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.
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