Placeholder Image

字幕表 動画を再生する

  • I'm gonna try and capture, contain and investigate a higher dimensional sphere.

  • Starting with a circle. So, if we begin in 2D, you've come across circles before. They look, you know, roughly like that.

  • We're gonna try and

  • contain a circle so we can look at how big it can be. So, what I'm gonna

  • do is start with a box, four by four, for reasons I'll explain in a moment. So that's four units that way,

  • it's four units that way. And now I'm going to put some safety padding spheres in there.

  • So if you kind of measure, if I was to actually split this into quarters, I'm gonna put safety spheres,

  • well, they're, they're circles. I'm gonna call them spheres no matter what dimension they're in, right? So when I say sphere,

  • I mean 2D sphere,

  • it's a circle. So, I'm not very good at drawing circles, so, there, there'll be one there, and it just touches,

  • it kisses all four sides of that box. There'll be one there, which just touches, or, so that just contacts

  • that sphere, so that just contacts there, and that one just contacts there. And so

  • Brady: "You really are bad at drawing circles."

  • That's adequate, okay, that's,

  • It'll do, it'll do. They're not overlapping

  • Brady: "Parker Circles."

  • That's a Par-

  • So. Meanwhile.

  • So this is why I made the box four by four, because it means these are now unit spheres or unit circles.

  • So they all have a radius, that's one in that direction, one, they've all got one in every, every direction.

  • I'm now gonna see, what is the biggest

  • circle I can fit in the middle.

  • So I've got my containing box, I've got my padding spheres, or circles,

  • I've now got, this is the subject, this is the one we care about, and I'm now gonna try and fit the biggest

  • circle possible in the middle. So, how big is that circle? Well, there's the very middle of our box.

  • Here is the center of this padding circle over here. We know that's one that way.

  • That's one that way. So what we have, if we zoom in a little bit, so that

  • complete length there, that whole thing is root 2, and that there is one. So that little bit there is

  • root 2 minus 1,

  • which equals around about

  • 0.414. So that's the biggest circle we can fit in the middle in 2D. Onwards and upwards to 3D. For 3D

  • we would have a cube with eight padding spheres, and so I've brought oranges.

  • These are the, the cheapest spheres I could find in the grocery store.

  • And, so that, that's the rough equivalent of what we had before.

  • You know what, I don't think they're gonna, I'm gonna tape them in place.

  • -kay, and those go on there. So I haven't got the

  • containing box, but you can imagine that there is,

  • the base is still four by four, that we had previously. That's four in every direction.

  • But now, as well as four by four, we'd have another length up here.

  • So we'd have another dimension this way, which is also four. And then here's the top of our four by four box,

  • and we can fit twice as many

  • spheres in as last time, so four spheres in

  • 2D, eight spheres in 3D. And you can see, again,

  • there's a gap in the middle, right?

  • So, inside our box, inside our padding spheres, in there is a void, and the question is, what is the biggest

  • 3D sphere

  • that we can fit inside, inside there?

  • Here are four adequate...

  • ...cir... ignore that. Okay, so,

  • Brady: "They've gone a bit beyond kissing?"

  • They've gone, yeah, yeah, they're getting to know each other very well. And so, who am I to judge?

  • So that is like the cross-section in here that lines up with the middle of the bottom layer of spheres.

  • So what we've done already is

  • for the center point

  • here in the middle, we know, from the middle of this one, it's one across to get there,

  • it's one up to get there, and so that diagonal

  • was root two. And then, before, we subtracted off the radius just to get the difference there.

  • But now, we want to, we need to go one, aren't we to go up, as well. And again,

  • it's another quarter of the whole box. So we have to go up a single unit, and so we need to do Pythagoras with

  • that, our base is now root two, and then our height is one. And so to work out what that is, root 2 squared is 2,

  • one squared is one. That there is root three. And actually, Pythagoras

  • generalizes to any number of dimensions.

  • It's just the square of all the different orthogonal directions, take the square, add them together, take the square root. Very straightforward.

  • So that's gonna make our lives very easy going forward. And once again, that diagonal, we want the difference between the radius of the sphere and

  • what's left. And so our new

  • radius of the biggest sphere we can fit in there is root 3 minus 1, which equals

  • 0.732 ... you know ... 05,

  • some stuff, okay. So actually, it's slightly bigger.

  • And that kind of makes sense, because, you know, there's not much room there to fit a circle in. There's a little bit more space

  • in there,

  • because, you know, there are more directions you can go in.

  • The question now is what happens in, in 4D? What happens in 5D? If we keep going

  • up and up and up and up and up, what's the maximum size?

  • The box is never getting any bigger.

  • It's always four in every direction. The spheres are never getting any bigger. The packing spheres are always radius one.

