to better, so we must pay them better and demand better standards.

[@henriquecoelho6594]

Thank you kindly for existing.

[@jokinglimitreached1503]

Well, that's a comment you don't see every day. Gotta remember that

[@batboyhood3]

I agree

[@Milxfanta]

Oh comment= 4 years ago, latest reply=4 days ago

[@sifilo]

You too!

[@rivertaig8703]

What I really appreciate about this channel - and it's so well exemplified in these last two videos - is that through the creative use of animation, geometry, and well thought out naration, you can spark in a non-math major like myself not just understanding of what would otherwise be esoteric and unapproachable concepts, but genuine excitement. I'm turning 50 this week, and I'm finding myself wanting to go back to college and get a degree in mathematics. This was superb. Thank you 3,141,592,653 times.

[@jpe1]

River Taig I turn 50 in three weeks and feel similarly. I’m not unhappy how my life has gone so far, but watching videos like this reinforces the fact that the only reason I dropped out of math classes was because they were poorly taught, not that I was stupid, and I imagine I will feel satisfaction if I go ahead and return to college to succeed in math classes where I previously failed. Good luck with your efforts.

[@Tony-nl6pf]

@@jpe1 You're absolutely right to want to learn but I wish there was a better place than our broken system.

[@bourgeoisremi9778]

It's never too late.

[@grinreaperoftrolls7528]

It's not too late to go back to school! DO IT! I am a physics major. I'm still deciding what I want my 2nd major to be.

[@ytkaci]

I can vouch for you about how this guy allows us to see stuff which at first seem so complex yet shown so concisely and logically. I'm 15 and this made almost perfect sense to me.

[@iamkiubi]

I understood precisely none of this, but it got me so interested that it makes me want to learn maths beyond what i did in high school. Great video!

[@krishgarg2806]

I watched this video a year ago, when I didn't know anything about, equations of circle and straight line, conservation of momentum and energy, and collisions. Now I have covered all the above topics, so I can finally say that I understood the video.

[@jeffreyleonard7210]

You came a long way in one year. That is excellent. Your progress + the value of sharing information + mentors + ethics = hope for a positive future. Yes, you can be that important.

[@Niganigaballsackboii]

ur in 11th studying for jee arent u?

[@krishgarg2806]

@@Niganigaballsackboii 😂😂 yup

[@gaquekk]

Yeah, i am in second grade of high school on brazil and i don't undertand almost nothing of the math but i hope i undertand onde day lol

[@jaiyank]

Didn't understand most of it but loved the sound of that collisionsss.

[@daniilgrigoryev6204]

Lol

[@jacobwright4653]

Have you set your calculator to base 10?

[@kishorekumarsathishkumar1562]

REEEEEEEEEEEEEEEEEE

[@anyonegotasnickersbar]

Have you set your calculator to calculate mode?

[@Cp.Chaos707]

Same?😂

[@welovfree]

11:51 That ghzzzzzt!! sound is satisfying for some reason.

[@user-cn5pm7zg1u]

Deja vu someone?

[@VENOM-tx6gp]

11:52

[@luucvinky2194]

welovfree yes

[@ericsbuds]

agreed :D

[@_trupples]

10h ASMR blocks colliding yes please

[@steelcash]

So glad I discovered your channel. Your contribution to the understanding of science and math is priceless. Thank you.

[@ABCDCBA1]

Hello! I'm a Korean student who loves your videos! I don't know how many times I've seen this video of you. When I was in the 3rd grade of middle school, I memorized it without any understanding. It was fun even though I didn't understand it at the time, but now that I'm in my second year of high school, I think I understand the principle a little bit, so I'm very happy If I learn a little more math, one day I can understand all the math videos in your videos, right? Your video is healing for me. Thank you for posting this video There may be misinterpretation using a translator I'm sorry😢

[@stemcell7200]

Thanks for mentioning me! Your videos are beautifully explained and if I had remembered the inscribed angle theorem I would have been saved a lot of trouble in my solution :)

[@3blue1brown]

It was a great video, and you made it so quickly too!

