All right, so we've got a sphere.
It's sort of a shell of a sphere, and then there's a thing for it to stand on, a little base,
there's a hole in the top,
and there's this pattern, this cool pattern of sort of pentagons, and so on.
And I've got a flashlight, or a torch,
if you haven't been in America for too long. You can take the sort of lens off
and you just get like a, it's supposed to be like, like, it's like a candle or something.
But what I want to use it for is
to show this thing's stereographic projection.
So, if I put the light right at the North Pole of the sphere
Then you get this pattern of shadows. So, right, so what is stereographic projection?
It's a map from the sphere
to the plane, and the light rays sort of do that map.
A light ray comes from the North Pole and
let's say this one here, right, so it hits the point on the sphere here
and it goes down to the the table here, to the plane. So and you can see like if I put my little finger
near to this point on the sphere, you can see the shadow
on the plane is where is where it's going to.
It was one of the first ways that people thought about, like, maps of the
Earth, or the heavens, right, the celestial stuff, so the Greeks knew about this stuff and a lot of the old
kind of star maps are done with stereographic projection. So one of the things that's really cool about it
is it preserves angles. And you can see on this model as well, these like three or six 60 degree angles here
It's like it's like a radiation symbol here
And the shadow has these 60 degree angles, and the sphere also has those 60 degree Angles
So not only does it preserve angles, right, angles stay the same, it also preserves circles.
You can see on this, there's all these
great circles like an equator of the sphere.
Well actually this one doesn't have an equator, if this is the North Pole, there isn't an equator drawn in,
but, like, the path that you take if you just start sailing in a straight line around the Earth is this circle that's like an equator.
If you look at where is this path on, down on the shadow, I think it's this one here
And you'll see that it's a circle there as well. So to see the circles a little bit better
maybe, let's, let's have a look at this one. It's got circles all over the place. It's kind of like a dodecahedron, icosahedron thing with circles.
Let's see what kind of shadow this makes, and if I hold the light in the right place you can see,
so this circle up here
goes to this circle down here. So, yes, somehow it knows that
circles are supposed to be circles, even though things are getting kind of weirdly distorted, right, the size of the circles changes a lot
on the plane, they're all the same size on the sphere, of course, but it still knows that they should be circles, right?
I mean, you know if you move it somewhere else,
then, like it has to be right at the north pole for it to work, and the same thing with the angles.
If it's not as a north pole, it's not a stereographic projection, it doesn't have all these nice properties.
So here's a yet another, right, pattern on a sphere, with this sort of weird curvy thing going on, so,
well, what's this gonna make? You may be surprised. Well, let's find out. And, put this in.
There's this perfectly regular
square grid hiding inside of this really curvy round thing.
Well, the way I made this is I started with the square grid and then I put it back up onto the sphere to see what's
going on. So I could just continue making this square grid going further and further out.
And then I'm gonna get closer and closer in to the North Pole,
but I'll never actually get to the North Pole, right? The North Pole somehow
corresponds to going infinitely far out this way, going infinitely many squares,
eventually, well I wouldn't ever get to the North Pole,
but that's what the North Pole corresponds to. The North Pole isn't there. It's a one-to-one thing. Every point on a sphere
gets a point on the plane, apart from the North Pole. So, so the plane, plus one point at infinity,
infinitely far away, is the same as the sphere. This is a two-dimensional surface goes to the two-dimensional plane. Let's go one dimension down.
Let's do the, so the equivalent of the
sphere the 2-dimensional sphere is the one-dimensional circle, right?
It looks like a line if you just look at a piece of it. Let's draw the, the table, and then
my light is gonna be here at the North pole of this
circle, right? A circle is sort of a one dimensional version of the, of the sphere.
And so, you know, a light ray comes down here, and it hits the sphere somewhere, and it hits the plane somewhere,
and that's the map. Stereographic projection takes you from here to here.
And so, you know, what we were doing is, okay, squares are marching off this way,
and like, you know, there's, there's another square,
and so there's another line that goes out here, and the corresponding point on the sphere
gets closer to the North pole, and you get closer, and this line goes off, and you get closer, and this line goes off,
but you never, like, the North Pole itself is somehow, like,
infinitely far away. Circle, same thing as the line plus a point.
And we had the sphere in ordinary three-dimensional space is the same thing as the two-dimensional plane
plus one point, the North Pole, infinitely far away. Sphere in two-dimensional space
sphere in three-dimensional space
sphere in 4-dimensional space? Okay,
so, if we're in four-dimensional space, we can still measure things, and we can still say, there's a sphere, right?
It's the thing which is the set of points which are at some distance from the center, and we can't see it
you can't, you know, because it's four dimensional. But, same thing. The sphere is the same thing as
the three-dimensional plane, space, whatever,
plus a point at infinity.
So, or in other words, I can't show you this sphere in 4-dimensional space,
but I can show you the shadow. The shadow is gonna be a three-dimensional thing, so we can see it,
or we can 3D print it, or we can do computer animations or something.
This gives you some idea of what this thing in four dimensions is.
