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Darts in Higher Dimensions (with 3blue1brown) - Numberphile

Apr 09, 2020
TRUE? I tell you to choose four random numbers. Let's add their squares. What is the probability that they are less than 1? And you think to yourself, oh, I don't know, you know, if they were all less than 0.5, I guess that would happen? But if one of them were 0.9, that doesn't rule out everything because the others could be quite small. And to think about this, let's think about the first situation where, instead of doing it for h1 and r1; when we're here doing it for what seems like a much simpler situation and it was, the probability that h zero is less than r zero, which simply means that x zero squared plus y zero squared is less than 1.
darts in higher dimensions with 3blue1brown   numberphile
We have a Purely analytical statement, choose two numbers from a hat according to this rule, what is the probability that the sum of their squares is less than 1? You answered it, geometrically. You immediately knew that we need to use pi. We're thinking about the area of ​​the circle, right? And it's actually quite subtle what's going on there, you have a question about a couple of numbers and you're gaining intuition and using facts that we've discovered in mathematics, like the area of ​​a circle. Thinking of it not as two separate entities but as a single point in a two-dimensional space.
darts in higher dimensions with 3blue1brown   numberphile

More Interesting Facts About,

darts in higher dimensions with 3blue1brown numberphile...

That may seem obvious, the reason I'm insisting on it is to look at what we have here. Four separate numbers, what you're asking for is a probabilistic property of these four numbers. It might be natural to think of them as a single point x zero, y zero. x1y1 within the 4-dimensional space. And it's basically a space that has four coordinates of real numbers. And now, Brady, let me ask you: what is the analogous question? In the same way that asking what is the probability that the sum of two random numbers is less than one, asks the question of what is the area of ​​a circle.
darts in higher dimensions with 3blue1brown   numberphile
When we ask what is the probability that the sum of four squares is less than 1. What do you want to know? (Brady: Well, I was how do you think we were going to end up using a) (cube and a sphere, but I feel like we've jumped?) We've jumped! We have jumped! The general idea that if I choose three random numbers x squared, y squared, z squared, what is the probability that they are less than 1? If they are chosen according to the same rule, what you will ask for is the volume of a sphere, because what it means that the sum of their squares is less than 1 is that they are inside a unit sphere.
darts in higher dimensions with 3blue1brown   numberphile
The volume of a sphere is 4 thirds pi r cubed. r will simply be equal to 1. So that's 4/3 pi. That is the volume of a sphere. The volume of the cube, which is 2 times 2 times 2, will be 2 cubed, which is 8. So if we were to ask an analogous question where 3

dimensions

appeared, you would simply divide these two numbers. You would have 4/3 pi, divided by 8. Which would be pi sixths. You'll see that it's actually quite useful that we omit the odd

dimensions

here. And here we go directly from 2 to 4, the question we want to ask now is what is the volume of a 4-dimensional sphere?
Or the four-dimensional equivalent of volume. Instead of saying hypervolume or making up a new word, they just say volume. You could say fantastically measured, but we will only say volume of a 4d sphere. And again, if he wanted to be pedantic, could the mathematician say ball? Because when they use the word sphere they normally mean the limit, right? It's just the peel of the orange. It is not the content of the orange. So what we really want is the ball. The other thing we need is the volume of a 4d cube. The cube is the easy one.
Just like in the other cases, the side lengths are all 2, 2, times 2, times 2, times 2. So that volume is 16. Volume of the sphere, there is a whole interesting discussion we could have about volumes of