[@maximan33]

@@3blue1brown thank you very much to explain many mathematical and physical problems deeply and out of box. Unfortunately many professors don't do

[@AalapShah12297]

I faced the same issue. I got stuck after I moved to the phase space and and represented the collisions as points along a zig-zag line. I was trying to get to pi somehow using area but the solution using inscribed angles is pure elegance.

[@kabivose]

Of course no one is going to like that comment any more!

[@kabivose]

They did! :-( 314 was perfect.

[@omooba00]

i always though i loved math, college taught me that i didn't, videos like this remind me that i did

[@joserafael9913]

same here, the struggle is real.

[@lordMaroza]

Where I'm from, kids work for the grades, not knowledge. That why we end up hating stuff that should be really, really fun.

[@TechSupportDave]

@@lordMaroza i'm pretty sure it's like this everywhere. people never do it for the knowledge. i did, and now i'm having the time of my life programming math-related software at home. and at first i hated maths. but the reality is, only the beginning is tedious. once you get into the intermediate or advanced mathematics, things start to become awesome even if you can't do much with that knowledge alone. it's videos like these that remind me that i love programming. this video even made me get back to doing what i used to constantly do. i'm currently recreating the collision program they made in this video and i'm looking forward to finishing it.

[@ekremdincel1505]

@@TechSupportDave Good! I made my own similation with python too. Did you see Code Train's video about that? He made it in javascript.

[@nyeeeeeee9346]

Same tbh, college is killing me atm

[@chimkinnugget7134]

Im going to be a physics major this upcoming year, and your videos are so interesting to me, and part of what inspired me in the first place. Love your videos! Thank you!

[@zeynolabedinsoleymani4591]

Watching your content reminds me good memories of my high school when I was preparing for math Olympiad. Very good days with lots of joys with solving challenging problems

[@blackpenredpen]

Purely amazing!!

[@redaabakhti768]

senpa

[@Albkiller22]

Heey blackpenredpen really happy to see you are a fan too!

[@bapolino733]

blackpenredpen YAY!

[@soufian2733]

CHEN LU

[@soumaknandi9218]

Oh my god! All my favourite channels are here!

[@GMPStudios]

This video is mechanics, calculus, trigonometry, geometry, co-ordinate geometry all in a nutshell!

[@Chungus67]

Welp, you just summoned kurzgesagt.

[@GMPStudios]

@@Chungus67 oh included "In a Nutshell" xD. But I don't think they'll do classical mechanics videos

[@ganondorfchampin]

Kurzgesagt is only smart enough to under half the things said in this vidoe.

[@GMPStudios]

@@ganondorfchampin Kurzgesagt is more of an inspiring channel than an educating channel.

[@colinsoileau5033]

And tbh if you think about it from an engineering aspect, vibrations and harmonics.

[@sonic5d]

I remember the humble beginnings of this channel vividly, with Euler's identity being the spark that first guided me here many years ago. Your adorable pi mascot has continually played an indispensable role, often unexpectedly appearing in the most surprising places. I'm incredibly grateful for the consistently stunning visual content you produce, which has succeeded in shedding light on complex ideas in an extraordinary way. Thank you for your tireless efforts in bringing such high-quality content to your viewers.

[@dasburstling]

I am not at all interested in maths, but this is definitely one of the coolest videos I’ve ever watched

[@funnyhackerman]

realest comment ever

[@JulianLudwig-uq8vq]

Is´nt that a contradiction?

[@bobtivnan]

The most brilliant part of this is making the connection between a physics formula and geometry. I'm a geometry teacher, and it makes me smile to see so many rich geometry concepts applied in a totally new way.

[@JackRule16]

Mean while my geometry teacher was out there clubbing during my high school days. Your kids are lucky to have someone who still wants to learn.

[@antares-the-one]

There is another way of geometry. Can be possible, that our entire world is just 11 dimensional geometry

[@ploniofludrasco2705]

Glad to read this, as I was stuck on this problem at the point just after drawing the circle and the lines 😅

[@Zghost276]

Of course. Geometry is commonly applied to physics. Such as finding net forces can require you to use cosine or sine of angels.

[@BlueyMcPhluey]

Mohr's Circle is another example of physics concepts visualised on a plane

[@tom_something]

0:51 - If your Geiger counter makes this sound, take your iodine.