So here, what we were doing, we were like,
drawing squares on the sphere in three-dimensional space, and then casting a shadow. So we're gonna draw something else on the sphere
in 4-dimensional space, and cast a shadow.
So what are we going to be drawing?
Monkeys. So this is a like a monkey design, like so, so you're sketching, you're sketching something on the surface of this sphere.
So just like, you know, you're drawing some two-dimensional picture here,
I need to draw a three-dimensional picture on this sphere in four-dimensional space. And the three-dimensional picture I'm gonna draw is
this monkey, or it's going to be a slightly distorted version of this monkey.
Brady: "So what, what, could you draw that with? Like, like, three-dimensional paint?"
Well, yeah.
I mean like, like the sphere in 4-dimensional
space is a three-dimensional thing, so anything threee-dimensional you can put it on there, in the same way for anything
two-dimensional on a two-dimensional sphere, or surface, whatever. So really this is drawn on one
cubical side of a hypercube,
whatever, and then it gets sort of
radially projected out onto the sphere in four-dimensional space. Once you're on the sphere then we can do this
stereographic projection, make a three-dimensional shadow of it in
three-dimensional space, which we can see. This is the monkey inside of one cube of a hypercube.
And he's, so this, this cube has six neighbors, so this monkey needs six neighbors
And they're all kind of like twisting around in various ways. This is showing kind of, some kind of crazy four-dimensional symmetry group.
That's why the monkey is there, is it's a motif that's showing this symmetry. We got our four-dimensional flashlight and/or torch,
whatever, depending on your commonwealth status,
and you project it down to three-dimensional space and we see what it looks like.
Brady: "We can see it then."
We can see it without poor three-dimensional brains that can't deal with four-dimensional stuff.
There are people who claim they can see in four dimensions, but I think they're lying. Right, so this is
eight monkeys,
and they're showing this particular kind of four-dimensional symmetry. So maybe you can see this, there's like,
there's a ring of four monkeys, one two three four, going around like this.
And they're all sort of standing on each other's heads.
And then there's another ring of four monkeys that goes through here, one two three four,
and there's a particular kind of sort of twisting symmetry that
sends, like, this monkey to this monkey to this monkey to this monkey, and they kind of screw, twist around
as you go through. There's an animation. You'll show the animation. Maybe it will make sense.
[monkey screeching]
Some people get like really weirded out by this.
They're like, you know, they're sort of disturbing, the shapes of them.
But it's just math. Like, this is just what it does to the monkey. There's no emotional content intended.
Take some it what you will. I guess that's art, right. You can do whatever you want with it.
Brady: "It's your fault you gave them the horrific faces though. That's not the fault of math."
Look, he's happy! Look at this happy monkey! I mean, maybe he's a little bit annoyed that his, his friend is standing on his head.
[snickering]
So this is the fun part. So this is a, this is a fun story, actually.
Brady: "It just pooped on you."
Well actually, so, oh, oh it's still going, huh.
So, so they're printed, right? 3D printed, and it's made with, like, plastic dust that gets melted. They're hollow.
But in shipping, some dust gets shaken out, so they, you know, gradually sort of lose dust, we can,
yeah, everything gets covered in white dust.
Brady: "That's from the fourth dimension!"
It is, it is four-dimensional monkey dust.
Now, I could have made this
less of a problem.
The issue is, like, the mouth is the only exit hole, right? If there were two exit holes in a monkey, then you could
blow air through it, and blow all of the dust out.
But where are you going to put another hole in a monkey?
I mean, you already have monkey butts all-. Like, you cannot take a photograph of this that doesn't involve a monkey butt. And if there
were eight, yeah, yeah. So this is eight monkeys,
which, and there's one monkey in each cube of the hypercube, that cube, the hypercube has eight, eight cubes in it
So this is a particular symmetry that comes sort of associated with hypercube
So there are these other four-dimensional versions of the Platonic solids. So there's something called a 24 cell, and you can do the same thing.
With 24 monkeys. So this one, there were these two chains of four monkeys each, head to, head to foot, head to foot
This one also has these, these chains of monkeys,
so the best one is probably this one right in the middle one two three four five
There's a chain of six monkeys here, going head to foot. And you can spot these other chains of monkeys.
So there's another chain of six monkeys. So there's all, there's I think, four different chains of six monkeys,
because 4 times 6 is 24, good? 24 cell. And just going one step further, because why not? You can do the same thing with
120 cell.
120 monkeys. This isn't actually the whole thing. So this is only up to the equator of the sphere in 4-dimensional space.
So this is only showing the southern
hyper-hemisphere, hemi-hypersphere, whatever
So there should be more monkeys out here.
You can see like, they're trying to grab on to things, but there are no more monkeys out there.
Just, it would get enormous, but okay, so it was like
chains of monkeys. This one has a chain of ten. And again, you can, you can do these animations where
those monkeys are sort of slowly moving along those chains, twisting as they go. And these are showing different four-dimensional symmetries
that don't really exist in three-dimensional space.
Even why, you know, this thing,
I mean,
it's not actually fair, because you know there's different,
like there's six dents taken out of this, and there's one, and you know, forgetting all that, suppose
it was an ordinary, you know, just a perfect mathematical cube,