higher

dimensions. spheres, the surface areas or surface area analogues of

higher

dimensional spheres, and where they come from. I'm not going to tell you that whole story now, maybe some other time. It happens that it is pi squared halves times r to the fourth power. Then take an extra pi. Which is pretty interesting, when you go from 2 to 3, you don't pick up that pi, but when you go from 3 to 4, you pick up that extra pi.
And since our radius is 1, in this case it will only be halves of pi squared. And let's remember what this is. This isn't the probability of hitting exactly two shots, right? It's that you hit those first two shots, so it's the probability that your score is at least two. Maybe there are three or four too, so we can write our answer here. The probability that our score is greater than or equal to two is equal to pi squared, and I'm actually going to write it like this, 16 as 2 to the power of four, just to remember where it came from.
So this was a kind of cube or hypercube. And then multiplied by 2. Where that was the denominator that we got from our volume of the 4d ball. And now we keep doing this for all the infinite different scores we can have. Very similarly, when we ask about the probability that our score is at least 3 or that it is greater than 3, it all comes down to asking: we are going to have 6 different coordinates: x zero, y zero, x1, y1, x2, y2; and that the sum of its squares is less than 1. (Brady: I ​​don't know how to calculate the volume of a six-dimensional ball) - No!
Luckily, mathematicians have discovered it for us. And again, it could be quite an interesting story, but if you go to Wikipedia, you can see a graph, there's a pretty nice pattern that we're going to point out where it comes from. But it turns out that the volume of a 6d ball is pi cubed, more than six times r to the power of 6. And so the volume of the 6d cube is, of course, 2 to the power of 6, so the answer to the probability of getting it right those first three throws, it will be pi cubed divided by that 2 to the power of 6 and then what was its denominator, which was 6.
Because that r ends up being 1. And now I'll tell you the most general fact, which is pretty clear. -blew it. The volume of a 2n ball is equal to pi to the power of n, so half the dimension, right? So in 2 dimensions you see pi, in 4 dimensions you see pi squared, and in six dimensions you see pi cubed. So pi to half the number of dimensions, divided by n factorial. Very clean, very beautiful. It's very reminiscent of another real celebrity number that a couple of people might be familiar with by now and that might be on their minds.
But if not, we'll get to that in a moment. Since this is a fact that we will simply convey from above, let's calculate the expected score. (Brady: Please, that's what I'm here for!) Great. First let me have a glass of water. Alright, are you ready for the grand finale? - (Brady: Yes!) - The final touch. It looks like it will be chaotic but it will collapse in the most wonderful way. So the expected score. Expected score, now remember what this means, let's have: it will be 1 times the probability that your score is 1, plus 2 times the probability that your score is 2, plus 3 times the probability that your score is 3. , etc.
Now all our expressions are bigger than things. They don't say exactly the same as, but that's completely fine. Because if I wanted to say, for example, what is the probability that your score is exactly equal to 2? So this will be your probability that your score is greater than 1, right? Minus the probability that the score is greater than 2. Because for it to be equal to 2 it is greater than 1, but it is not greater than 2 and this completely covers the possibilities, right? If you probably score higher than 1, that covers all possibilities of scoring higher than 2, so it's okay to just subtract this.
