How colliding blocks act like a beam of light...to compute pi.

[@3blue1brown]

Some added notes (I may come back to add more later): 1) There's a slight error in the video where I say the answer is floor(pi/theta). Really it should be ceil(pi/theta) - 1, to account for the cases where pi/theta is an integer. For example, when the mass ratio is 1, theta will be pi/4, so pi/theta is 4, but there are 3 clacks. As stated in the video, though, you still think about this is "how many times can you add theta to itself before it surpasses (or equals) pi?" 2) What I animate here as I say "angle of reflection" and "angle of incidence" differs a little with convention. Typically in optics, you look at the angle between the beam and a line *perpendicular* to the mirror, rather than with the mirror itself. 3) 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. 4) 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. This is common in the context of math, but you should know, 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 did here. I hope you enjoyed this little sequence. I still get happy whenever I think about the phenomenon and the various explanations for it.

[@Spibidydkdushusbwns]

Makes sense

[@baerlauchstal]

Superb.

[@baerlauchstal]

@Ryan Tavenner Right, and there are loads of others: 2, 3, 6, 8, 10, 12, 14, 17, 29, 36, 40, 49, 53, 58, 63, 68, 79, 85, 91, 97, 110, 117, 124, 131, 146, 154, 162, 170, 187, 196, 205, 214, 233, 243, 253, 263, 284, 295, 306, 329, 377, 389, 402, 415, 428, 441, 468, 482, 496, 525, 585, 616, 632, 648, 681, 698, 732, 749, 784, 802, 839.... But seemingly, none that is a power of 100, that's the thing.

[@johnnicholson8811]

@Ryan Tavenner How about a ratio of 710 to 113

[@jabejabe1424]

There's one more error: 2:46 those two angles arent the angle of incidence and the angle of reflection

[@Minstorm34]

I'm continually impressed on how this channel can make me feel completely lost in the first 5 minutes but completely enlightened in the last 5

[@PuzzleQodec]

Let's hope that all of his videos will be longer than 10 minutes. ;-)

[@WhiteSpatula]

Me too! And it keeps getting better with each additional viewing. However, I also find that my comprehension only ever approaches a certain maximum (which I believe is around 66.67%) without ever actually surpassing it. So, I know that by the time I’ve seen any given 3B1B video more than a dozen times or so, I’ve already reached maximum cerebral absorption, and my continued viewings are for drool-and-wonder purposes only. -Phill, Las Vegas

[@iamdino0]

could not relate to this more

[@RaglansElectricBaboon]

@@WhiteSpatula Surely your comprehension is 100% / Pi ?

[@Xcyiterr]

or you could be like me and be lost in both the first 5 and the last 5 minutes

[@Joseph125]

The year is 2119. 3blue1brown has become an immortal overlord, having been able to simulate the entire universe using a pair of hypothetical, frictionless blocks that make a cool clicking sound.

[@HidekiShinichi]

they dont make that clacking sound. Its imaginary :v

[@1996Pinocchio]

With your profile picture, you're only allowed to write 3 words comments

[@codetoil]

lolz

[@HidekiShinichi]

@@1996Pinocchio me? why?

[@gamma-bv6ty]

One of these days 3blue1brown is gonna upload a video that solves the Goldbach Conjecture by drawing circles in a clever way.

[@bigbakerboi3872]

_for the record this guy is making _*_literally two squares hitting each other_*_ that interesting_

[@donielf1074]

Parth Sankhe And you thought math was boring? Have you *never* watched a video on this channel before?

[@ttanfield5616]

@@donielf1074 You either mean, "... Interesting". Not "that".

[@jigartalaviya2340]

@@donielf1074 I think u misunderstood him.

[@Morgwic]

@@ttanfield5616 "that" in this context means "super" or "really", also if you use the word "either" you should give two options, for example: You either mean "Interesting" OR "That"

[@alexboyer2247]

FOR THE THIRD TIME

[@truppelito]

Honestly, when I read the original paper, the trick of counting the number of reflections by having the line be straight was one of the most extraordinary "mind-blown" moments I've ever had

[@Shakespeare563]

I feel so incredibly proud of myself for coming up with this solution on my own independently when the first video dropped. I normally suck at coming up with this kind of problem solving, but maybe all the math videos I've been watching are finally paying off

[@ascaniuspotterhead2484]

It’s really cool that you managed that. That isn’t simple.