[@arienkano6002]

Not good not bad either

[@KingHalbatorix]

that's when you take out your 9mm You do NOT want to end up like the japanese guy who got lethal radiation poisoning and lived for almost 3 months with his organs literally melting

[@Kasser_KSR]

@@KingHalbatorix Wasn't that guy kept alive by doctors who wanted to do some experiments on him or smth (his name was Hisashi Ouchi btw)

[@krozjr5009]

3,141,592 collisions. Not great, not terrible.

[@akasakasvault7597]

hum. well i guess i need to get to my local drug store.

[@lilcyolo]

More teachers need to use you. I tried studying using my teachers method and got nowhere. I watched a couple of your videos and now I feel like I actually understand it.

[@flyingbirds6794]

This is truly one of the BEST videos on KZbin, maybe the best one!!! Thank you so much!!! What an excellent explanation

[@mohitkulkarni943]

I like this idea of giving a problem and then introducing the corresponding topic. Thinking about the problem and then watching the solution actually interests me more than watching a video on a random topic. I really hope you continue this..

[@mdtalhaansari1096]

Hope this becomes a trend.

[@jinjunliu2401]

I honestly love both just as much

[@fpereyra]

Seria interesante que haga un video acerca del problema de la braquistocrona en la esfera. La solucion, para nada trivial, es una epicicloide

[@rokus1145]

Oh yeah yeah

[@poopcatapult2623]

Okay I need to reinstall Deus Ex.

[@BigDan85]

I am always stunned when I see such content. You must be so gifted, when you - are able to understand this physics stuff - are able to talk about it in a way that others can understand it AND still keep it at a high level - are sooo good at animating and editing - have a great voice that people like to listen to IHML

[@mikefraser4513]

I am always jealous.

[@nahema9079]

True his voice is cute

[@matheuscabral9618]

he is not “gifted” he is skilled, he learn all that

[@r1pperduck]

​@@matheuscabral9618statistically, not everyone has the ability to aquire these skills if they tried. Its both.

[@matheuscabral9618]

@@r1pperduck what is your point? Sure a lobster can’t learn physics, but anyone without a major disability surely can

[@chezlizzle]

Highly recommend for any classical mechanic enthusiasts. Great video.

[@GeorgeZoto]

Beautiful, just beautiful 😊 The connection between dynamics and geometry is beautiful 😍

[@adude6568]

As Archimedes once said, "give me an infinite mass object and an immovable wall and I can compute all of pi" Wait...

[@anveshkhode3794]

Did he predict Joker and Batman tho

[@aloysiusvo318]

Did he actually say that?

[@Soarvivor]

No, the real quote is “Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.”

[@LukasAnell]

lol

[@Win090949]

Finally, we know what happens when an unstoppable force meets an immovable object

[@diegokoivukangas6461]

I have no idea what you're talking about but damn I love those collision sounds

[@LiamList7]

Amen

[@NedInYaHead]

Which is funny, because the reason it's so satisfying probably has something to do with the fact that the frequency results in circular shape, exactly what 3b1b is explaining.

[@LiamList7]

@@NedInYaHead …circular shape?

[@samuelcrawford8055]

Thank you very much for this Grant. Phase space is often indecipherably defined by places like wikipedia so this was incredibly helpful and clear.

[@ThorHC11]

This is sickeningly beautiful. Grant, your content is fabulous.

[@aditt.7475]

3:14 - "We are on a hunt for PI" Notice the timestamp...

[@high2407]

I refuse to believe that's an accident lmao

[@sabiti5428]

This whole thing reminds me that everything is Pi, and therefore waves.

