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.
@@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"
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.
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
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
@@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."
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
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.
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!
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.
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)
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.
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.
Your animation is legitimately the absolute best on RU-vid. It's always obvious what you're doing, and makes a difficult subject seem plainly intuitive. Wow!
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!
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!
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.
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 ^^
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!!!
I envy today's children who have access to videos like these when learning math, geometry and physics. A good visualization like this is better than hours of lecture.
"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.
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.
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.
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
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)....
Imma just point to the start of the video and go off in a little tangent... The sight of Grant's face is such a refreshing think to have. Not only are you a genius, you're also a very beautiful man
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!
I just logged in to say: thank you SO much for letting that last collision finish just before changing scene at 1:10. I was genuinely quite stressed for a moment that I wouldn't get to see that last collision. :') I'm surprised you didn't mention in the last video how it's a bit annoying that a lot of the powers of (factors of) 100 result in a final collision that takes quite a long time! (Thank you for making these awesome, beautiful videos. I'm just exploring and reexploring a few of your videos, and loving it :))
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!
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 :) ).
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
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.
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?
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!
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.
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.
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.
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!
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.
ALRIGHT I GIVE IN!! I’ve finally subscribed haha. I kept hearing about 3Blue1Brown as this cool mathsy-sciency channel but didn’t really get hooked until now. This series of videos has blown my mind in the best way: I didn’t understand half of the content I watched, but I now understand more than I did before watching it. That’s good enough for me - you have earned your place in my sub box.
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!
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!
If it were the case... I would have an IQ of 2000 😂
5 лет назад
for me it does the opposite, it makes me think about how fucking bad i'm at math and makes me want to give up on learning because i would never be able to think about such an awesome process like the one shown in this video. fucking hell, some people are incredible.
@ Same here... I start feeling bad about how stupid I am and how much more there is to learn and understand in this universe but I am so dumb I don't understand most of the things I find interesting like math and science but I can't stop watching RU-vid videos by these smart people. Wish I was smart enough to contribute something to this beautiful world😔😢😢🙁...
The material of the string is a bit reflective. The environment is of high contrast and therefore the resulting color changes. In a equally lit environment or with a lambertian material the color would stay the same
