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The fourth dimension.
My name is Ludwig Schläfli.
I am a Swiss geometer.
I lived during the nineteenth century
and I'm going to open the door to the fourth dimension for you!
Even if I say so myself, I was a visionary.
I was one of the very first
to understand that
spaces with many dimensions really exist
and that their geometry can be studied.
If flat creatures living in a plane
can understand 3-dimensional polyhedra,
then why shouldn't we understand polyhedra in four dimensions?
One of my main achievements
was to describe all regular polyhedra in four dimensions.
What is the fourth dimension?
A lot has been written on the subject;
Science fiction writers never tire of talking about it!
I'm going to explain things on the blackboard.
You'll see that this blackboard has a bit of magic about it.
What's important is to prepare yourself to forget about the world
which is familiar to us
and to imagine a new world
that our eyes and our senses have no direct access to.
We’ll have to be smart, just like the lizards were before.
I'm going to climb up to a viewpoint
that, unfortunately, you cannot see
and I'll try to describe what I see from there.
But before we begin I'll draw a straight line on the board.
Let me just mark the origin here.
Each point on this line
can be located by its distance from the origin,
with a minus sign, if it is on the left
or a plus sign if it is on the right.
Usually the number is denoted by x
and is called the abscissa.
Since the position of a point on a line
can be described by a single number,
we say that the line has 1 dimension.
Now, I draw a second axis,
perpendicular to the first one.
Each point in the blackboard plane
is now completely described by two numbers,
usually denoted x and y : the abscissa and the ordinate.
The plane has 2 dimensions.
If you had to explain to some being living on a line
what it is to be a point in the plane, that is unknown to him,
you could simply say
"a point in the plane is just a pair of numbers."
Let's go to the third dimension.
The chalk now writes in the air
and draws a third axis, perpendicular to the two previous ones.
A point in space is described by three numbers,
x, y and z.
One could say to the reptiles
that are curious to know about our world
"A point in space is just three numbers"
Let's go to the fourth dimension.
One could try to draw a fourth axis
perpendicular to the others, but that's impossible!
So, we have to do something else instead.
Of course, we might just say
that a point in the fourth dimension
is nothing other than 4 numbers, x,y,z,t.
That doesn't help us a lot!
In spite of the difficulties, we are going to try to get
a feeling for this geometry,
As a first attempt at understanding
we shall proceed by analogy.
Here is a segment...
and an equilateral triangle...
and finally a regular tetrahedron.
Our magical blackboard enables us to draw in space.
How can we keep this up in 4 dimensions?
Observe that the segment, the triangle and the tetrahedron,
have 2, 3 and 4 vertices, respectively.
Therefore, we can try to continue with 5 vertices!
Let's go then.
For the segment, the triangle or the tetrahedron,
an edge connects each pair of vertices.
So we have to connect the 5 vertices in pairs.
We count
one edge
two, three, four, 5, 6, 7, 8, 9, and 10 edges.
In the tetrahedron
there is a triangular face for each triple of vertices
We proceed the same way,
which gives us
2, 3, ..., 10 faces.
But, if we keep going, by analogy,
we have to add a tetrahedral face
for each set of four vertices.
There are 5 sets.
That's it! We've constructed our 4-dimensional object.
We'll call this the "simplex".
Let's spin it round in space a little
as we did with the tetrahedron.
Of course, you have to imagine the simplex spinning
in a 4-dimensional space,
what you see is only its projection on the blackboard.
What makes things a touch complicated
is that faces get tangled and that they cross each other.
Well, some experience is required to be able to see in four dimensions.
We're going to take the simplex,
which is in 4D space
and move it gradually so that different cross sections of it meet
"our" 3-dimensional space.
In the same way that reptiles
could see a polygon appearing and disappearing,
we'll see a 3-dimensional polyhedron
which appears, changes shape and then vanishes.
Here is the simplex passing through our 3-dimensional space.
We're now going to meet
more 4-dimensional polyhedra
passing through our own 3-dimensional world.
Here is the hypercube, a member of the family that starts with
the segment and continues up through the square and the cube.
I must confess that getting a feeling for the geometry
from the slice method like this is rather tricky...
I discovered the analogues of the icosahedron and the dodecahedron.
They have complicated names
but I'll just call them 120 cell and 600 cell
since the former has 120 faces and the latter 600.
Look at the 120 cell, it's just passing through our space.
And now here's the 600 cell.
Of course, when I say that a 4-dimensional polyhedron has 600 faces,
I mean 3-dimensional faces.
Yes, these 600 faces are 600 tetrahedra.
As for the 120 cell, it consists of 120 dodecahedra!
In a minute, we'll see how we can get to know them better.
To observe these 4-dimensional objects
with our 3-dimensional eyes,
we can look at their shadows.
The objects are still in 4D space
but they are projected onto our 3D space
exactly like a painter might project a landscape onto his canvas.
We've already done just this with the simplex.
Here is the hypercube.
Of course, it's spinning in space
so that we can appreciate all the details.
Notice for instance that the hypercube has 16 vertices.
Here's a little newcomer.
It's the most beautiful of my discoveries.
An object that I call the 24 cell,
it has absolutely no analogue in three dimensions.
It's a purely 4-dimensional creature.
I am very proud of my discovery.
Look how wonderful it is ! 24 vertices, 96 edges, 96 triangles and 24 octahedra.
A real little gem!
Here is the shadow of the 120 cell
in all its majesty!
A rather complicated majesty, you have to agree!
Let's get inside and have a look at its structure.
Look: 600 vertices, 1200 edges.
4 edges start at each vertex.
A completely regular structure.
All vertices, all edges play the same role.
It's a pity that the projection breaks the symmetry.
Let's work your imagination a little.
Imagine the object in 4D space
where a huge group of rotations
permutes all these vertices and edges.
The champion is... the 600 cell.
Like a gigantic macromolecule
with its 720 edges and 120 vertices,
and 12 edges starting from each vertex.
Our exploration of 4-dimensional
polyhedra won't stop here
as their stereographic projections are bound to
give us a better feeling for their geometry.