[@scottbelcourt5280]

Power to you bro!

[@Verlisify]

I want a behind the scenes of how the graphics work. It seems easier to just program it than actually animate it

[@melody_florum]

@@00O3O1B there’s a fork of the library designed for public use

[@Jesin00]

@@melody_florum where?

[@Tiessie]

@@Jesin00 its called manim community

[@coldaddysupreme2956]

He programs most of his animations

[@jean-baptiste6479]

The thing is so well explained that the software does the animation automatically.

[@GUITARHERO9720]

I could not be more excited for this. Every time you upload, it's like unwrapping a Christmas gift I've wanted all year, except it's better because it happens much more frequently.

[@puskajussi37]

And right after you won't have to go "well this socks"

[@xxcsokixx]

@@puskajussi37 be glad for socks too my friend. You can appriciate the better gifts more if you get some socks sometimes.

[@hauler-hx6gw]

I think it's more of that present you didn't ask for but am really happy you got

[@FinBoyXD]

Ryan Kelly I just gave you your 314 th like. Pi it all the way!

[@theycallme_nightmaster]

donate on patreon for each video then

[@brianevans4]

This video series has got my A level maths class, not to mention teacher, very intrigued. We can't stop talking about how satisfying the clacking sound is when the blocks collide. Love your videos Grant!

[@evaristegalois6282]

3Blue1Brown and Kurzgesagt both just uploaded on the same day again... _This is beyond science_

[@vigneshdesmond]

Best day to be alive

[@tapwater424]

Kurzgesagt is pop science entertainment for people who don't like to learn

[@GMPStudios]

@@tapwater424 Learning doesn't always mean solving a bunch of equations. Both channels are good

[@adamxue6096]

@@tapwater424 3Blue1Brown is more focused, more in depth already, where Kurzegesagt is more of a, trying to start peoples interest in certain subjects, and then direct those who becomes more interested in Science to like Brilliant where they can learn more in depth, like "We got the big picture now, and it is awesome/bad, how do we make it happen/get it fixed."

[@aaronkoch3273]

brb, looking up kurzgesagt... :)

[@denelson83]

10:08 - Nice square root sign you just made there.

[@himanshsachdeva]

It was actually 𝜃'th root. :)

[@RieMUisthegoaT]

how to generate the square root notation from how to generate digits of pi...

[@IStMl]

@@RieMUisthegoaT "How colliding blocks act like a beam of light... to draw a yellow square root"

[@mech0s]

You're getting many likes for this comment, mostly because your time link is just above the like button and our fingers are fat!

[@IStMl]

@@mech0s Hum are you ok ?

[@Pablo360able]

It makes perfect sense that laws of kinematics would translate to laws of optics, when you remember that light is a physical thing that’s ultimately governed by the same meta-rules as solid blocks - just with very different properties that make it behave differently in almost every situation.

[@idjles]

This is beautiful anyway you look at it. Presentation, Voice, Animation, Content, Optics, Geometry, Energy, Momentum. Euler would have loved it.

[@nadivkaspi6211]

What I've learned from Numberphile and 3Blue1Brown is that maths is 100% about perspectives and beauty. Learning maths in school was a ball ache. We never learned about interesting topics, and while what we learned was important to understand ideas like this, it almost squashed my love of math. Thanks to these kinds of videos, I think everyone can find something interesting in maths.

[@STD282]

Wow. Just wow. Learning maths is like reading well-crafted series of detective stories. At the beginning it's all strange and mysterious, and at the end everything becomes clear and obvious.

[@philippw4769]

and the truly awesome part is that you dont think "oh, that was so obvious..." but "holy gosh is that beautiful and well working" (just like a good sherlock episode)

[@ganondorfchampin]

The truth is actually the opposite. At first you think math is simple, but the more you learn you realize just how complex math is, and mysterious.

[@raimuu14]

So you could say this is *"Blue's Math Clues"*

[@PrincessEev]

This is certainly a candidate for one of the most elegant solutions to a problem I've ever seen.

