Tip:
Highlight text to annotate it
X
Circles in space...
that are arranged so to create beautiful ornaments.
In order to understand better the 3-dimensional sphere
in 4-dimensional space,
I will show you how to fill the space with circles
and thus create what mathematicians call a "fibration".
By the way, my name is Heinz Hopf
and I am one of the main contributors to the development of topology
during the first half of the twentieth century.
Look at this toric surface,
filled with circles that appear to be linked.
Let me explain this picture to you.
Circles, spheres, and tori are among the simplest objects
studied by topologists.
A topologist tries to understand the connections between these objects.
I worked in Berlin, Princeton and Zurich,
and one still comes across my name often in contemporary mathematics:
Poincaré-Hopf theorem, Hopf invariant, Hopf algebra, Hopf fibration.
Let me paint my portrait for you.
I published the discovery of "my" fibration in 1931
but, as always, I have to say that I relied upon
many predecessors, like Clifford for instance
who you see here, and who worked in England during the nineteenth century.
Let's begin with some explanations on a blackboard, well a whiteboard this time!
What do you see?
A 2-dimensional plane?
Well, yes and no!
This is indeed a 2-dimensional plane but
it is a plane of complex dimension 2,
or in other words, a space with real dimension 4.
Go on, make an effort!
Each point in this plane is determined by two coordinates;
but each of these two coordinates is a complex number,
which, remember, is itself defined by two real numbers...
Each of the axes is a complex line;
so that each point on these axes has one coordinate
which is a complex number.
For instance, here you see the point 2-i on the first axis.
The same is true for the other axis, the y-axis.
Here, we can see the point 1-2i on this axis.
Now our whiteboard board is magical,
but not enough to be able to show us the two planes simultaneously!
If we try to depict them in 3-dimensional space, they will intersect along a line
but in 4-dimensional space, they intersect only at the origin :
after all, they are axes!
Now what do you see?
A circle? Yes... and no!
What you see, or rather, what you should imagine,
is the set of points in 4-dimensional space
that are at distance 1 from the origin.
In other words, this is nothing other than the 3-sphere S^3.
Well, of course, you need to have a little imagination...
Let's try to see at least how this sphere intersects the first axis.
The 3-sphere intersects the first axis
in the set of points on this axis which are at distance 1 from the origin.
You see, the 3-sphere intersects the first axis in a circle.
The same is true for the second axis
which intersects the 3-sphere in a circle as well, the blue circle.
Now, what is true for the horizontal line and the vertical line
is equally true for all lines going through the origin.
Here you can see the line with equation z2=-2z1
but we could do the same with any line z2 = a z1,
for any complex number a.
In this manner, the 3-sphere in 4-dimensional space
is filled with circles;
one for each complex line going through the origin in our
plane of complex dimension 2.
Careful though! In the picture, you get the impression that the red circles intersect each other
but this is not the case in the reality of dimension 4.
Lines only meet at the origin,
so their intersections with the unit sphere
don't intersect at all, in fact.
I was the one who discovered
this decomposition of the sphere into circles.
and ever since, it is known as the Hopf fibration.
Why fibration?
Well, you should think of the fibres of fabric.
We are going to look at all that using stereographic projection.
Imagine that we project the 3-sphere from the north pole
onto the tangent space at the south pole, which is our 3-dimensional space.
Here is the projection of one of the circles which as we have seen,
is the intersection of one complex line and the 3-sphere.
But there are many such circles,
one for each complex line going through the origin.
For each complex number a,
we can consider the line z2=az1 and its associated circle.
Let us vary this number a or, what will amount to the same thing,
let us rotate this line in order to see how the circle changes.
Notice that sometimes the circle appears to be a straight line
but this is simply because it passes through the north pole of our 3-sphere.
Let's look at two of these circles simultaneously.
In the lower left-hand corner, they are the two moving complex points, one red, the other green.
You can see the circles
associated to the red and green points.
Notice that these two circles are linked together
like two links of a chain:
it is impossible to separate them without breaking them.
For the fun of it, let's consider three circles...
Look at the dance of these three linked circles.
Now let's take many more complex lines,
chosen randomly,
and let's look at them all at once.
The circles fill up the space,
and no two of them intersect:
this is an example of a fibration.
Let's try to understand this better
by returning to the board for a moment.
Look, we have a Hopf circle for each line
Each one of these lines has an equation of the form z2 = a z1
where a is a complex number,
the slope of the line,
and is indicated by the red point moving on the green line...
Actually, the vertical axis does not have such an equation
but in this case, we may say that a is infinite.
Don't forget that a is a complex number.
The green line is also a complex line,
so it is a real plane, of course.
Summing up: the complex lines that we are interested in,
are completely described
by a point on the green line
and an additional point at infinity.
But we already saw that if one adds a point at infinity
to the complex line, we get the usual 2-sphere.
Once more, this is stereographic projection.
So the complex lines that interest us
are described by points on the yellow sphere;
the 2-dimensional sphere S^2.
So we have a circle for each point on the 2-sphere.
But a circle is
a sphere of dimension 1, isn't it?
All these circles fill up the 3-sphere.
Each point on the 3-sphere belongs to a single circle
and therefore defines a point on the 2-sphere.
In this way, we get a projection
from the 3-sphere to the 2-sphere.
Complicated, isn't it?
Mathematicians say that above any point of the base S^2
there is a fibre which is a circle S^1
and that the total space of this fibration is the sphere S^3.
I am very proud of my fibration
all the more so because
it has become a fundamental object in topology!