We don't have to do anything more than that. So I'm going to expand all of this a little bit before everything collapses. So this would be 1 times the probability that the score is greater than 0, minus the probability that the score is greater than 1, plus 2 times the probability that the score is greater than 1 minus the probability that the score is greater than 2. And just one more for good measure, the probability that the score is greater than 2 minus the probability that the score is greater than 3. So at this point we have a lot of cancellations happening. So we have this probability that its score is greater than 0 and it stands alone.
For the probability of it being greater than 1, we subtract one here but then add two here. So we have probability that the score is greater than 1 and then similarly for a score greater than 2, we subtract 2 here but add 3 again here. So we're just going to add the probability that the score is greater than 2. And generally, all we do is add all the numbers that we just saw. All those fun things involving pi. Because with 3 we are subtracting 3, but in the next one we will be adding 4, and so on. So in every puzzle we just had, every micro puzzle, we just added up all of their answers.
So the probability that your score is at least zero, that's 1, because your score is definitely greater than zero because you get a point just for playing. Probability that it is greater than 1, that is the one you answered me first. What I was saying is pi divided by 4. Next up, so I'm going to write this slightly differently because we always have some power of pi divided by some power of two. I'm going to write this as PI quarters squared divided by 2. Okay? Then came 4 squared, that's the same as 2 to the fourth power and then pi squared just went in there.
So I'm going to write it here as pi quarters squared times 1 over 2. And then the next one we had here. And again, I'll write it in terms of quarter pi. Instead of writing pi cubed over 2 to the sixth power, which is the same as pi cubed over 4 cubed, I'll write it as pi over 4 cubed. And 6 is actually 3 factorial, like the reason we see the 2 and then the 6, comes from this general pattern of hanging, of having a factorial. So here we have pi over 4 cubed times 1 over 3 factorial, this is actually two factorial. And maybe you see where this is going or maybe at this point you're saying that you promised me this would seem simple.
It actually seems extremely complicated, you have factorials, you're grouping the pi's together for some reason. So there are some viewers right now, especially if maybe they just got out of a calculus class, or if they're particularly ingratiated with the number e, who will see something screaming in their head right now. There is something known as a Taylor series for e to the power of x. Which is that it's 1 plus x and okay, it's x raised to the 1 factorial over 1. x squared over 2 factorial, x cubed over 3 factorial and so on. Where you evaluate it as an infinite polynomial where each of the terms is 1 over n factorial.
It's not just that e to the x is equal to this, in fact I think it's the healthiest way to look at the exponential function and what e to the x is: this defines it, right? This is what should come to your mind when you think about exponential growth and e to the power of x is this particular infinite series. This particular polynomial. This is where it will come up, especially in probability, it lends itself to an easier interpretation of why it is its own derivative, there are all kinds of good things, it extends, makes it easier to understand things like e al (pi )(i) ; all of that.
This is the healthy way to think about e to the power of x. So if you looked at this and weren't already thinking about e, what it means is that there is a healthier relationship with e waiting for you in the future. - (Brady: Or if you didn't see it you have an unhealthy relationship) You currently have an unhealthy relationship with e, that is absolutely true. If you didn't see it. Or you have no relationship with e. But if we compare this to the series, which I uselessly drew a little far away, but you can put them together quite easily, what's playing the role of x right now are the quarter pi.
So all of this is e to the power of pi quarters. Which, if we plug it into a calculator, is roughly equivalent to two point one nine three two, go on, go on, go on. That is, if you're a terrible random

