3Blue1Brown - 2019-02-03
The third and final part of the block collision sequence. Part 1: https://youtu.be/HEfHFsfGXjs Part 2: https://youtu.be/jsYwFizhncE Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/clacks-thanks Error correction: I wrote the answer as floor(pi/theta), when really it should be ceiling(pi/theta) - 1 t account for values of theta perfectly dividing pi. For example, the case of equal masses gives an angle of pi/4, and 3 total clacks. This beautiful result, and the solution shown here, are due to Gregory Galperin: https://www.maths.tcd.ie/~lebed/Galperin.%20Playing%20pool%20with%20pi.pdf And here's a lovely interactive built by GitHub user prajwalsouza after watching this video: https://prajwalsouza.github.io/Experiments/Colliding-Blocks.html Speaking of looking glass universes... https://www.youtube.com/user/LookingGlassUniverse NY Times blog post about this problem: https://wordplay.blogs.nytimes.com/2014/03/10/pi/ The plushie pi shown at the video's start: https://www.3blue1brown.com/store 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: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
Pi is like Rome... everything leads to it
@Alapan Das. No need to re-connect everything to itself, really
Why would we connect? Pi is connected itself.
No, that's e
Hahaha
pi is so romantic
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.
@Hideki Shinichi because there're only three segments that light is passing through in your PfP, making it, at most, "Clack Clack Clack". Turn those clacks into other words, and you've got three words.
@CaTastrophy427 those ment to be sheets of green paper with different shades stack on each other
@Hideki Shinichi you broke the laws!!!
@AleXander Russell i am a bad boi
@Hideki Shinichi O NOøŐØÕÖÓÔÒ Ö
Me: Yo pass the aux cord
Friend: you better play some fire
Me: clackclackclack
ayyyyy
Clack clack clack Rheeeeee clack clack clack
Oh clack clack.
The car: AW YEAH CLACK CLACK CLACKCKCKCKCLACK CLACK
what music do you listen to? It’s complicated.
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.
After seeing that pi first 100 000 000 only has a maximum of 8 consecutive 9... Thats (1/10)^8 which is 1 option in 100 000 000. Just how much randoms are the numbers of pi. And much weigted are each of the 10 bases. {1 2 3 4 5 6 7 8 9} in pie?
Just gave the 314th like to this comment :)
OP, so optimally optical
Looking for the first correction. Would you also add it on the video as annotations?
why do we stop to explain a dot product, but you did not stop to explain the relevance of the Taylor series of arc-tan. I used one of those way more then the other in undergraduate math/physics and I remember one of them way better than the other, unfortunately, it was the one you explained again, and now I have to go and google information about Taylor series approximations after watching all 3 of these amazing videos.
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
Let's hope that all of his videos will be longer than 10 minutes. ;-)
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
could not relate to this more
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.
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)
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.
So you could say this is "Blue's Math Clues"
this is what he said at his ted talk!
10:08 - Nice square root sign you just made there.
@abdullah almasri "How colliding blocks act like a beam of light... to draw a yellow square root"
You're getting many likes for this comment, mostly because your time link is just above the like button and our fingers are fat!
@mech0s Hum are you ok ?
@I StM I Not really. Thanks for asking. Fat fingers and delerious with a cold. I still like the comment.
A moment of symbolic reflection
Who’s here for the light clacking noise?
Insanely satisfying
Yes.. but how does one make it ?
Yea
Python anyone ?
Oh hello there kenny
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 ^^
@wick jon thanks for the link
Commentjng for later
I had a feeling these videos were too polished to be made with a generic animation engine.
@Sam Welter lonk xD
It’s gotta be with those formulas he shows us! Or I’ll riot!
A click on this video is worth 80 IQ points
Clicking on this video made me feel like my IQ is 80...
@Lattamonsteri if you start with 0 IQ points... I guess you have more than 20, because you hit the @Maxim G REPLAY link. That on its own is worth 20 points ;-)
@Lattamonsteri I think he means it's worth plus 80. So for example if you normally have 100 IQ, you can change your perspective, now it's as if you have 180 IQ.
Hahahah classic.
Level does not meet the minimum requirements, therefore 80 IQ points is not received.
Grant, is there a similar way to produce e? Perhaps a feedback mechanism that increases a measurement exponentially?
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).
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.
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.
You could use Euler's identity to get e out of this. You'd wind up with an equation involving ln(-1) and i.
@Paul Paulson That was pretty neat
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.
Yup! My mistake.
Nice catch, floor is the same as ceiling-1 everywhere except at integers
Nice feedback!
I only listen to real music
clackclackclackclackclack
My brother makes music like that
😂
Click clack click click clack
Hörst du die Maschine in der Nacht
This is a legit song quote
My favorite part is the chorus
nzZzzZzzZZZZZZZZZZZZZZTTttt
Aphex Twin made this song in 1997. It's called "Bucephalus Bouncing Ball."
3b1b: sjjabauskckdbsbdjfjsjshfk math
my monkey brain: square make good sound
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?
@mejecamsharp observation! if there were no small block or the big one is infinitely heavy(both are basically the same scenario) it's not hard to imagine the big block would just take a sharp turn at the wall.
@M. Sierra yeah I don't have any proof that it's gaussian. It's just a go-to guess. tbh i cant seem to wrap my head around it when it comes to proving it's a specific function :P
Isn't there a Pi in everything that goes to and fro?
@Henry Yang I was thinking about this, and the cardioid represents the energy needed to cancel the momentum of the large block, not simply the energy of the small block + the big block. As the block gets smaller it will approximate this curve better. It's really easy to see why pi shows up if you think of the small block as an additional circle rolling along the cardioid and approximating the curve better and better the smaller it gets.