[@Sparsh_05]

Damn

[@binarywizard69420]

Ah ahhhggggggggb

[@playboibcrazy3290]

@@high2407 me too

[@user-hh4xs7ml7s]

Can't wait to listen to 15 minutes of that sweet sweet clacking

[@hecko-yes]

[asmr] calculating pi

[@viniciuslambardozzi4358]

10 hours of blocks colliding

[@cielvague]

Agreed. We need to organize in order to get our hands on that stuff. I only see 2 options : 1. Find out where 3Blue1Brown lives and kidnap him to some overseas basement where he would be forced to make colliding blocks animations for the rest of his life in order to satisfy our now insatiable addiction for high-frequency clacking 2. Get a similar sound file and actually code and upload the damn thing on github. Anyone volunteering for option 2 ? (or option 1 if it's more to your fancy)

[@JNCressey]

Come on 3b1b... give us a slower 15 minute simulation of clacking. :)

[@jiffylou98]

ahh, I remember going down into my grandmother's irradiated childhood basement in Nagasaki and hearing that oh so satisfying sound on the geiger counter as rogue hadrons pierced through my DNA. now I have a tail.

[@danielthemaniel3856]

What a whimsical and inviting nature!

[@DylBilch]

I Can’t explain how amazing these videos are. I failed precalc in college and wrote off mathematics completely but somehow I can understand was you talking about and I can follow in your videos. I’m having epiphanies just watching.

[@meeksource4047]

I love the awkward wait for that last click

[@jwadaow]

OOf... awkward

[@Shubham-qk8fw]

That clack sound is just from another world. So satisfying 👌

[@aidanokeeffe7928]

It sounds like a Geiger counter

[@Shubham-qk8fw]

@@aidanokeeffe7928 Exactly

[@baguettegott3409]

Nooooo the sound is horrible! It's fine if it plays just once, but if a lot of these clacks play very fast it makes me really squirmy and uncomfortable, I hate it

[@chloroplast8611]

Baguette Gott ocd

[@aslaksockcraft]

The clack violates the laws of thermodynamics.

[@theresamclaird1572]

What a wonderfully surprising way to find pi! And so interesting to dig into the explanations.

[@rainidrops]

This is so fascinating, i love this connection. Super described man!

[@masskonfuzion]

I'm not sure what I can say here that hasn't already been said.. but I'll try 😁 I studied computer engineering in college - some of my 3rd & 4th year classes covered the calculus of electricity (AC circuits and such), which dealt heavily with phase diagrams, phase angles, etc.. Never, at any point in my education, did any teacher come anywhere close to explaining any of that math as clearly and concisely as this (or any other 3Blue1Brown video, for that matter), that I can recall. This seriously is on another level.

[@chriscatto-smith7701]

masskonfuzion I agree totally. Seven years it took to gain a BSc in Maths, double masters in engineering and post graduate aerodynamics. I have never had something as complex described in such an understandable and interesting, concise manner.

[@Pawbreakers_com]

Explain things as simply as possible..but do not oversimplify. :)

[@akasakasvault7597]

agreed (lol i was taking a test and it was so annoying i came over to this video instead (i'm in home education so i can do this))

[@saketh123gaming9]

Daaamn I'm in grade 10

[@TechSupportDave]

This is exactly what happened to me. I take Engineering and we too are dealing with AC electricity and trigonometry, phasor diagrams and such. It feels so nice and refreshing having someone teach these topics to you so well and in such a satisfying manner. i'm definitely subscribing and cannot wait to see more of his videos!

[@theCodyReeder]

Now to figure how tau fits this better. ;)

[@sdcard9649]

What do Cody's lab and 3blue1brown have in common? Both have gold videos

[@-_Nuke_-]

Clever comment but I won't give it to you and here's why. All you people that like Tau more than Pi don't even know how to pronounce Tau correctly and as a Greek it pisses me off. The letter " τ " is not pronounced "Tau" that's just another thing that was lost in translation from Ancient Greek to latin and then to English. The Greek letter " τ " is pronounced " Ταυ " see the resemblance? T = τ A = α U = υ That's why you pronounce Tαυ as Tau though just because u = υ doesn't mean that in Greek the letter " υ " is pronounced as a " u " every single time. In some occasions the Greek letter " υ " is pronounces as an " f " sound. And in this case we have Ταυ = Taf *T H E R E F O R E* Tau should be pronounces as *Taf* because that's the correct Greek pronunciation of the name. Calling "Taf" as "Tau" is horrible and can only be done by people that have no idea of the Greek alphabet. I'm so mad that people call this beautiful " τ " letter as "tau" that sounds horrible and makes me wanna vomit. Now maybe you in particular do know the correct pronunciation of the word so I apologise if that's so, but anyway, I'm leaving this comment here for other people to stop calling τ as Tau. yikes.

