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The S2 sphere is in three dimensional space, and has dimension 2.
In the same way, we can study the sphere
of dimension 4.
It contains all the points that are at the same distance from a centerpoint.
But now, to determine the position of a point on this sphere,
we need three numbers.
This means that the sphere has dimension three,
and of course we will call it S3.
You will not be able to see this sphere
in four dimensional space
because your space has only three dimensions,
and the screen only has two!
I can only call upon your imagination.
To get a better understanding of 4 dimensional polyhedra
we can do just what the lizards did
with the 3 dimensional polyhedra,
we first inflate them so that they lie on a sphere
and then project this stereographically onto the plane.
This time, we'll inflate the polyhedron
until its faces lie on a hypersphere
in 4 space
and project stereographically back into our own 3 space.
I'm going up to the north pole
of the sphere in 4 space
and I'll project it down to you
in your 3 space.
You can't see where I am,
just remember how the lizards couldn't see
their kinsman right up on his viewpoint either.
Now we're in exactly the same situation.
Here's the simplex.
You can see its 5 vertices
and its 10 edges.
Of course, in this view, edges are circular arcs.
So now we have a situation like
that of the 3 dimensional polyhedra
projected stereographically onto a plane.
Here's the hypercube.
It's easy to recognize it
from its 32 edges and its 16 vertices.
Seeing things this way is so much easier than
with the shadow method or the 3 dimensional cross-sections.
Here's the 24 cell
with 24 vertices and 96 edges!
Finally, the 120 cell
and the 600 cell.
Let's add the 2 dimensional faces, to get an even better view!
The simplex,
with its 10 triangular faces.
Of course, these 2 dimensional faces are pieces of spheres,
just as before when we saw that the edges were circular arcs.
The simplex is spinning in 4 space,
before being projected stereographically;
remember when the Earth was spinning like a ball
and we saw the motion of the continents.
Now and again, a face passes through the projection pole
and the projection becomes infinite:
it looks like it blows up on the screen.
Let's take a quick look at the hypercube.
You see that the space is divided
into 8 cube-shaped zones,
these are the 3 dimensional faces of the hypercube.
As for the 2 dimensional faces,
they are squares (though rather bloated and twisted).
There are 24 of them.
Ah my favorite! The 24 cell.
Look at that!
The 24 cell is really wonderful.
24 vertices, 96 edges, 96 triangles and 24 octahedra.
8 edges start at each vertex.
Here's the 120 cell,
let's try to understand its geometry better.
4 edges start at each vertex.
The two-dimensional faces are pentagons.
There are 720 of them!
These 720 pentagons form 120 dodecahedra.
Look at all those dodecahedra
fitting nicely together.
Isn't that amazing?
Let's finish with the 600 cell
with its 600 3-dimensional tetrahedral faces,
its 1200 triangular faces
its 720 edges and its 120 vertices.
Trust me, there are 14400 symmetries
of 4 space which preserve this object!
Well there you are, we're done
with our first voyage into the fourth dimension...
It's a dimension full of amazing things.
Of course, the mathematician's imagination
isn't limited to the fourth dimension.
There are the fifth, the sixth,
the n-th dimension, and even...
the infinite dimension!
Each dimension has its own character;
but it has to be said that the fourth is the prettiest.
Why ? Maybe because, after all,
it has a sort of physical reality.
Einstein's relativity theory,
dating from the early twentieth century,
postulates that space and time are somehow bound together
into a 4 dimensional space-time.
A point in this space-time is an event,
characterized by its position in space x,y,z
and by the time t when it occurs.
Dealing with relativistic physics therefore requires
an understanding of 4 dimensional geometry.
It is interesting to notice
that the discovery of this 4 dimensional geometry
precedes by some fifty years
the discovery of relativity.
It's one of the many interactions between mathematics and physics
that the history of science delights in.