[@user-hh4xs7ml7s]

colliding blocks I never thought you would bring it back

[@refusoagaino6824]

Pong

[@martinsimbona6145]

Man you're quite a thinker. Your illustration of math and science is an art on it's own. I'm fascinated by your intuitive way of representation stuff.

[@AzureFlash]

Pi is like Rome... everything leads to it

[@Magnus_Deus]

Next video: How to solve pi with memes Next next video: How to find pi with a toothbrush

[@dhoyt902]

e specially.

[@redbeam_]

exactly!

[@yaksher]

Finding pi with a toothbrush is 'easy'. You just need a tile floor with tiles the same length as your toothbrush.

[@David-km2ie]

The king is φ

[@TheSucread]

Shouldn't the final formula be Ceiling[pi/arctan(sqrt(m2/m1))] -1 rather the one with the floor? Formula shown in the video gives a wrong answer (4) for the number of collisions in case of equal masses.

[@3blue1brown]

Yup! My mistake.

[@karolakkolo123]

Nice catch, floor is the same as ceiling-1 everywhere except at integers

[@rpifb18]

Nice feedback!

[@danielkharnas3298]

nice♥

[@Kenlimepie]

Who’s here for the light clacking noise?

[@proghostbusters1627]

Exactly

[@idjles]

And who is waiting for that final, delayed clack?

[@fluent_styles6720]

Kenlimepie asmr for my eyes and ears

[@BothHands1]

Yess!! Lol

[@FindLiberty]

But what if that _light clacking noise_ remains trapped within the Aether (the medium the light waves propagate through)?

[@Aquillyne]

Omg, that way of rephrasing the light bounce problem as reflection gave me shivers!

[@florianquerniard1084]

The same for me... I almost cry...

[@jojoboss3917]

Isn't it amazing how the number pi shows up in places that look random at first... How beautiful it is to unravel the connection between so arbitrary fields in science... I truly admire the owner of the channel for making this kind of content. Those videos explain really abstract topics in such an easy way! I am a big fan of the 3 blue 1 brown channel.

[@Porglit]

Your animation is legitimately the absolute best on KZbin. It's always obvious what you're doing, and makes a difficult subject seem plainly intuitive. Wow!

[@gabrieleciccarello4876]

It's incredible. Awesome. You're inspiring me even more to study maths. This nice change in prospective litterally amazed me. You're a wonderful lecturer Grant. I believe that lots and lots of people have reevaluated math after watching your videos. I hope one day to be as good as you are. Keep on lecturing!

[@davegrox3150]

This kind of videos/questions makes me feel so glad that I studied calculus and stuff. Because I can understand and appreciate the beauty of it

[@billrussell3955]

That was seriously awsome. Newtonian math. Gravity, Optics, differential geometry.

[@NoNTr1v1aL]

differential geometry? where?

[@flxdrv5020]

it's not gravity but conservation of energy.

[@penus7639]

Lol gravity

[@mrosskne]

are you just saying random words?

[@Verlisify]

The block clicking is the most satisfying thing ever uploaded to youtube

[@nurneues1662]

Me: Yo pass the aux cord Friend: you better play some fire Me: clackclackclack

[@nestorv7627]

Oh yeah yeah

[@Solrex_the_Sun_King]

Clack clack clack Rheeeeee clack clack clack

[@HungryTacoBoy]

Oh clack clack.

[@jo_nm9484]

The car: AW YEAH CLACK CLACK CLACKCKCKCKCLACK CLACK

[@temirlankasmaliev9322]

what music do you listen to? It’s complicated.

[@chielonewctle7601]

I think one of the greatest significances of this solution, as you say in the video, is that it gives out a way to simulate the collision progress. Spectacular!

[@vypxl]

Could you make a video on how you create your animations and simulations? I always wonder how it is done and if your are coding it yourself or have a software or whatever. Also I would like to mimic it a bit for fun ^^

[@wickw]

3b1b has its own animation engine: github.com/3b1b/manim

[@samwelter6083]

@@wickw oh hell yeah! Thanks for the lonk:)

[@cannot-handle-handles]

@vypxl Shmebulock. :-)

[@vypxl]

@@cannot-handle-handles yeeees

[@misugijun]

@@wickw thanks for the link

[@mertgokgoz4073]

To think that I would never have known any of these without this channel is very scary actually. It reminds me of my stupidity since I couldn't have figure these out myself. So much beauty in life is lost simply because we are incompetent. Anyways, thank you very much for your work!

