Name: Strange Spheres in Higher Dimensions - Numberphile
Uploaded: 2020-03-27T18:03:58.000Z
Duration: 10 min 33 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

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

Adverbs of manner

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

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,

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."

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

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.

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

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,

spheres in as last time, so four spheres in

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

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

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.

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

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

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

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

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?

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

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,

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.

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

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

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

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

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

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

Subtitles ListPlay Video

Strange Spheres in Higher Dimensions - Numberphile

stuff

ultimate

slightly

grocery