[@vinyak123rohatgi]

Love your videos

[@LeoStaley]

@@-_Nuke_- **NO**. That "F" isn't an F. it's a DIGAMMA. en.wikipedia.org/wiki/Digamma. you pronounce a digamma as, basically, a w sound, except when it is functioning as a vowel, in which case you pronounce it very much like a u. So YES. "TAU" is the correct pronunciation. I took 4 years of ancient Greek. Pedants like *you* who have no idea what you're talking about, but ACT like you do, are the reason the world sucks.

[@NotaWalrus1]

Well, the reason tau unironically fits better is that you are adding copies of 2theta to get to 2pi.

[@Jonathan-ru9zl]

Grant, your place in the great educators of all time is granted 🙌

[@barsanyibela4027]

This was beautiful. I have no words.

[@aidanokeeffe7928]

The use of sound effects in this video makes it infinitely more fun

[@silentexcorcist453]

3 strongest things in the universe: The old family computer A nokia phone That wall in the simulation

[@evetheeevee2977]

Lol

[@villagernumber7882]

the sideways gamer. What about bedrock.

[@antonioandreigiani6532]

nope the floor

[@xenoduck3189]

Those are in order

[@NovaWarrior77]

Meme worthy.

[@hrz6di]

This is the most beautiful video I've watched this year. I love it.

[@the1jessebrown]

This is an example of a title I’d never be searching for on purpose but absolutely happy to find

[@EricNicholsdesign]

I've said it before on your videos but seriously... my career path, and life for that matter, would have been radically different if my visual learning mind was taught math with videos like this. Its amazing how instantaneously I can comprehend ideas that would have taken hours from just a textbook or verbal explanation. Thank you for your videos, they're fantastic!

[@QuantumPhyZ]

Psychologists have been saying things like that forever. Not everybody have the same learning style. School systems all around the world could be way better, but the thing is things like this are common in private schools.

[@EricNicholsdesign]

Was commenting directly to the content creator. But in response - of course they have, exactly my point, duh, and sorry I went to public school?

[@anLTproduction]

I was about to write the exact same thing, thanks for doing it for me!

[@AppliedScience]

Wow. My favorite video of yours yet! Very nice work.

[@3blue1brown]

Thanks!

[@odw32]

Everyone: Check the Applied Science channel. This man is an absolute mastermind when it comes to engineering & high-end DIY science experiments.

[@akilansundaram2181]

@@3blue1brown Awesome video. You really need more subscribers. Every single video of yours is so good for understanding mathematical concepts. Especially the animations, they are just godly.

[@hoomaneshghi253]

Perfect video Thanks

[@amuzak9063]

I wanted to like this comment, but it has exactly pi likes on it right now...

[@CrazyGaming-ig6qq]

Very fascinating. Ending at the step just before the step that would increase the value beyond pi. What an amazing and funny way to approximate this constant.

[@Beara-sd9vg]

Solid memes, glad to hear you’re feeling okay. You just earned yourself a new subscriber my guy, can’t wait for future videos

[@thoughtpolease7183]

The quality of this channel is insanely high

[@JasonSun386]

"Most notably, the conservation of energy plants the circular seed that ultimately blossoms into the pi we find in the final count." - this sentence is like poetry

[@soumyasishbhattacharyya2805]

♥️

[@these_handles_are_stupid]

They’d teach us about graphing stuff like reflections and translations or how to make a circle in school but never really applied it in an understandable, non-abstract way. It made math frustrating. This demonstration puts a face to the name of so many concepts I learned. Very nicely done 👏

[@Rizwan1875]

Presentation and brilliance at its best! 💫

[@LuxuryDigitalAgenci]

Whenever i watch 3blue1brown I realize my ultimate goal is to find this man and hug him!!

[@ranbirroy4627]

I would be surprised if that doesn't result in a pi.

[@vaibhavsingh6760]

@@ranbirroy4627 very very cheeky

[@vaibhavsingh6760]

@@ranbirroy4627 may result a pie

[@DmitryShevkoplyas]

This is crazy!! Now I've seen it all! Thank you...