[@timmijhonson9263]

I only listen to real music clackclackclackclackclack

[@markorezic3131]

Clack clack clack clackclackclaclaclclkkkkksqeeeeeeukclacla clack clack clack clack...... clack

[@otakarbeinhauer]

This is Earth radio. And now here is human music. clack clack clack.

[@seungjunrhee]

do u no da wae?

[@neroyuki241]

clack x 3.14

[@rcb3921]

@@markorezic3131 -- you crushed it with that written representation. Brilliant.

[@gnikola2013]

Simply astonishing. I'm still young fortunately, so I think that with hard work, I will eventually be able to have the amazing ability to apply drastic changes of perspective when aproaching a problem. Apparently this is the way to get to do some beautiful maths

[@thenotflatearth2714]

I think 3b1b has an addiction

[@MickHaggs]

We need a colliding blocks intervention

[@user-hh4xs7ml7s]

We all do

[@XtremeQuantumSrength405]

Lol true like me

[@3blue1brown]

It's true. I'm a clack addict.

[@BothHands1]

3Blue1Brown and now you've turned me on to your poison. Quick, i need another fix 😅

[@luxaeterna5281]

I am an Italian student that can understand english mathematical language with some efforts, and I am currently dealing with the basics of Physics, like Equations of Motion, Uniform Circular Motion and so on, but I like to see what I will have to study even If I don’t understand almost anything, Yet I want to Stimulate my Brain, Hoping that doing so I will learn these concepts faster in the future. Thank You for these videos, they are a real resource for students all around the World!

[@brooksvenuti7490]

I'm a PhD student in physics and I have never seen problems solved so cleverly than I have in your videos.

[@arpitagarwal82]

This 3 videos series blew my mind.... I am mesmerized by the way maths and physics connects... I will remember this video series till my last breath (hopefully)....

[@TheScienceBiome]

Grant, you’re a wizard.

[@ahmadness6995]

man... you need to write a book on the beauty of mathematics, you have made mathematics a spiritual experience for me, i deeply thank you

[@aidarosullivan5269]

Never saw a solution more beautiful than this!

[@SocksWithSandals]

I thought you lost me for a moment there but I jumped down the rabbit hole and reboarded your train of thought. Great animations helped a lot.

[@P4n0r4mA]

How your able to visualize complex mathematics is simply mindblowing. Your my favourite teacher so far :) Really inspiring!

[@joshyoung1440]

Did you ever find a more inspiring English teacher to help you with YOUR syntax? YOU'RE a bit rusty...

[@enahpincer6233]

I must say after watching 3b1b's videos , I have improved drastically in math. I learned how to think through the problems. How and why something the way it is. I am extremely grateful to these videos. Thank you so much Grant (It has made my life a lot better :) ).

[@repeat_stop]

Hi I am a high school student who got many help from 3b1b! Your video was very helpful to our mini-research. You’re the best math youtuber,and wish you get more subs!!!

[@jrl53]

These videos have to be the most beautiful ones ever to be uploaded to KZbin

[@laradimello5791]

I wasn't expecting to be educated on this sunday evening. Thank you 3b1b for that cool analogy!

[@itisALWAYSR.A.]

This is beautiful and satisfying. There was a point 3/4 of the way through where the math lost me.... but then it came round to an elegant visualisation and I actually feel physically calmer now for it. It's an odd sensation to describe.

[@johnchessant3012]

I notice you're worrying a lot more in this video about viewers who just ask, "Why is this useful?", instead of appreciating math for its own sake. Math is an art, and this channel is proof.

[@yondaime500]

I know, right? Asking what's the point of math is like asking what's the point of music. Math just happens to be incredibly useful for solving practical problems and understanding how the universe works, but that's just a bonus.

[@insafidris2366]

You are right! In one of his TED talks, he questions how would anyone ever use Wingardium Leviosa so that's a nice analogy.

[@soumyashreepatra6224]

I really like the combination of creativity and science the guy used to define these perspectives.......

[@mrhiran29]

This is absolutely BEAUTIFUL!!!