darts

player, who unrealistically hits inside a square with uniform probability, which you wouldn't do because it would be rotationally symmetric, but whatever you're hitting inside a square. And you continue like this and then you play a thousand games, on average your score would be 2.1932. And remember my score was 4. (Brady: So aren't you twice as good as someone who has absolutely no skills?) - I couldn't ask for a better recommendation Yes, that's right.
So there are a couple of things I like about this puzzle, the reason I want to pass it on. Hopefully the most unexpected or mind-blowing component is that we're even talking about higher-dimensional geometry. Naturally, you found yourself asking, in the middle, what is the volume of a six-dimensional ball? Good? And nowhere in that was it asked: Does the universe have six dimensions? You know, as if string theory invokes 6... no, no, no, as if that's not the reason mathematicians necessarily worry about higher dimensions. What was happening is that you had six numbers and you were encoding a property of those six numbers with something that we like to describe geometrically.
Instead of saying that the sum of its squares is less than 1, you say that it appears inside a six-dimensional ball. That also means it saved you some work. I submitted this question so you guys would like it.some followers of the channel as an early preview of things, and one of them responded to me saying: you know, I've been working on it and it's just, like, really hard. I've been working on these integrals. What is very strange is that even in the case of the second dart I get pi squared for some reason. So it is working on all these integrals, under the hood what is happening is that it is rediscovering the volume of a four dimensional ball, TRUE?
So it provides a more universal language for people to talk about it. It emerged from two-dimensional geometry. Nothing in the bullseye is four-dimensional, and I think it's just a common misconception that people who listen to mathematicians describe things like four-dimensional manifolds or that Poincaré's conjecture has been answered for everything except four dimensions. They don't really care about something where you can move in four dimensions, it's about encoding quadruplets into points. And the second is that I think it helps build a healthier relationship with e. Because this series is much more important than the number itself.
I could, I could even just ask the riddle and then we won't answer it, but it's a good thing to end with. (Brady: Yes) So here we were choosing these random numbers between negative 1 and 1 and what they asked us: What we were asking is something about when their squares add up. When is that less than 1? You can play a much simpler probability game where let's say I'm going to pick numbers from 0 to 1 with uniform probability. And I'll continue until the sum of the numbers I chose ends up being greater than 1. So, for example, if the first number you choose is 0.3 and then the second is 0.6, their sum is 0.9.
And if the next one you choose is 0.5, that's the point at which you pass. So the question you may ask yourself is what is the expected number of samples you should take before you exceed one? e appears in the answer, right? And the way it appears is quite similar here and it's a distilled form because it doesn't involve circles, so there's not the confusing factor of pi. You see the factorials much more impeccably. It is very important that you mention where this puzzle comes from. I saw this on Twitter, I think it was Greg Egan. He specifically designed a puzzle such that the answer would be like adding the volumes of higher dimensional balls, because we have this wonderful formula for the volume of higher dimensional balls, when the number of dimensions is even.
And everyone who has a healthy relationship with e looks at this, a thing to the power of n divided by n factorial, they scream in their mind the exponential function with e to the power of x, right? And in a way it asks you to add them. Which is kind of strange, why would you do that? It's very strange. How do you interpret the sum of the area of ​​a circle with the volume of a four-dimensional sphere? So he specifically designed a puzzle for this to be the answer, which I think is beautiful and clever. Here is something very interesting.
We are adding all these volumes, right, it converges. And what that means is that higher and higher, the volume of a high-dimensional sphere is pretty small, right? In fact, we could calculate it. If I said hello, what is the volume of a 100 dimensional ball? Well, that would be pi to the power of 50, divided by 50 factorial. Well, the thing is that with 50 factorials the numbers you are multiplying are 1 times 2 times 3 times 4 times 5 times 6; you are multiplying larger and larger numbers. Pi to the power of 50, you're always multiplying a pi, right? Then the denominator starts winning because in the end you added an extra pi at the top, but an extra 50 at the bottom.
Then go back quite a bit. About 2.368 times 10 to the power of negative 40. If you think about what that means, if you're in one hundred dimensions, right, and you look at a 100-dimensional cube and you say, let's look at the sphere that touches each edge of that cube, what proportion of the square does that sphere occupy? It's about 10 to the minus 40, right? Which... to a lot of people might seem pretty counterintuitive, right, because if you think about a two-dimensional circle, it fills most of the square. Or a three-dimensional sphere that fills most of the cube. But if you think about what he's really asking... (Brady: Wait, are you doing what you just scolded people for and imagining them as real spaces?) (Instead of...) - Well, I do. are!
Very sorry. When I say real, I mean they are exactly as real as the numbers, right? Like the real number line, you won't find it in nature. You're not walking through the woods and there's the real number line. In the same way as 100-dimensional space, it is a useful abstraction. What I would suggest people not to ask is whether the universe has 100 dimensions. And is the only way it makes sense to ask questions about 10 dimensions, if the universe has that wiggle room? But people find it quite contradictory how small the balls are, in a sense, in higher dimensions.
But if you interpret it much more literally by saying pick 100 numbers, okay, all chosen evenly between negative 1 and 1, add their squares, what is the probability that the sum of all those squares is less than 1? Like, oh, we have 100 of them; of course they will be bigger than 1, there are so many numbers. The same thing that is happening. There are so many dimensions. That is why it is useful to have a back-and-forth between analytical thinking and geometric thinking. If you haven't already, be sure to check out Grant's awesome math channel:

3blue1brown

. And if you want to hear more from Grant on the Numberphile podcast, check out the links on the screen and in the video description.

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