  • The middle sphere is

  • always

  • inside

  • the void left between the box, which isn't getting bigger,

  • it's getting more directions within it, but it's not getting bigger lengths, and the spheres, which, again,

  • they go in more directions,

  • but they're not getting bigger. And a sphere is always, in any dimension, all the points which are the

  • same distance from some central point, you just have more directions in which that can happen.

  • So this is the dimension D that

  • we're currently working in, and this is the radius of the middle

  • sphere in

  • dimension D.

  • So, we've already done the first couple. We, in 2D, it was 0.4142,

  • 3D, which we just did, 0.7320,

  • 4D, it's still the square of one, square of one, square of one,

  • square of one, for the fourth direction, add them together, take the square root,

  • which is 2! That's not so straightforward. So in fact, the radius of

  • the central sphere in 4D is

  • root 4,

  • which is 2 minus 1, which is 1. So now, the middle sphere

  • has exactly the same radius as the padding spheres around it. So there will be 16 padding spheres in 4D, and the single one

  • in the middle is exactly the same size.

  • Brady: "That's awesome!"

  • It's pretty, it's pretty cool. 4D, lovely stuff happens.

  • 5D - let's see what happens. You have now got

  • 32 padding spheres, the box is 4 in all of 5 different directions, and the middle one has a radius of 1.2360,

  • so now it's bigger. The middle one is is bigger than the other ones around it. That's odd.

  • Let's keep going. In 6 dimensions,

  • it is 1.4494 bunch of stuff, a little bit bigger again,

  • but hopefully slowing down. 7 dimensions, 1.6457.

  • 8 dimensions, 1.8284.

  • 9 dimensions, it's 2.

  • So, in 9D, the middle sphere

  • has the same radius as

  • the distance from its center to the outside of the containing box. So in 9D, the middle sphere has just

  • contacted

  • the outer

  • surface of the box. It's, it's somehow got past the padding spheres, which are all over it,

  • it's got through them, even though it's limited

  • by them, like, they, they define how big it is.

  • Brady: "It's not allowed to touch them."

  • It's not, it kisses them, it's not going through them. Somehow,

  • it's big enough that it's contacting the box without

  • going through the spheres touching it.

  • An in 10D, slightly disturbingly, it escapes the box completely. It's now 2.162.

  • So, as of ten dimensions, and indeed ten dimensions onwards, it is bigger than the original box which was containing it.

  • I mean, the short moral of the story is that higher dimensional spheres are really weird.

  • And difficult to contain.

  • Brady: "How are you reconciling this in your head? How does it make sense to you?"

  • Ah!

  • There, there's a trick you can use in mathematics called not worrying about it.

  • Everything we have learnt

  • no longer really works.

  • And so, people who do have to come to grips with higher dimensional spheres will describe them as being spiky.

  • Higher dimensional spheres are spiky.

  • There's some of those spikes are making it out of the box.

  • Brady: "Like that ship that Superman went up in in the original Superman movie?"

  • That, I mean, I was trying to think of a good example

  • which everyone will be able to, you know, really connect with. And I think you've nailed it with that

  • 70s?

  • Superman reference, if I remember the film? Well, good job, good job.

  • I can see how you're the master of bringing science to the masses, Brady.

  • Brady: "I'm gonna put the picture on the screen!"

  • Okay, you put the picture on the screen. Good point, good point.

  • No, yeah, so, so imagine that, imagine your favorite film with a spiky thing in it. It's a bit like that.

  • And so, people don't get hung up on that. So you just go, higher dimensional spheres are spiky.

  • And our word "spiky" is not perfect, but in some way it grasps what's going on in higher dimensions.

  • It just turns out, the more dimensionss the sphere is in, the pointier it gets.

  • Well, somewhat appropriately, this video about fitting circles and spheres

  • into a square space has been brought to you by Squarespace.

  • Squarespace is the ultimate online service for creating and managing websites. I use it myself for various projects, including blogs,

  • calculator unboxings, of course, and podcasts. If you've got a bright idea, something you want to share with the world, maybe a website all about

  • spiky spherical objects in movies, then you really need to be checking out Squarespace.

  • They make everything easy, from choosing a domain, picking a great template to get started, and making sure your site looks good on all devices.

  • You can try before you buy,

  • setting up a site for free, just to make sure it's as good as I'm saying. You want to go to squarespace.com

  • slash numberphile, and that's gonna get you 10% off your first domain or website. There's the address up there,

  • squarespace.com

  • slash numberphile.

I'm gonna try and capture, contain and investigate a higher dimensional sphere.

字幕と単語

動画の操作 ここで「動画」の調整と「字幕」の表示を設定することができます

B1 中級

高次元の奇妙な球体 - Numberphile (Strange Spheres in Higher Dimensions - Numberphile)

  • 0 0
    林宜悉 に公開 2021 年 01 月 14 日
動画の中の単語