[@cbranalli]

every time i watch one of Grant's videos 1) i'm very glad i studied engineering so i can get the gist of what he's saying and 2) i'm very glad i don't still work in that highly competitive field. very much reminiscent of the old "Watch Mr. Wizard".

[@buzzy7601]

Was struggling with a similar problem in how to solve it by Paul Zeitz! An absolutely beautiful and elegant proof. Thank you so much 3Blue1Brown

[@hal6yon]

Grant puts out videos so innocent and so insightful that i cry everytime i watch a new one

[@Bibibosh]

this is so interesting,... please make more about this phenomenon

[@andtherefore8076]

I just studied this topic in the ray optics. It's amazing that this no. of images thing could be this much related to the pi collision problem. Such a great video!!

[@jadissa3841]

I love how it got to the exact same part of the proof at the end...

[@MegaCreeper]

This problem demonstrates the beauty of mathematics and physics. Well done.

[@TariqAdil]

Your channel makes me love math.. math before you=😵... Math after you=😍

[@robertshelnutt5081]

That's BEAUTIFUL! Thank you. Nature and the world around us gets better when you understand it.

[@murgmir]

my mind is blown everytime 3B1B uploads

[@himanshsachdeva]

3blue1blown

[@HasolIm]

* mind blown moment *

[@JochCool]

I absolutely love your proofs. They give you the kind of "OMG this works out so perfectly" feeling.

[@YostPeter]

This video has transcended into mathematical art.

[@Stelios.Posantzis]

I may be biased but for me these two episodes are the most elegant problem(s) presented in the series so far.

[@rahulshaw9051]

You really is a magician,showing us the magics of mathematics.😉

[@bassboossaful]

Your youtube channel, and specifically this series of videos talking about this question gives me so much inspiration, i cant even express in words. I love it so much that i feell the need to share it with my friends and experience the shock of discovery over and over. Thank you so much! Please keep on posting problems like this, and the various creative analogies. Thank you so much!

[@N12458]

"What many people do not realise is that the ideas underlying these solutions can help in solving serious problems in maths"- Shows two blocks colliding.

[@jorgealexandre4616]

It's not about the blocks, it's about the problem solving tool of using a phase space to convert some kinds of problems into others. The blocks are just an easy way of illustrating the principle.

[@N12458]

@@jorgealexandre4616 hey man, I was kidding, ofcourse you are right, but the above wasnt meant to be an attempt at criticism.

[@GeraldIrsiegler]

I always seem a bit lost in the middle of your videos, the screen is filled with formulas and I think I'll need some time to process all the things written, but then you pull out an amazing visual aid and everything suddenly makes sense! Love your videos, seriously such great content!

[@spencertaylor6910]

Awesome video grant, thanks! You should make a video about differential forms, orientation, and the generalized stokes’ theorem

[@bluellamaslearnbeyondthele2456]

This. This is one of the few things that managed to strike me as new and worthwhile information in like 4 years? Thank you for existing mate.

[@adamhrankowski1298]

Grant, is there a similar way to produce e? Perhaps a feedback mechanism that increases a measurement exponentially?

[@nigeldupaigel]

I think not. That means perpetual motion, right?

[@ericluo4900]

I wish there was something physical but it actually comes down to interest rates. But I hope someone does figure out the origin of e through a feedback mechanism similar to this

[@adamkrasuski4743]

There are a bunch of places where e appears in physics. One of the mechanical ones would be damped oscillator (think: counting number of oscillations before halving the amplitude or something similar).

[@PaulPaulPaulson]