[@Fitzeroniii]

I’m not sure why this popped up on my shorts, but I’m in love with it. I love learning though and especially learning stuff I would’ve never bothered to try and learn on my own.

[@evilotto9200]

Maths I had to power through by rote memorization, you are able provide with meaningful understanding. Thank you. You are great. I want my tuition refunded.

[@JorgetePanete]

rote?

[@NolorW]

@@JorgetePanete "by rote" = "by heart", learn something without understanding it (or being required to), I think :D (not native)

[@TheMan83554]

@@NolorW Correct

[@kuro13wolf]

​@@JorgetePanete Is it really easier to leave a reply and look ignorant than to write "rote" on a new tab? Genuinely baffles me.

[@mushroomcube623]

@@kuro13wolf He probably didn't think it was a word and was trying to correct OP

[@shebahammy]

All the good collision scenes Comment if I missed anything 0:20 64 kg 11:04 10 thousand kg 9:26 10 thousand kg 11:48 1 million 14:28 1 million 0:32 1 million kg 0:50 1 trillion kg

[@slim531]

THANK YOU

[@abc3631]

@asdf hjkl 😀 really

[@scott420]

Legend

[@ghabsterlol7768]

we have a fellow memer here i see :)

[@jakechurcher5837]

The 1kg and 10,000kg sounds the most satisfying

[@greasycheese1957]

Im pretty proud of myself for actually getting this, I hope you know you are an amazing teacher and the internet is lucky to have you

[@georgohm_DC]

This is truly a great video. Thank you👏🏻

[@pewdepie995]

I was so looking forward to this

[@UseAfterFreee]

"And if this solution leaves you feeling satisfied... It shouldn't" Well there goes my excitement ... "Trust me I've saved the best for last so I hope to see you again in the next video" And my excitement just came back doubled ! Thanks 3B1B you are awesome!!

[@3blue1brown]

Looking forward to seeing you there!

[@elementsofphysics7324]

@@3blue1brown He won't be the only one back on your channel, trust me!!!!!!

[@demi_ryka1306]

I love videos like this- for some reason I can't learn well (Especially physics my god) in school- but videos like this where I'm actually interested in the stuff and equations and whatnot just holds my attention really well- makes me wish education could be more personalized

[@diy_wizard]

This is the best thing I’ve seen for a while, thank you!😮

[@xdo3865]

3:14 Pi people: Hunt for pi! Me: aren’t you just hunting yourselves?

[@LongboardtTrail]

Was it a coincidence that you used 3:14 as a time stamp. It's the first three digits of pi

[@VivekKumar-td3me]

@@LongboardtTrail pi is everywhere

[@VivekKumar-td3me]

@@LongboardtTrail that's scary

[@benjiboi69_vlogz87]

@@VivekKumar-td3me i know

[@rollingrocky3608]

@@LongboardtTrail Santa Maria!

[@YouTubeTryingToBeTwiter31581]

This video needs to have 3,141,592 views

[@jbtechcon7434]

...then one more guy watches it and everyone is like goddammit Kyle!

[@preppen78]

I'd need an explanation for why Pi unexpectedly shows up in the youtube count if that happens. Preferably in a geometry lesson with clacking noises.

[@evanwilliams2048]

Excuse me the 2 would round up to a 3

[@ObjectsInMotion]

I think you mean 31,415,926 views.

[@loresoong8207]

@@ObjectsInMotion wouldnt the 6 round up to a 7?

[@user-fp9kz6xv6l]

Amazing! This is your first video I have seen and it is definitely not going to be the last.

[@wellfolks9096]

It's one thing to understand math But explaining and visualising it like you do is really something else Keep up the good work

[@MarkLucasProductions]

I have no mathematical ability whatsoever but this video (like many others) is absolutely orgasmic. No wonder some people love mathematics.

[@PhysicsHelps]

Most of mathematical ability is finding this kind of stuff orgasmic, plus patience.

[@berkaykartopu]

it’s physhics

[@navaneethsujesh850]

Hard work beats talent

[@bobon123]

If you like this video, it is simply not true that you do not have mathematical ability. You have been probably just exposed to mathematics in the wrong way.

