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• 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

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

• 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,

• 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?

• ...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.

• 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

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

B1 中級

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

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林宜悉 に公開 2021 年 01 月 14 日