A car that automatically sets its speed to the value of the distance to a reference point would give you e^x, and therefore e. But let me try to provide some new perspective first. For the general formula y=a^x the only value for a where the formula is equal to its derivation is e. That's kind of the definition, but by that we don't know the actual value of e yet for which this works. But we can conclude that if you draw the function e^x and take any point on it, and you draw a tangent to it through that point, it will always intersect the x axis exactly one to the left. So for a point P with coordinates (x_p, y_p), it would intersect at (x_p - 1, 0). Why? Because the slope of the tangent is the derivation of e^x which is e^x again by definition, which has the value y_p at this point. Going one to the left results in going y_p down, and we started at height y_p so we end up at zero. In theory, you could draw e^x with this knowledge, and read the value at x=1 to find the value for e. Drawing it would work like this: Start by drawing a point at (0, 1), because a^0 is always 1 for all values of a. Go one unit to the left and mark the x axis, in this case at (-1, 0). Draw a line through these two points. This is the slope at your starting point. Follow the slope a tiiiny little bit to the right beyond where we started, mark another point and repeat the process for this new point: Your x-axis intersection also moves a tiny bit to the right (in order to always be one unit left of our current point), draw a line through it and through our new point, and mark an even newer new point to the right of our old new point. Repeat. It feels like taking a triangle with a base of constant size 1 and shifting it to the right while its top corner keeps touching the function. Repeat until you reach x=1. Read the y value. Congratulations, this is e. Well, it would be, if you could truly do infinitesimal small steps to the right. You can approximate it by using smaller and smaller steps. But what if we start by taking big steps instead? Lets find of what happens at stepsize 1. Big leaps. Going one to the right with the same slope without "updating" it in between. Start is at (0, 1) again, thus the first slope is 1. Following the same steps, your next points will be (1, 2), (2, 4), (3, 8) etc. which all lie on y=2^x. So 2 would be our first approximation of e. Think of it as someone sitting in a car, one meter away from a reference point. Once every second he looks at the distance to the reference point and adjusts his speed to this value. So he starts with 1m/s, drives 1s to total distance 2, then (instantaneously) sets his speed to 2, drives one second to distance 4 etc. The pattern is 2^x again. If he updates his speed faster, his trajectory will get closer to e^x. If he manages to construct a perfect control engineering device that could measure the distance with no delay and could instantaneously adjust the speed, the distance of the car over time would be a perfect e^x. Just measure the distance at t=1s and you get the value for e.

[@JNCressey]

If you take a radioactive sample and record every decay then, at the mean average time of the decays, the radioactivity will be 1/e as much as the radioactivity at the start.

[@NoriMori1992]

I was so confused for most of this, but that solution at the end is so elegant! That method is so easy!

[@henryyang478]

hi 3B1B, during my attempt to solve this problem, I encountered both of your solutions. In addition, I found that the velocity-time curve of the lighter block looks like a Gaussian/normal distribution curve, while that of the heavier block looks like a sigmoid/logistic function. can you explain the connection in this case? How do you get pi from these functions?

[@loganstrong5426]

I know the formula for normal distribution has in it a 2π factored into the standard distribution, so this likely comes from the same place. I'd love to know more about that equation, so it'd be a great video topic!

[@loganstrong5426]

Just looked it up to clarify: in front of the exponential function is a factor of 1/sqrt((σ^2)(2π)).

[@henryyang478]

Also, if you draw the position of the heavier block in x-coordinates and time in y-coordinates, you'll get something resembles a parabola, just a funny little side note. I didn't manage to go anywhere with this so I hope you guys can make sense out of it.

[@filmliebhaber5283]

Spontaneously I see two connections to a circle: First the normal destribution is of the form exp(-x²) with some more parameters. If you integrate it (the Integral looks like a sigmoid function), you usually integrate in polar coordinates (they are circle like) and the Integral from negativ infinity to infinity is pi. Second, in Physics the gaussian function has a special connection to Fourietransformation, as it is the function in which the product of the standard derivation before and after the transformation is minimal. 3B1B explained Fourietransformation with a circle, so maybe you can start there.

[@skaterlehrling]

Woah probs to you that you figured out both soluntions. I also encountered a similar curve, but I think it has nothing to do with the gaussian distribution and the sigmoid function. The exponential factor was defined to 1/sqrt((σ^2)(2π)) for practical reasons. Moreover it is a weirdly stretched cosine and sine function which maybe can be proven by looking at the v1 v2 diagram. The weird stretch might be a result of the varying time periods between the collisions.

[@richardrigling4906]

This is a great trio of videos. I volunteer teaching physics at the local high school, at least resume after the COVID mess abates. I'll pass the original problem, and the twofollow ups to the teachers and see if they are interested in presenting it to the students. It links/couples so many areas of physics and math!

[@flav6350]

Amazing channel ! Thank you very much for this ultra-high quality content.