[@prozzezion]

i´m with you

[@flyingwhale157]

Why am i finding myself watching this video every few months? And find it interesting and mesmerizing each time again and again

[@MrMegaManFan]

Brilliantly explained. 👍

[@elementsofphysics7324]

I am an astrophysicist, and I can say that this video is one of my favourite maths video of all time. I hope the future will show that people with such a talent will have the opportunity to have a lifelong career as "teachers" on KZbin, with their videos available worldwide, instead of being stuck in a job they don't like. Some researchers are wonderful teachers (and some are not...) transmitting to a handful of people in a specific university. Some genius teachers won't get a researcher position. This can be the future.

[@roeisameach592]

I remember watching this a year and a half and ago going "oh, I don't get it, but cool" now after being halfway into high-school physics and finishing with geometry I think I got some of it, see you in a year when I actually understand everything and can fully appreciate it.

[@aglassofwater8582]

So…. You fully understand it yet?

[@overfotu]

@@aglassofwater8582 lol

[@Cyba_IT]

How about now?

[@ontopofbottom]

You won’t ever understand all of it. Get your PhD and you still won’t understand all of physics. We know the answers to many things, but some reasons for why things are stay a mystery

[@ledgerlinelife]

I don't usually understand the reasoning behind problems like this, but I think I get this one, and it's pretty cool. Thanks 3Blue1Brown!

[@darciestimpson8904]

thankyou for inspiring and fueling the spark of excitement i get from maths ! you've built a truly wonderful community

[@rmclean3]

Your explanations are so clear and pleasant, even someone like myself with virtually no math background beyond highschool, can understand it... mostly ;)

[@tymofei8586]

what tf im shocked as mechanical engineer,literally how this is possible and even more interesting who might have thought of this ? Linking pi to dynamics,geometry,trigonometry,mathematics at the same time..I mean its really different point of view. Brilliant channel deserves to be subscribed..

[@NetheriteMiner]

I should be doing Trigonometry homework rn. This is more interesting.

[@chupasaurus]

Mechanics has the tightest link to mathematics, even models in geometrical optics (which is basically made of geometry and trigonometry) are just describing the mechanics of photons.

[@c4p4c1t1v3]

Your animations are something else... I wish I had access to this when I was in high school and college

[@Isaak.4]

I use u to learn in the background while I’m doing other things and I learn so much

[@muzaffercaginipekoglu105]

Hello there from Turkey, being an x-mechanical engineer and now a private maths and physics tutor, this totally amazed me, thank you for this wonderful channel and all the explanations made...this is absolutely mind blowing. I'm not very good at maths and physics but I'm getting there :). I'd have been so happy if just math and physics were explained to me in high school and in my university years like this...applied mathmematics etc...not just by using the textbook and the board, but just like this via insight, real word applications...hope we can teach the youngsters or guide them after us how beautiful and amazing math is. Thanks a million for the channel Grant. Thanks to teacher Edria Murray from MEF Int. School of Izmir, Turkey for recommending this youtube channel to me and to her students.

[@kylecow1930]

1:08 holy hell my thing got in the video

[@dexterdev]

congrats :)

[@yourlordandsaviouryeesusbe2998]

Well done my friend.

[@duckymomo7935]

Which solution was urs

[@solarnaut]

consider yourself to now have 1.5 M jealous friends ;-) and welcome to faceborg.

[@kylecow1930]

I made the p5* simulation @@duckymomo7935

[@nafiurpial7209]

It's unbelievably amazing!!!!

[@cheshire1]

There's another nice way to see why all the arc lengths are the same. We get arc 2 as a mirror image of arc 1 across the x-axis, so it must have the same length. Then arc 3 is a mirror of arc 2 across the diameter through the midpoint of arc 1. And we continue like this, at each point mirroring across one of these two lines.

[@nesslam4832]

I understand everything in this video but I can't wrap my head around how they all fit together perfectly, like wow... how did anyone figure this out?