@3blue1brown tracking the positions of blocks that add their energies in a limited way and also skate off into infinity... isn't thinking about it in this way getting awfully close to describing the Mandelbrot set using a physics engine?
That was seriously awsome. Newtonian math. Gravity, Optics, differential geometry.
differential geometry? where?
it's not gravity but conservation of energy.
"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.
3Blue1Brown and Kurzgesagt both just uploaded on the same day again...
This is beyond science
@Tim what did Kurzgesagt cherry pick in the marijuana video?
There is one Kurzgesagt video everyone should view and that is "How Far Can We Go".
You are insulting 3Blue1Brown. Kurzgesagt has nothing to do with science.
This is science
Is Kurzgesagt from germany?
This is beautiful anyway you look at it. Presentation, Voice, Animation, Content, Optics, Geometry, Energy, Momentum. Euler would have loved it.
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
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!
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.
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.
10:43
Holy shit, Let the light pass through the mirror and flip the whole world.
3b1b thinks in the reflected world.
Grant, you’re a wizard.
The reason why I study English is to watch 3blue1brown's video.
But there is a limit.
Please anyone translate to korean😢😢
I don't speak Korean, sorry ): 🇰🇷
You can try asking for help on reddit. between the maths subreddit, the 3B1B subreddit and the translators subreddit, there's maybe some people that can use both languages well enough to translate this.
Shujaun...?
@Anurag Thakur I know it's four months late but its Sujeon
"When you think about it", life, the Universe and everything, people do form habits of practical thinking that obliterate the true origins of actual knowledge in the general accumulated histories of "whatever works".
If you recognise the strategic emergence of dimensionality coordination/interference positioning resonance properties of the number sequences inherent in Quantum-fields of e-Pi-i omnidirectional-dimensional connection continuity, then "at the limit" .dt orthogonality of sequences, 0->1+/- Inflation condensation-containment, a 1st law of Thermodynamics superposition is aligned as scalar proportioning orbital, "radially", equivalent to "geometrical" tangency to this here-now-forever, zero-infinity axis.
This is a sequence-derivivation of all frequency multiples of aligned scalar proportioning, observable in Lyman-Balmer type emission-absorbtion Spectroscopy of e-Pi-i multiples of phase-locked orbital-spacing.
So the Pure Dynamic Mathematics basis of Logarithmic Time Communication orthogonality sequences is everywhere evident in fractal condensation by i-reflection total internal reflection containment, that is the functional cause-effect "emanation"/Exclusion Principle of Logarithmic Time, ..and the above video is a "discovery" type problem solving technique that can illuminate the inherent processes of time duration timing.
8:12
"Duck products" can't be unheard
This video has transcended into mathematical art.
my mind is blown everytime 3B1B uploads
3blue1blown
* mind blown moment *
Omg, that way of rephrasing the light bounce problem as reflection gave me shivers!
I won't be suprised if 3Blue1Brown comes with other solution of the same problem using Faraday's laws next week!!!!!!!!
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
I wasn't expecting to be educated on this sunday evening. Thank you 3b1b for that cool analogy!
Around the seven minute mark he literally proves the relationship between energy, mass, and the speed of light and even pi using kinematic --> optic simplification by plotting vectors. I don't know why but taking a completely different discipline and a barebones problem to prove fundamental properties of the universe is so interesting!
It's not important, but it's technically not the floor of pi/theta, it's the ceiling minus 1. Using the 45 degree example for a moment, pi/theta would be pi/(pi/4) or 4 and floor(4) is just four, but it only bounces 3 times.
Just a nitpick.
You really is a magician,showing us the magics of mathematics.😉
I came here because I was told that there were loads of satisfying clicking sounds in 3blue1brown's latest videis.
Your channel makes me love math.. math before you=😵... Math after you=😍
I love how it got to the exact same part of the proof at the end...
no u
Oh also can you upload the colliding blocks thing? Please :c
Edit: just looked in the desc :P
Totally above my head...until the last two minutes, when it all kinda clicks together. (I don't know many of the terms, though.)
This is awesome.
Don't think you're just going to get away with that pun.
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
"is worth 80 IQ points"
Me: ok then let's collect IQ
These animations are so satisfying, completed by this well explained reasonning .... Thank you !!
Never saw a solution more beautiful than this!
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
I think 3b1b has an addiction
@Sthaman Sinha Haha - it's an actual dumb flat-earther. I've never met one of you before.
Sthaman Sinha sounds like something a sheep would say
@PCreeper394 just what I was thinking
314th like....
@Grin Reaper Of Trolls me too
11:20
and that's how you can model refraction
when the practical/appearent speed - is not constant
2:46 those two angles arent the angle of incidence and the angle of reflection
BigBakerBoi - 2019-02-03
for the record this guy is making literally two squares hitting each other that interesting
Sarthak Kumar - 2019-05-19
I THINK YOUR ENGLISH IS WEAK....
Sarthak Kumar - 2019-05-19
@TTANFIELD AND YOURS TOO... LOL
Sarthak Kumar - 2019-05-19
@TTANFIELD INDEED ..
Mononoke Hime - 2019-06-15
@Doniel F But he said "for the record"
This is clearly implying that he's not surprised.
Edit: actually, I don't even now why saying "for the record" would imply that he's not surprised. I just feel like it is.
So I would be thankful if someone could explain it to me, because I'm far from being in the mindset to think about it at the moment.
U K - 2019-11-12
True said. It is very satisfying to watch.