[@iridium8562]

This solution is more elegant than the first, just wow

[@ayushsatyam2616]

10:43 Holy shit, Let the light pass through the mirror and flip the whole world. 3b1b thinks in the reflected world.

[@vfxlord]

absolutely amazing, I didn't major in math yet I completely fallowed the video. I love your enthusiasm for the subject, and also now share it

[@pretzelroll451]

Please continue making this many videos

[@MateHegyhati]

How much I wish, I had such channels, when I was still a kid in school. Thank you.

[@macahjus7069]

8:12 "Duck products" can't be unheard

[@michaelwang1730]

quack

[@axelanderson2030]

This is like the first mathematical 3b1b video I fully understood and followed.

[@NoneOne_]

So Part 2 has 10 times the mass of Part 1? 0:35

[@elementsofphysics7324]

The best maths channel on earth uploaded the 3rd episode of one of its best series for my birthday... *This is beyond science too*

[@heal-ing]

The reason why I study English is to watch 3blue1brown's video. But there is a limit. Please anyone translate to korean😢😢

[@anselmschueler]

I don't speak Korean, sorry ): 🇰🇷

[@anuragthakur9376]

Shujaun...?

[@nowneothanielverse]

@@anuragthakur9376 I know it's four months late but its Sujeon

[@tiagopadua]

Wow this is beyond cool. The fact that the momentum line keeps going straight through the reflected spaces just blew my mind!

[@KC-wm6tr]

There is a point I am quite confused: in 3:16 it says angle of incidence does not equal angle of reflection, it is understandable as the two blocks are not in the same mass and they will eventually move together at the same time; however, in 6:53, it appears that angle of incidence is equal to angle of reflection, complying with conservation of momentum. I am wondering the reason behind. Is it because the axes change from distance to square root of mass times distance? Thank you a lot!

[@KC-wm6tr]

haha as i continue, i found the answer of my somehow silly Q in 13:13 :D yay

[@DavidPimentel]

A little more than a year and a half later, and this three-video series is still fascinating.

[@jabejabe1424]

2:46 those two angles arent the angle of incidence and the angle of reflection

[@Kreweta_Karrbon]

I'm just getting more and more amazed with your every video. Thank you for your work!

[@TheScienceBiome]

I was just about to go to bed, when I saw this upload. I kid you not that I said, “This, I’ll allow”

[@Piemaster1123]

Great video! In addition, if you use this perspective, you can easily find criteria for when the stationary block will come to a full stop after the last hit. Try it out!

[@CaTastrophy427]

14:15 Well then, a (web)novel I'm writing is, so far, worth 1120 IQ points (14 PoV/perspective changes. There are multiple main characters we see through the eyes of). And it's not even CLOSE to done.

[@KafshakTashtak]

Wow, you teach people things that no school could do. Mind blown.

[@eurovisioncyan9550]

*Clacks intensifies* The Walmart meme is back...

[@XtremeQuantumSrength405]

Yuuuuuuu

[@edderiofer]

*drill noises*

[@Jmcgee1125]

These videos are great if you want a eureka moment. Such incredible math with a surprising solution that doesn't feel awkward at all.

[@Mrtroll258]

6:39 When second block is 16kg, the first one bounces 12 times. 12 is 1100 in binary and pi is 11.001001 in binary. Does it work for every base?

[@xway2]

It should do, yeah.

[@amj.composer]

It does

[@pedronunes3063]

Yes, just remember, 10 is not a special base.

[@amj.composer]

@@pedronunes3063 Exactly! I wonder in what way the world would've been different if we all counted in, say, base 12.

[@andymcl92]

Yep, I think he mentioned this in a pinned comment in one of the previous videos, or maybe in the previous video.

[@gaurangagarwal3243]

The beauty of your videos ,maths , Physics is overwhelming.(no imogi can express this feeling)

[@TheScienceBiome]

Sneaky reference at 13:34 ;)

[@cybervoid8442]

of what?

[@BothHands1]

Through the looking glass?

[@AnonymousAnonymous-ht4cm]

@@cybervoid8442 Its a reference to the sequel to the most popular piece of mathematical satire: Alice in Wonderland

[@thephysicistcuber175]

OMG

[@TheFerdi265]

He referenced the channel "Looking Glass Universe"

[@visantibanez]

I love the way you make math look so easy and obvious