[@nadavron5460]

This isnt a very hard question in regard to physics, but it is a very elegant and nice looking presentation

[@jonathanmartin2326]

First of all, thanks for asking this, Second of all, this is less of figuring anything out specifically, but more so explaining how certain concepts relate to one other. In other words, when working with equations, or concepts it is sometimes important to look at the information from another perspective, this is similar to a lot of university physics equations tie in circles to things like spring constants and pendulums. You can solve problems without the other perspective, but the ability to see it from another, may allow you to solve even more problems down the line, hence why we have standard quadrant system, spherical coordinates, and that other one I don't remember anymore :)

[@nesslam4832]

@@jonathanmartin2326 damn thanks for that in depth reply! I do see the purpose of translating scenarios into different mathematical concepts. My comment was more of an emotional "WOW" moment than anything, mostly because I've never seen mathematics presented in such a way in school before. (Also regarding coordinate systems I think you mean polar and cylindrical. I kind of see how that analogy translates to these systems too, but on a much narrower scale since they mostly still deal with the same scenario except with a different "notation" with the way you perceive the dimensions, I guess? In this video it relates a system of colliding bodies with circular "phases" which I found to be much more impressive haha)

[@sanchu6335]

So lets say there is an infinite amount of universes that go by the butterfly effect, our universe is the one that discovery happens in, also by the same logic everything and anything can happen

[@albin1816]

idk much about math (I suck at it). But I can intuit that the small mass and an immovable wall is equivalent to a perfect spring when returning the large mass, which means it will have a curvature of a perfect circle. And the larger the ratio between the small and large mass, the closer to perfect it is.

[@user_hat]

How to find *The End* Of Pi: Set moving block to INF mass.

[@oliverfeuer9323]

IUnderated coment

[@ondrazposukie]

The end of pi doesn't exist. Nor can infinitely many collisions occur in that time.

[@paulmccartney8293]

@@ondrazposukie you tried to be smart but actually, dumb lol

[@numbers4685]

@@paulmccartney8293 lol

[@AnkurKumar-yx5sq]

If you're homeless, just buy a home

[@Joseanaslin]

This is very interesting and I hope I learn more about this soon.

[@banerjeeproductions4625]

Omg, opened my eyes.... Thank you so much

[@Raketemensch-fl3sv]

This channel is really extraordinary. I haven't done math in like 15 years, and then it was highschool, so mostly just rote memorization/plugging things in/that kind of b.s., never anything like getting to fundamentals or deeper understanding. Why can't math curricula in primary/secondary school aim at inspiring awe or seeing the beauty of math? And... unfortunately most of the content put out here is beyond my immediate ken, but enough of it is such that, if i make the effort, i can at least start to grasp some of what's going on, and it's wonderful. You're a very good teacher, and i like that these videos don't (seem to) dumb things down too much. They're intimidating to the layperson, but you know that it'll be worth it to try.

[@fiveoneecho]

I routinely forget about drawing the relations when solving a system of equations in physics like momentum and energy, but it really is an awesome trick for understanding a system. Sometimes when doing a quick solve by hand, it can be very helpful to just sketch the relations and see where they are equal. I love this channel because no matter how many times I watch a video or how well I understand a topic, there is always some humbling reminder of something SUPER helpful I have forgotten.

[@Warszawianin]

I love this! Take a surprising and baffling observation, which one could ignore because it doesn't seem to matter (why should I care about the number of collisions of hypothetical objects in an idealized frictionless setting with a wall of infinite mass?), then develop an elegant way of describing the phenomenon to arrive at a transparent explanation leading us to the point that the observation is no longer surprising, but obvious. Why? Because the explanation exists, and it's beautiful! And as bonus we have a toolset to apply elsewhere. ❤

[@prawnydagrate]

When I was in fourth grade, a guy who worked at my school (not sure if he was a teacher, but I don't think he was a janitor) told me about this problem knowing I liked math. I was fascinated. I love that guy.

[@dodobow]

이건 정말 상상도 못한 개념에 대한 영상이네요. 원주율과 물리적 시뮬레이션 사이의 연관이 있을 줄은 상상도 못했는데 이 영상을 통해 하나 배워갑니다. 수학과 물리학에 대해서 다시 한 번 생각해보게 되네요.

[@nooooheyyy]

skibidi toilet maths

[@slimeslayer5195]

​@@nooooheyyy grow up